statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
has_constant_speed_on_with.has_locally_bounded_variation_on
(h : has_constant_speed_on_with f s l) : has_locally_bounded_variation_on f s | λ x y hx hy,
by simp only [has_bounded_variation_on, h hx hy, ne.def, ennreal.of_real_ne_top, not_false_iff] | lemma | has_constant_speed_on_with.has_locally_bounded_variation_on | analysis | src/analysis/constant_speed.lean | [
"data.set.function",
"analysis.bounded_variation",
"tactic.swap_var"
] | [
"ennreal.of_real_ne_top",
"has_bounded_variation_on",
"has_constant_speed_on_with",
"has_locally_bounded_variation_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_constant_speed_on_with_of_subsingleton
(f : ℝ → E) {s : set ℝ} (hs : s.subsingleton) (l : ℝ≥0) : has_constant_speed_on_with f s l | begin
rintro x hx y hy, cases hs hx hy,
rw evariation_on.subsingleton f (λ y hy z hz, hs hy.1 hz.1 : (s ∩ Icc x x).subsingleton),
simp only [sub_self, mul_zero, ennreal.of_real_zero],
end | lemma | has_constant_speed_on_with_of_subsingleton | analysis | src/analysis/constant_speed.lean | [
"data.set.function",
"analysis.bounded_variation",
"tactic.swap_var"
] | [
"ennreal.of_real_zero",
"evariation_on.subsingleton",
"has_constant_speed_on_with",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_constant_speed_on_with_iff_ordered :
has_constant_speed_on_with f s l ↔
∀ ⦃x⦄ (hx : x ∈ s) ⦃y⦄ (hy : y ∈ s), (x ≤ y) →
evariation_on f (s ∩ Icc x y) = ennreal.of_real (l * (y - x)) | begin
refine ⟨λ h x xs y ys xy, h xs ys, λ h x xs y ys, _⟩,
rcases le_total x y with xy|yx,
{ exact h xs ys xy, },
{ rw [evariation_on.subsingleton, ennreal.of_real_of_nonpos],
{ exact mul_nonpos_of_nonneg_of_nonpos l.prop (sub_nonpos_of_le yx), },
{ rintro z ⟨zs, xz, zy⟩ w ⟨ws, xw, wy⟩,
cases le_... | lemma | has_constant_speed_on_with_iff_ordered | analysis | src/analysis/constant_speed.lean | [
"data.set.function",
"analysis.bounded_variation",
"tactic.swap_var"
] | [
"ennreal.of_real",
"evariation_on",
"evariation_on.subsingleton",
"has_constant_speed_on_with",
"mul_nonpos_of_nonneg_of_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_constant_speed_on_with_iff_variation_on_from_to_eq :
has_constant_speed_on_with f s l ↔ (has_locally_bounded_variation_on f s ∧
∀ ⦃x⦄ (hx : x ∈ s) ⦃y⦄ (hy : y ∈ s), variation_on_from_to f s x y = l * (y - x)) | begin
split,
{ rintro h, refine ⟨h.has_locally_bounded_variation_on, λ x xs y ys, _⟩,
rw has_constant_speed_on_with_iff_ordered at h,
rcases le_total x y with xy|yx,
{ rw [variation_on_from_to.eq_of_le f s xy, h xs ys xy,
ennreal.to_real_of_real (mul_nonneg l.prop (sub_nonneg.mpr xy))], },
... | lemma | has_constant_speed_on_with_iff_variation_on_from_to_eq | analysis | src/analysis/constant_speed.lean | [
"data.set.function",
"analysis.bounded_variation",
"tactic.swap_var"
] | [
"ennreal.of_real_to_real",
"ennreal.to_real_of_real",
"has_constant_speed_on_with",
"has_constant_speed_on_with_iff_ordered",
"has_locally_bounded_variation_on",
"mul_comm",
"variation_on_from_to",
"variation_on_from_to.eq_of_ge",
"variation_on_from_to.eq_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_constant_speed_on_with.union {t : set ℝ}
(hfs : has_constant_speed_on_with f s l) (hft : has_constant_speed_on_with f t l)
{x : ℝ} (hs : is_greatest s x) (ht : is_least t x) : has_constant_speed_on_with f (s ∪ t) l | begin
rw has_constant_speed_on_with_iff_ordered at hfs hft ⊢,
rintro z (zs|zt) y (ys|yt) zy,
{ have : (s ∪ t) ∩ Icc z y = (s ∩ Icc z y), by
{ ext w, split,
{ rintro ⟨(ws|wt), zw, wy⟩,
{ exact ⟨ws, zw, wy⟩, },
{ exact ⟨(le_antisymm (wy.trans (hs.2 ys)) (ht.2 wt)).symm ▸ hs.1, zw, wy⟩, }, ... | lemma | has_constant_speed_on_with.union | analysis | src/analysis/constant_speed.lean | [
"data.set.function",
"analysis.bounded_variation",
"tactic.swap_var"
] | [
"ennreal.of_real_zero",
"evariation_on.subsingleton",
"evariation_on.union",
"has_constant_speed_on_with",
"has_constant_speed_on_with_iff_ordered",
"is_greatest",
"is_least",
"le_rfl",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_constant_speed_on_with.Icc_Icc {x y z : ℝ}
(hfs : has_constant_speed_on_with f (Icc x y) l)
(hft : has_constant_speed_on_with f (Icc y z) l) : has_constant_speed_on_with f (Icc x z) l | begin
rcases le_total x y with xy|yx,
rcases le_total y z with yz|zy,
{ rw ←set.Icc_union_Icc_eq_Icc xy yz,
exact hfs.union hft (is_greatest_Icc xy) (is_least_Icc yz), },
{ rintro u ⟨xu, uz⟩ v ⟨xv, vz⟩,
rw [Icc_inter_Icc, sup_of_le_right xu, inf_of_le_right vz,
←hfs ⟨xu, uz.trans zy⟩ ⟨xv, vz.tra... | lemma | has_constant_speed_on_with.Icc_Icc | analysis | src/analysis/constant_speed.lean | [
"data.set.function",
"analysis.bounded_variation",
"tactic.swap_var"
] | [
"has_constant_speed_on_with",
"is_greatest_Icc",
"is_least_Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_constant_speed_on_with_zero_iff :
has_constant_speed_on_with f s 0 ↔ ∀ x y ∈ s, edist (f x) (f y) = 0 | begin
dsimp [has_constant_speed_on_with],
simp only [zero_mul, ennreal.of_real_zero, ←evariation_on.eq_zero_iff],
split,
{ by_contra',
obtain ⟨h, hfs⟩ := this,
simp_rw evariation_on.eq_zero_iff at hfs h,
push_neg at hfs,
obtain ⟨x, xs, y, ys, hxy⟩ := hfs,
rcases le_total x y with xy|yx,
... | lemma | has_constant_speed_on_with_zero_iff | analysis | src/analysis/constant_speed.lean | [
"data.set.function",
"analysis.bounded_variation",
"tactic.swap_var"
] | [
"ennreal.of_real_zero",
"evariation_on.eq_zero_iff",
"evariation_on.mono",
"has_constant_speed_on_with",
"le_rfl",
"zero_le'",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_constant_speed_on_with.ratio {l' : ℝ≥0} (hl' : l' ≠ 0) {φ : ℝ → ℝ}
(φm : monotone_on φ s)
(hfφ : has_constant_speed_on_with (f ∘ φ) s l)
(hf : has_constant_speed_on_with f (φ '' s) l')
⦃x : ℝ⦄ (xs : x ∈ s) : eq_on φ (λ y, (l / l') * (y - x) + (φ x)) s | begin
rintro y ys,
rw [←sub_eq_iff_eq_add, mul_comm, ←mul_div_assoc,
eq_div_iff (nnreal.coe_ne_zero.mpr hl')],
rw has_constant_speed_on_with_iff_variation_on_from_to_eq at hf,
rw has_constant_speed_on_with_iff_variation_on_from_to_eq at hfφ,
symmetry,
calc (y - x) * l
= l * (y - x) : by rw mul_... | lemma | has_constant_speed_on_with.ratio | analysis | src/analysis/constant_speed.lean | [
"data.set.function",
"analysis.bounded_variation",
"tactic.swap_var"
] | [
"eq_div_iff",
"has_constant_speed_on_with",
"has_constant_speed_on_with_iff_variation_on_from_to_eq",
"monotone_on",
"mul_comm",
"variation_on_from_to",
"variation_on_from_to.comp_eq_of_monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_unit_speed_on (f : ℝ → E) (s : set ℝ) | has_constant_speed_on_with f s 1 | def | has_unit_speed_on | analysis | src/analysis/constant_speed.lean | [
"data.set.function",
"analysis.bounded_variation",
"tactic.swap_var"
] | [
"has_constant_speed_on_with"
] | `f` has unit speed on `s` if it is linearly parameterized by `l = 1` on `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_unit_speed_on.union {t : set ℝ} {x : ℝ}
(hfs : has_unit_speed_on f s) (hft : has_unit_speed_on f t)
(hs : is_greatest s x) (ht : is_least t x) : has_unit_speed_on f (s ∪ t) | has_constant_speed_on_with.union hfs hft hs ht | lemma | has_unit_speed_on.union | analysis | src/analysis/constant_speed.lean | [
"data.set.function",
"analysis.bounded_variation",
"tactic.swap_var"
] | [
"has_constant_speed_on_with.union",
"has_unit_speed_on",
"is_greatest",
"is_least"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_unit_speed_on.Icc_Icc {x y z : ℝ}
(hfs : has_unit_speed_on f (Icc x y)) (hft : has_unit_speed_on f (Icc y z)) :
has_unit_speed_on f (Icc x z) | has_constant_speed_on_with.Icc_Icc hfs hft | lemma | has_unit_speed_on.Icc_Icc | analysis | src/analysis/constant_speed.lean | [
"data.set.function",
"analysis.bounded_variation",
"tactic.swap_var"
] | [
"has_constant_speed_on_with.Icc_Icc",
"has_unit_speed_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_unit_speed {φ : ℝ → ℝ} (φm : monotone_on φ s)
(hfφ : has_unit_speed_on (f ∘ φ) s) (hf : has_unit_speed_on f (φ '' s))
⦃x : ℝ⦄ (xs : x ∈ s) : eq_on φ (λ y, (y - x) + (φ x)) s | begin
dsimp only [has_unit_speed_on] at hf hfφ,
convert has_constant_speed_on_with.ratio one_ne_zero φm hfφ hf xs,
simp only [nonneg.coe_one, div_self, ne.def, one_ne_zero, not_false_iff, one_mul],
end | lemma | unique_unit_speed | analysis | src/analysis/constant_speed.lean | [
"data.set.function",
"analysis.bounded_variation",
"tactic.swap_var"
] | [
"div_self",
"has_constant_speed_on_with.ratio",
"has_unit_speed_on",
"monotone_on",
"nonneg.coe_one",
"one_mul",
"one_ne_zero"
] | If both `f` and `f ∘ φ` have unit speed (on `t` and `s` respectively) and `φ`
monotonically maps `s` onto `t`, then `φ` is just a translation (on `s`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_unit_speed_on_Icc_zero {s t : ℝ} (hs : 0 ≤ s) (ht : 0 ≤ t)
{φ : ℝ → ℝ} (φm : monotone_on φ $ Icc 0 s) (φst : φ '' (Icc 0 s) = (Icc 0 t))
(hfφ : has_unit_speed_on (f ∘ φ) (Icc 0 s))
(hf : has_unit_speed_on f (Icc 0 t)) : eq_on φ id (Icc 0 s) | begin
rw ←φst at hf,
convert unique_unit_speed φm hfφ hf ⟨le_rfl, hs⟩,
have : φ 0 = 0, by
{ obtain ⟨x,xs,hx⟩ := φst.rec_on (surj_on_image φ (Icc 0 s)) ⟨le_rfl, ht⟩,
exact le_antisymm (hx.rec_on (φm ⟨le_rfl,hs⟩ xs xs.1))
(φst.rec_on (maps_to_image φ (Icc 0 s)) (⟨le_rfl, hs⟩)).1, },
s... | lemma | unique_unit_speed_on_Icc_zero | analysis | src/analysis/constant_speed.lean | [
"data.set.function",
"analysis.bounded_variation",
"tactic.swap_var"
] | [
"has_unit_speed_on",
"monotone_on",
"tsub_zero",
"unique_unit_speed"
] | If both `f` and `f ∘ φ` have unit speed (on `Icc 0 t` and `Icc 0 s` respectively)
and `φ` monotonically maps `Icc 0 s` onto `Icc 0 t`, then `φ` is the identity on `Icc 0 s` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
natural_parameterization (f : α → E) (s : set α) (a : α) : ℝ → E | f ∘ (@function.inv_fun_on _ _ ⟨a⟩ (variation_on_from_to f s a) s) | def | natural_parameterization | analysis | src/analysis/constant_speed.lean | [
"data.set.function",
"analysis.bounded_variation",
"tactic.swap_var"
] | [
"function.inv_fun_on",
"variation_on_from_to"
] | The natural parameterization of `f` on `s`, which, if `f` has locally bounded variation on `s`,
* has unit speed on `s`
(by `natural_parameterization_has_unit_speed`).
* composed with `variation_on_from_to f s a`, is at distance zero from `f`
(by `natural_parameterization_edist_zero`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
edist_natural_parameterization_eq_zero {f : α → E} {s : set α}
(hf : has_locally_bounded_variation_on f s) {a : α} (as : a ∈ s) {b : α} (bs : b ∈ s) :
edist (natural_parameterization f s a (variation_on_from_to f s a b)) (f b) = 0 | begin
dsimp only [natural_parameterization],
haveI : nonempty α := ⟨a⟩,
let c := function.inv_fun_on (variation_on_from_to f s a) s (variation_on_from_to f s a b),
obtain ⟨cs, hc⟩ := @function.inv_fun_on_pos _ _ _ s
(variation_on_from_to f s a) (variation_on_from_to f s a b) ⟨b, bs, rfl⟩,
... | lemma | edist_natural_parameterization_eq_zero | analysis | src/analysis/constant_speed.lean | [
"data.set.function",
"analysis.bounded_variation",
"tactic.swap_var"
] | [
"function.inv_fun_on",
"function.inv_fun_on_pos",
"has_locally_bounded_variation_on",
"natural_parameterization",
"variation_on_from_to",
"variation_on_from_to.edist_zero_of_eq_zero",
"variation_on_from_to.eq_left_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_unit_speed_natural_parameterization (f : α → E) {s : set α}
(hf : has_locally_bounded_variation_on f s) {a : α} (as : a ∈ s) :
has_unit_speed_on (natural_parameterization f s a) (variation_on_from_to f s a '' s) | begin
dsimp only [has_unit_speed_on],
rw has_constant_speed_on_with_iff_ordered,
rintro _ ⟨b, bs, rfl⟩ _ ⟨c, cs, rfl⟩ h,
rcases le_total c b with cb|bc,
{ rw [nnreal.coe_one, one_mul, le_antisymm h (variation_on_from_to.monotone_on hf as cs bs cb),
sub_self, ennreal.of_real_zero, Icc_self, evariation_... | lemma | has_unit_speed_natural_parameterization | analysis | src/analysis/constant_speed.lean | [
"data.set.function",
"analysis.bounded_variation",
"tactic.swap_var"
] | [
"edist_natural_parameterization_eq_zero",
"ennreal.of_real_to_real",
"ennreal.of_real_zero",
"evariation_on.eq_of_edist_zero_on",
"evariation_on.subsingleton",
"has_constant_speed_on_with_iff_ordered",
"has_locally_bounded_variation_on",
"has_unit_speed_on",
"natural_parameterization",
"nnreal.coe... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convolution_integrand_bound_right_of_le_of_subset
{C : ℝ} (hC : ∀ i, ‖g i‖ ≤ C) {x t : G} {s u : set G} (hx : x ∈ s) (hu : - tsupport g + s ⊆ u) :
‖L (f t) (g (x - t))‖ ≤ u.indicator (λ t, ‖L‖ * ‖f t‖ * C) t | begin
refine le_indicator (λ t ht, _) (λ t ht, _) t,
{ refine (L.le_op_norm₂ _ _).trans _,
apply mul_le_mul_of_nonneg_left (hC _) (mul_nonneg (norm_nonneg _) (norm_nonneg _)) },
{ have : x - t ∉ support g,
{ refine mt (λ hxt, _) ht,
apply hu,
refine ⟨_, _, set.neg_mem_neg.mpr (subset_closure h... | lemma | convolution_integrand_bound_right_of_le_of_subset | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"mul_le_mul_of_nonneg_left",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support.convolution_integrand_bound_right_of_subset (hcg : has_compact_support g)
(hg : continuous g) {x t : G} {s u : set G} (hx : x ∈ s) (hu : - tsupport g + s ⊆ u) :
‖L (f t) (g (x - t))‖ ≤ u.indicator (λ t, ‖L‖ * ‖f t‖ * (⨆ i, ‖g i‖)) t | begin
apply convolution_integrand_bound_right_of_le_of_subset _ (λ i, _) hx hu,
exact le_csupr (hg.norm.bdd_above_range_of_has_compact_support hcg.norm) _,
end | lemma | has_compact_support.convolution_integrand_bound_right_of_subset | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"continuous",
"convolution_integrand_bound_right_of_le_of_subset",
"le_csupr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support.convolution_integrand_bound_right (hcg : has_compact_support g)
(hg : continuous g) {x t : G} {s : set G} (hx : x ∈ s) :
‖L (f t) (g (x - t))‖ ≤ (- tsupport g + s).indicator (λ t, ‖L‖ * ‖f t‖ * (⨆ i, ‖g i‖)) t | hcg.convolution_integrand_bound_right_of_subset L hg hx subset.rfl | lemma | has_compact_support.convolution_integrand_bound_right | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.convolution_integrand_fst [has_continuous_sub G] (hg : continuous g) (t : G) :
continuous (λ x, L (f t) (g (x - t))) | L.continuous₂.comp₂ continuous_const $ hg.comp $ continuous_id.sub continuous_const | lemma | continuous.convolution_integrand_fst | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"continuous",
"continuous_const",
"has_continuous_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support.convolution_integrand_bound_left (hcf : has_compact_support f)
(hf : continuous f) {x t : G} {s : set G} (hx : x ∈ s) :
‖L (f (x - t)) (g t)‖ ≤ (- tsupport f + s).indicator (λ t, ‖L‖ * (⨆ i, ‖f i‖) * ‖g t‖) t | by { convert hcf.convolution_integrand_bound_right L.flip hf hx,
simp_rw [L.op_norm_flip, mul_right_comm] } | lemma | has_compact_support.convolution_integrand_bound_left | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"continuous",
"mul_right_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convolution_exists_at [has_sub G] (f : G → E) (g : G → E') (x : G) (L : E →L[𝕜] E' →L[𝕜] F)
(μ : measure G . volume_tac) : Prop | integrable (λ t, L (f t) (g (x - t))) μ | def | convolution_exists_at | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [] | The convolution of `f` and `g` exists at `x` when the function `t ↦ L (f t) (g (x - t))` is
integrable. There are various conditions on `f` and `g` to prove this. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convolution_exists [has_sub G] (f : G → E) (g : G → E') (L : E →L[𝕜] E' →L[𝕜] F)
(μ : measure G . volume_tac) : Prop | ∀ x : G, convolution_exists_at f g x L μ | def | convolution_exists | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_exists_at"
] | The convolution of `f` and `g` exists when the function `t ↦ L (f t) (g (x - t))` is integrable
for all `x : G`. There are various conditions on `f` and `g` to prove this. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convolution_exists_at.integrable [has_sub G] {x : G} (h : convolution_exists_at f g x L μ) :
integrable (λ t, L (f t) (g (x - t))) μ | h | lemma | convolution_exists_at.integrable | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_exists_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measure_theory.ae_strongly_measurable.convolution_integrand'
[has_measurable_add₂ G] [has_measurable_neg G] [sigma_finite ν]
(hf : ae_strongly_measurable f ν)
(hg : ae_strongly_measurable g $ map (λ (p : G × G), p.1 - p.2) (μ.prod ν)) :
ae_strongly_measurable (λ p : G × G, L (f p.2) (g (p.1 - p.2))) (μ.prod ν) | L.ae_strongly_measurable_comp₂ hf.snd $ hg.comp_measurable measurable_sub | lemma | measure_theory.ae_strongly_measurable.convolution_integrand' | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"has_measurable_add₂",
"has_measurable_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measure_theory.ae_strongly_measurable.convolution_integrand_snd'
(hf : ae_strongly_measurable f μ) {x : G}
(hg : ae_strongly_measurable g $ map (λ t, x - t) μ) :
ae_strongly_measurable (λ t, L (f t) (g (x - t))) μ | L.ae_strongly_measurable_comp₂ hf $ hg.comp_measurable $ measurable_id.const_sub x | lemma | measure_theory.ae_strongly_measurable.convolution_integrand_snd' | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measure_theory.ae_strongly_measurable.convolution_integrand_swap_snd'
{x : G} (hf : ae_strongly_measurable f $ map (λ t, x - t) μ)
(hg : ae_strongly_measurable g μ) : ae_strongly_measurable (λ t, L (f (x - t)) (g t)) μ | L.ae_strongly_measurable_comp₂ (hf.comp_measurable $ measurable_id.const_sub x) hg | lemma | measure_theory.ae_strongly_measurable.convolution_integrand_swap_snd' | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above.convolution_exists_at' {x₀ : G}
{s : set G} (hbg : bdd_above ((λ i, ‖g i‖) '' ((λ t, - t + x₀) ⁻¹' s)))
(hs : measurable_set s) (h2s : support (λ t, L (f t) (g (x₀ - t))) ⊆ s)
(hf : integrable_on f s μ) (hmg : ae_strongly_measurable g $ map (λ t, x₀ - t) (μ.restrict s)) :
convolution_exists_at f g x₀ ... | begin
rw [convolution_exists_at, ← integrable_on_iff_integrable_of_support_subset h2s],
set s' := (λ t, - t + x₀) ⁻¹' s,
have : ∀ᵐ (t : G) ∂(μ.restrict s),
‖L (f t) (g (x₀ - t))‖ ≤ s.indicator (λ t, ‖L‖ * ‖f t‖ * ⨆ i : s', ‖g i‖) t,
{ refine eventually_of_forall _,
refine le_indicator (λ t ht, _) (λ t h... | lemma | bdd_above.convolution_exists_at' | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"bdd_above",
"convolution_exists_at",
"le_csupr_set",
"measurable_set",
"mul_le_mul_of_nonneg_left"
] | A sufficient condition to prove that `f ⋆[L, μ] g` exists.
We assume that `f` is integrable on a set `s` and `g` is bounded and ae strongly measurable
on `x₀ - s` (note that both properties hold if `g` is continuous with compact support). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convolution_exists_at.of_norm' {x₀ : G}
(h : convolution_exists_at (λ x, ‖f x‖) (λ x, ‖g x‖) x₀ (mul ℝ ℝ) μ)
(hmf : ae_strongly_measurable f μ)
(hmg : ae_strongly_measurable g $ map (λ t, x₀ - t) μ) :
convolution_exists_at f g x₀ L μ | begin
refine (h.const_mul ‖L‖).mono' (hmf.convolution_integrand_snd' L hmg)
(eventually_of_forall $ λ x, _),
rw [mul_apply', ← mul_assoc],
apply L.le_op_norm₂,
end | lemma | convolution_exists_at.of_norm' | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_exists_at",
"mul_assoc"
] | If `‖f‖ *[μ] ‖g‖` exists, then `f *[L, μ] g` exists. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
measure_theory.ae_strongly_measurable.convolution_integrand_snd
(hf : ae_strongly_measurable f μ) (hg : ae_strongly_measurable g μ)
(x : G) : ae_strongly_measurable (λ t, L (f t) (g (x - t))) μ | hf.convolution_integrand_snd' L $ hg.mono' $
(quasi_measure_preserving_sub_left_of_right_invariant μ x).absolutely_continuous | lemma | measure_theory.ae_strongly_measurable.convolution_integrand_snd | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measure_theory.ae_strongly_measurable.convolution_integrand_swap_snd
(hf : ae_strongly_measurable f μ) (hg : ae_strongly_measurable g μ)
(x : G) : ae_strongly_measurable (λ t, L (f (x - t)) (g t)) μ | (hf.mono' (quasi_measure_preserving_sub_left_of_right_invariant μ x).absolutely_continuous)
.convolution_integrand_swap_snd' L hg | lemma | measure_theory.ae_strongly_measurable.convolution_integrand_swap_snd | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convolution_exists_at.of_norm {x₀ : G}
(h : convolution_exists_at (λ x, ‖f x‖) (λ x, ‖g x‖) x₀ (mul ℝ ℝ) μ)
(hmf : ae_strongly_measurable f μ)
(hmg : ae_strongly_measurable g μ) :
convolution_exists_at f g x₀ L μ | h.of_norm' L hmf $ hmg.mono'
(quasi_measure_preserving_sub_left_of_right_invariant μ x₀).absolutely_continuous | lemma | convolution_exists_at.of_norm | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_exists_at"
] | If `‖f‖ *[μ] ‖g‖` exists, then `f *[L, μ] g` exists. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
measure_theory.ae_strongly_measurable.convolution_integrand
(hf : ae_strongly_measurable f ν) (hg : ae_strongly_measurable g μ) :
ae_strongly_measurable (λ p : G × G, L (f p.2) (g (p.1 - p.2))) (μ.prod ν) | hf.convolution_integrand' L $ hg.mono'
(quasi_measure_preserving_sub_of_right_invariant μ ν).absolutely_continuous | lemma | measure_theory.ae_strongly_measurable.convolution_integrand | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measure_theory.integrable.convolution_integrand (hf : integrable f ν) (hg : integrable g μ) :
integrable (λ p : G × G, L (f p.2) (g (p.1 - p.2))) (μ.prod ν) | begin
have h_meas : ae_strongly_measurable (λ (p : G × G), L (f p.2) (g (p.1 - p.2))) (μ.prod ν) :=
hf.ae_strongly_measurable.convolution_integrand L hg.ae_strongly_measurable,
have h2_meas : ae_strongly_measurable (λ (y : G), ∫ (x : G), ‖L (f y) (g (x - y))‖ ∂μ) ν :=
h_meas.prod_swap.norm.integral_prod_rig... | lemma | measure_theory.integrable.convolution_integrand | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"real.norm_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measure_theory.integrable.ae_convolution_exists (hf : integrable f ν) (hg : integrable g μ) :
∀ᵐ x ∂μ, convolution_exists_at f g x L ν | ((integrable_prod_iff $ hf.ae_strongly_measurable.convolution_integrand L
hg.ae_strongly_measurable).mp $ hf.convolution_integrand L hg).1 | lemma | measure_theory.integrable.ae_convolution_exists | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_exists_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support.convolution_exists_at {x₀ : G}
(h : has_compact_support (λ t, L (f t) (g (x₀ - t)))) (hf : locally_integrable f μ)
(hg : continuous g) : convolution_exists_at f g x₀ L μ | begin
let u := (homeomorph.neg G).trans (homeomorph.add_right x₀),
let v := (homeomorph.neg G).trans (homeomorph.add_left x₀),
apply ((u.is_compact_preimage.mpr h).bdd_above_image hg.norm.continuous_on).convolution_exists_at'
L is_closed_closure.measurable_set subset_closure (hf.integrable_on_is_compact h),
... | lemma | has_compact_support.convolution_exists_at | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"continuous",
"continuous_on.ae_strongly_measurable_of_is_compact",
"convolution_exists_at",
"equiv.coe_fn_mk",
"equiv.to_fun_as_coe",
"homeomorph.homeomorph_mk_coe",
"homeomorph.trans_apply",
"measurable_set",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support.convolution_exists_right
(hcg : has_compact_support g) (hf : locally_integrable f μ) (hg : continuous g) :
convolution_exists f g L μ | begin
intro x₀,
refine has_compact_support.convolution_exists_at L _ hf hg,
refine (hcg.comp_homeomorph (homeomorph.sub_left x₀)).mono _,
refine λ t, mt (λ ht : g (x₀ - t) = 0, _),
simp_rw [ht, (L _).map_zero]
end | lemma | has_compact_support.convolution_exists_right | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"continuous",
"convolution_exists",
"has_compact_support.convolution_exists_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support.convolution_exists_left_of_continuous_right
(hcf : has_compact_support f) (hf : locally_integrable f μ) (hg : continuous g) :
convolution_exists f g L μ | begin
intro x₀,
refine has_compact_support.convolution_exists_at L _ hf hg,
refine hcf.mono _,
refine λ t, mt (λ ht : f t = 0, _),
simp_rw [ht, L.map_zero₂]
end | lemma | has_compact_support.convolution_exists_left_of_continuous_right | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"continuous",
"convolution_exists",
"has_compact_support.convolution_exists_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above.convolution_exists_at [has_measurable_add₂ G] [sigma_finite μ] {x₀ : G}
{s : set G} (hbg : bdd_above ((λ i, ‖g i‖) '' ((λ t, x₀ - t) ⁻¹' s)))
(hs : measurable_set s) (h2s : support (λ t, L (f t) (g (x₀ - t))) ⊆ s)
(hf : integrable_on f s μ) (hmg : ae_strongly_measurable g μ) :
convolution_exists_at ... | begin
refine bdd_above.convolution_exists_at' L _ hs h2s hf _,
{ simp_rw [← sub_eq_neg_add, hbg] },
{ have : ae_strongly_measurable g (map (λ (t : G), x₀ - t) μ), from hmg.mono'
(quasi_measure_preserving_sub_left_of_right_invariant μ x₀).absolutely_continuous,
apply this.mono_measure,
exact map_mono... | lemma | bdd_above.convolution_exists_at | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"ae_measurable",
"bdd_above",
"bdd_above.convolution_exists_at'",
"convolution_exists_at",
"has_measurable_add₂",
"measurable_id'",
"measurable_set"
] | A sufficient condition to prove that `f ⋆[L, μ] g` exists.
We assume that the integrand has compact support and `g` is bounded on this support (note that
both properties hold if `g` is continuous with compact support). We also require that `f` is
integrable on the support of the integrand, and that both functions are s... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convolution_exists_at_flip :
convolution_exists_at g f x L.flip μ ↔ convolution_exists_at f g x L μ | by simp_rw [convolution_exists_at, ← integrable_comp_sub_left (λ t, L (f t) (g (x - t))) x,
sub_sub_cancel, flip_apply] | lemma | convolution_exists_at_flip | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_exists_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convolution_exists_at.integrable_swap (h : convolution_exists_at f g x L μ) :
integrable (λ t, L (f (x - t)) (g t)) μ | by { convert h.comp_sub_left x, simp_rw [sub_sub_self] } | lemma | convolution_exists_at.integrable_swap | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_exists_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convolution_exists_at_iff_integrable_swap :
convolution_exists_at f g x L μ ↔ integrable (λ t, L (f (x - t)) (g t)) μ | convolution_exists_at_flip.symm | lemma | convolution_exists_at_iff_integrable_swap | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_exists_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support.convolution_exists_left
(hcf : has_compact_support f) (hf : continuous f) (hg : locally_integrable g μ) :
convolution_exists f g L μ | λ x₀, convolution_exists_at_flip.mp $ hcf.convolution_exists_right L.flip hg hf x₀ | lemma | has_compact_support.convolution_exists_left | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"continuous",
"convolution_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support.convolution_exists_right_of_continuous_left
(hcg : has_compact_support g) (hf : continuous f) (hg : locally_integrable g μ) :
convolution_exists f g L μ | λ x₀, convolution_exists_at_flip.mp $
hcg.convolution_exists_left_of_continuous_right L.flip hg hf x₀ | lemma | has_compact_support.convolution_exists_right_of_continuous_left | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"continuous",
"convolution_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convolution [has_sub G] (f : G → E) (g : G → E') (L : E →L[𝕜] E' →L[𝕜] F)
(μ : measure G . volume_tac) : G → F | λ x, ∫ t, L (f t) (g (x - t)) ∂μ | def | convolution | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [] | The convolution of two functions `f` and `g` with respect to a continuous bilinear map `L` and
measure `μ`. It is defined to be `(f ⋆[L, μ] g) x = ∫ t, L (f t) (g (x - t)) ∂μ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convolution_def [has_sub G] : (f ⋆[L, μ] g) x = ∫ t, L (f t) (g (x - t)) ∂μ | rfl | lemma | convolution_def | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convolution_lsmul [has_sub G] {f : G → 𝕜} {g : G → F} :
(f ⋆[lsmul 𝕜 𝕜, μ] g : G → F) x = ∫ t, f t • g (x - t) ∂μ | rfl | lemma | convolution_lsmul | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [] | The definition of convolution where the bilinear operator is scalar multiplication.
Note: it often helps the elaborator to give the type of the convolution explicitly. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convolution_mul [has_sub G] [normed_space ℝ 𝕜] [complete_space 𝕜] {f : G → 𝕜} {g : G → 𝕜} :
(f ⋆[mul 𝕜 𝕜, μ] g) x = ∫ t, f t * g (x - t) ∂μ | rfl | lemma | convolution_mul | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"complete_space",
"normed_space"
] | The definition of convolution where the bilinear operator is multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_convolution [smul_comm_class ℝ 𝕜 F]
{y : 𝕜} : (y • f) ⋆[L, μ] g = y • (f ⋆[L, μ] g) | by { ext, simp only [pi.smul_apply, convolution_def, ← integral_smul, L.map_smul₂] } | lemma | smul_convolution | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_def",
"pi.smul_apply",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convolution_smul [smul_comm_class ℝ 𝕜 F]
{y : 𝕜} : f ⋆[L, μ] (y • g) = y • (f ⋆[L, μ] g) | by { ext, simp only [pi.smul_apply, convolution_def, ← integral_smul, (L _).map_smul] } | lemma | convolution_smul | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_def",
"pi.smul_apply",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_convolution : 0 ⋆[L, μ] g = 0 | by { ext, simp_rw [convolution_def, pi.zero_apply, L.map_zero₂, integral_zero] } | lemma | zero_convolution | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convolution_zero : f ⋆[L, μ] 0 = 0 | by { ext, simp_rw [convolution_def, pi.zero_apply, (L _).map_zero, integral_zero] } | lemma | convolution_zero | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convolution_exists_at.distrib_add {x : G} (hfg : convolution_exists_at f g x L μ)
(hfg' : convolution_exists_at f g' x L μ) :
(f ⋆[L, μ] (g + g')) x = (f ⋆[L, μ] g) x + (f ⋆[L, μ] g') x | by simp only [convolution_def, (L _).map_add, pi.add_apply, integral_add hfg hfg'] | lemma | convolution_exists_at.distrib_add | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_def",
"convolution_exists_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convolution_exists.distrib_add (hfg : convolution_exists f g L μ)
(hfg' : convolution_exists f g' L μ) : f ⋆[L, μ] (g + g') = f ⋆[L, μ] g + f ⋆[L, μ] g' | by { ext, exact (hfg x).distrib_add (hfg' x) } | lemma | convolution_exists.distrib_add | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convolution_exists_at.add_distrib {x : G} (hfg : convolution_exists_at f g x L μ)
(hfg' : convolution_exists_at f' g x L μ) :
((f + f') ⋆[L, μ] g) x = (f ⋆[L, μ] g) x + (f' ⋆[L, μ] g) x | by simp only [convolution_def, L.map_add₂, pi.add_apply, integral_add hfg hfg'] | lemma | convolution_exists_at.add_distrib | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_def",
"convolution_exists_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convolution_exists.add_distrib (hfg : convolution_exists f g L μ)
(hfg' : convolution_exists f' g L μ) : (f + f') ⋆[L, μ] g = f ⋆[L, μ] g + f' ⋆[L, μ] g | by { ext, exact (hfg x).add_distrib (hfg' x) } | lemma | convolution_exists.add_distrib | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convolution_mono_right {f g g' : G → ℝ}
(hfg : convolution_exists_at f g x (lsmul ℝ ℝ) μ)
(hfg' : convolution_exists_at f g' x (lsmul ℝ ℝ) μ)
(hf : ∀ x, 0 ≤ f x) (hg : ∀ x, g x ≤ g' x) :
(f ⋆[lsmul ℝ ℝ, μ] g) x ≤ (f ⋆[lsmul ℝ ℝ, μ] g') x | begin
apply integral_mono hfg hfg',
simp only [lsmul_apply, algebra.id.smul_eq_mul],
assume t,
apply mul_le_mul_of_nonneg_left (hg _) (hf _),
end | lemma | convolution_mono_right | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"algebra.id.smul_eq_mul",
"convolution_exists_at",
"mul_le_mul_of_nonneg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convolution_mono_right_of_nonneg {f g g' : G → ℝ}
(hfg' : convolution_exists_at f g' x (lsmul ℝ ℝ) μ)
(hf : ∀ x, 0 ≤ f x) (hg : ∀ x, g x ≤ g' x) (hg' : ∀ x, 0 ≤ g' x) :
(f ⋆[lsmul ℝ ℝ, μ] g) x ≤ (f ⋆[lsmul ℝ ℝ, μ] g') x | begin
by_cases H : convolution_exists_at f g x (lsmul ℝ ℝ) μ,
{ exact convolution_mono_right H hfg' hf hg },
have : (f ⋆[lsmul ℝ ℝ, μ] g) x = 0 := integral_undef H,
rw this,
exact integral_nonneg (λ y, mul_nonneg (hf y) (hg' (x - y))),
end | lemma | convolution_mono_right_of_nonneg | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_exists_at",
"convolution_mono_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convolution_congr [has_measurable_add₂ G] [has_measurable_neg G]
[sigma_finite μ] [is_add_right_invariant μ] (h1 : f =ᵐ[μ] f') (h2 : g =ᵐ[μ] g') :
f ⋆[L, μ] g = f' ⋆[L, μ] g' | begin
ext x,
apply integral_congr_ae,
exact (h1.prod_mk $ h2.comp_tendsto
(quasi_measure_preserving_sub_left_of_right_invariant μ x).tendsto_ae).fun_comp ↿(λ x y, L x y)
end | lemma | convolution_congr | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"has_measurable_add₂",
"has_measurable_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_convolution_subset_swap : support (f ⋆[L, μ] g) ⊆ support g + support f | begin
intros x h2x,
by_contra hx,
apply h2x,
simp_rw [set.mem_add, not_exists, not_and_distrib, nmem_support] at hx,
rw [convolution_def],
convert integral_zero G F,
ext t,
rcases hx (x - t) t with h|h|h,
{ rw [h, (L _).map_zero] },
{ rw [h, L.map_zero₂] },
{ exact (h $ sub_add_cancel x t).elim }
... | lemma | support_convolution_subset_swap | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"by_contra",
"convolution_def",
"not_and_distrib",
"not_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measure_theory.integrable.integrable_convolution (hf : integrable f μ) (hg : integrable g μ) :
integrable (f ⋆[L, μ] g) μ | (hf.convolution_integrand L hg).integral_prod_left | lemma | measure_theory.integrable.integrable_convolution | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support.convolution [t2_space G] (hcf : has_compact_support f)
(hcg : has_compact_support g) : has_compact_support (f ⋆[L, μ] g) | is_compact_of_is_closed_subset (hcg.is_compact.add hcf) is_closed_closure $ closure_minimal
((support_convolution_subset_swap L).trans $ add_subset_add subset_closure subset_closure)
(hcg.is_compact.add hcf).is_closed | lemma | has_compact_support.convolution | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"closure_minimal",
"is_closed",
"is_closed_closure",
"is_compact_of_is_closed_subset",
"subset_closure",
"support_convolution_subset_swap",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_convolution_right_with_param'
{g : P → G → E'} {s : set P} {k : set G} (hk : is_compact k) (h'k : is_closed k)
(hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : locally_integrable f μ) (hg : continuous_on (↿g) (s ×ˢ univ)) :
continuous_on (λ (q : P × G), (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ) | begin
assume q₀ hq₀,
replace hq₀ : q₀.1 ∈ s, by simpa only [mem_prod, mem_univ, and_true] using hq₀,
have A : ∀ p ∈ s, continuous (g p),
{ assume p hp,
apply hg.comp_continuous (continuous_const.prod_mk continuous_id') (λ x, _),
simpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true] using hp },
have... | lemma | continuous_on_convolution_right_with_param' | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"bound",
"closure_minimal",
"continuous",
"continuous_at",
"continuous_const",
"continuous_id'",
"continuous_on",
"continuous_within_at",
"continuous_within_at.comp",
"convolution_integrand_bound_right_of_le_of_subset",
"filter.prod_mem_prod",
"generalized_tube_lemma",
"has_compact_support.c... | The convolution `f * g` is continuous if `f` is locally integrable and `g` is continuous and
compactly supported. Version where `g` depends on an additional parameter in a subset `s` of
a parameter space `P` (and the compact support `k` is independent of the parameter in `s`),
not assuming `t2_space G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on_convolution_right_with_param
[t2_space G] {g : P → G → E'} {s : set P} {k : set G} (hk : is_compact k)
(hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : locally_integrable f μ) (hg : continuous_on (↿g) (s ×ˢ univ)) :
continuous_on (λ (q : P × G), (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ) | continuous_on_convolution_right_with_param' L hk hk.is_closed hgs hf hg | lemma | continuous_on_convolution_right_with_param | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"continuous_on",
"continuous_on_convolution_right_with_param'",
"is_compact",
"t2_space"
] | The convolution `f * g` is continuous if `f` is locally integrable and `g` is continuous and
compactly supported. Version where `g` depends on an additional parameter in a subset `s` of
a parameter space `P` (and the compact support `k` is independent of the parameter in `s`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on_convolution_right_with_param_comp'
{s : set P} {v : P → G} (hv : continuous_on v s)
{g : P → G → E'} {k : set G}
(hk : is_compact k) (h'k : is_closed k) (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : locally_integrable f μ) (hg : continuous_on (↿g) (s ×ˢ univ)) :
continuous_on (λ x, (f ⋆[L, μ... | begin
apply (continuous_on_convolution_right_with_param' L hk h'k hgs hf hg).comp
(continuous_on_id.prod hv),
assume x hx,
simp only [hx, prod_mk_mem_set_prod_eq, mem_univ, and_self, id.def],
end | lemma | continuous_on_convolution_right_with_param_comp' | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"continuous_on",
"continuous_on_convolution_right_with_param'",
"is_closed",
"is_compact"
] | The convolution `f * g` is continuous if `f` is locally integrable and `g` is continuous and
compactly supported. Version where `g` depends on an additional parameter in an open subset `s` of
a parameter space `P` (and the compact support `k` is independent of the parameter in `s`),
given in terms of compositions with ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on_convolution_right_with_param_comp [t2_space G]
{s : set P} {v : P → G} (hv : continuous_on v s)
{g : P → G → E'} {k : set G}
(hk : is_compact k) (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : locally_integrable f μ) (hg : continuous_on (↿g) (s ×ˢ univ)) :
continuous_on (λ x, (f ⋆[L, μ] g x) (... | continuous_on_convolution_right_with_param_comp' L hv hk hk.is_closed hgs hf hg | lemma | continuous_on_convolution_right_with_param_comp | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"continuous_on",
"continuous_on_convolution_right_with_param_comp'",
"is_compact",
"t2_space"
] | The convolution `f * g` is continuous if `f` is locally integrable and `g` is continuous and
compactly supported. Version where `g` depends on an additional parameter in an open subset `s` of
a parameter space `P` (and the compact support `k` is independent of the parameter in `s`),
given in terms of compositions with ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_compact_support.continuous_convolution_right
(hcg : has_compact_support g) (hf : locally_integrable f μ)
(hg : continuous g) : continuous (f ⋆[L, μ] g) | begin
rw continuous_iff_continuous_on_univ,
let g' : G → G → E' := λ p q, g q,
have : continuous_on (↿g') (univ ×ˢ univ) := (hg.comp continuous_snd).continuous_on,
exact continuous_on_convolution_right_with_param_comp' L
(continuous_iff_continuous_on_univ.1 continuous_id) hcg (is_closed_tsupport _)
(λ p... | lemma | has_compact_support.continuous_convolution_right | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"continuous",
"continuous_id",
"continuous_iff_continuous_on_univ",
"continuous_on",
"continuous_on_convolution_right_with_param_comp'",
"continuous_snd"
] | The convolution is continuous if one function is locally integrable and the other has compact
support and is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bdd_above.continuous_convolution_right_of_integrable [second_countable_topology G]
(hbg : bdd_above (range (λ x, ‖g x‖))) (hf : integrable f μ) (hg : continuous g) :
continuous (f ⋆[L, μ] g) | begin
refine continuous_iff_continuous_at.mpr (λ x₀, _),
have : ∀ᶠ x in 𝓝 x₀, ∀ᵐ (t : G) ∂μ,
‖L (f t) (g (x - t))‖ ≤ ‖L‖ * ‖f t‖ * (⨆ i, ‖g i‖),
{ refine eventually_of_forall (λ x, eventually_of_forall $ λ t, _),
refine (L.le_op_norm₂ _ _).trans _,
exact mul_le_mul_of_nonneg_left (le_csupr hbg $ x - ... | lemma | bdd_above.continuous_convolution_right_of_integrable | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"bdd_above",
"continuous",
"continuous_at",
"continuous_const",
"le_csupr",
"mul_le_mul_of_nonneg_left"
] | The convolution is continuous if one function is integrable and the other is bounded and
continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
support_convolution_subset : support (f ⋆[L, μ] g) ⊆ support f + support g | (support_convolution_subset_swap L).trans (add_comm _ _).subset | lemma | support_convolution_subset | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"support_convolution_subset_swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convolution_flip : g ⋆[L.flip, μ] f = f ⋆[L, μ] g | begin
ext1 x,
simp_rw [convolution_def],
rw [← integral_sub_left_eq_self _ μ x],
simp_rw [sub_sub_self, flip_apply]
end | lemma | convolution_flip | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_def"
] | Commutativity of convolution | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convolution_eq_swap : (f ⋆[L, μ] g) x = ∫ t, L (f (x - t)) (g t) ∂μ | by { rw [← convolution_flip], refl } | lemma | convolution_eq_swap | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_flip"
] | The symmetric definition of convolution. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convolution_lsmul_swap {f : G → 𝕜} {g : G → F}:
(f ⋆[lsmul 𝕜 𝕜, μ] g : G → F) x = ∫ t, f (x - t) • g t ∂μ | convolution_eq_swap _ | lemma | convolution_lsmul_swap | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_eq_swap"
] | The symmetric definition of convolution where the bilinear operator is scalar multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convolution_mul_swap [normed_space ℝ 𝕜] [complete_space 𝕜] {f : G → 𝕜} {g : G → 𝕜} :
(f ⋆[mul 𝕜 𝕜, μ] g) x = ∫ t, f (x - t) * g t ∂μ | convolution_eq_swap _ | lemma | convolution_mul_swap | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"complete_space",
"convolution_eq_swap",
"normed_space"
] | The symmetric definition of convolution where the bilinear operator is multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convolution_neg_of_neg_eq (h1 : ∀ᵐ x ∂μ, f (-x) = f x) (h2 : ∀ᵐ x ∂μ, g (-x) = g x) :
(f ⋆[L, μ] g) (-x) = (f ⋆[L, μ] g) x | calc ∫ (t : G), (L (f t)) (g (-x - t)) ∂μ
= ∫ (t : G), (L (f (-t))) (g (x + t)) ∂μ :
begin
apply integral_congr_ae,
filter_upwards [h1, (eventually_add_left_iff μ x).2 h2] with t ht h't,
simp_rw [ht, ← h't, neg_add'],
end
... = ∫ (t : G), (L (f t)) (g (x - t)) ∂μ :
by { rw ← integral_neg_eq_self, ... | lemma | convolution_neg_of_neg_eq | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [] | The convolution of two even functions is also even. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_compact_support.continuous_convolution_left [first_countable_topology G]
(hcf : has_compact_support f) (hf : continuous f) (hg : locally_integrable g μ) :
continuous (f ⋆[L, μ] g) | by { rw [← convolution_flip], exact hcf.continuous_convolution_right L.flip hg hf } | lemma | has_compact_support.continuous_convolution_left | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"continuous",
"convolution_flip"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above.continuous_convolution_left_of_integrable [second_countable_topology G]
(hbf : bdd_above (range (λ x, ‖f x‖))) (hf : continuous f) (hg : integrable g μ) :
continuous (f ⋆[L, μ] g) | by { rw [← convolution_flip], exact hbf.continuous_convolution_right_of_integrable L.flip hg hf } | lemma | bdd_above.continuous_convolution_left_of_integrable | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"bdd_above",
"continuous",
"convolution_flip"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convolution_eq_right' {x₀ : G} {R : ℝ}
(hf : support f ⊆ ball (0 : G) R)
(hg : ∀ x ∈ ball x₀ R, g x = g x₀) : (f ⋆[L, μ] g) x₀ = ∫ t, L (f t) (g x₀) ∂μ | begin
have h2 : ∀ t, L (f t) (g (x₀ - t)) = L (f t) (g x₀),
{ intro t, by_cases ht : t ∈ support f,
{ have h2t := hf ht,
rw [mem_ball_zero_iff] at h2t,
specialize hg (x₀ - t),
rw [sub_eq_add_neg, add_mem_ball_iff_norm, norm_neg, ← sub_eq_add_neg] at hg,
rw [hg h2t] },
{ rw [nmem_supp... | lemma | convolution_eq_right' | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_def"
] | Compute `(f ⋆ g) x₀` if the support of the `f` is within `metric.ball 0 R`, and `g` is constant
on `metric.ball x₀ R`.
We can simplify the RHS further if we assume `f` is integrable, but also if `L = (•)` or more
generally if `L` has a `antilipschitz_with`-condition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_convolution_le' {x₀ : G} {R ε : ℝ} {z₀ : E'}
(hε : 0 ≤ ε)
(hif : integrable f μ)
(hf : support f ⊆ ball (0 : G) R)
(hmg : ae_strongly_measurable g μ)
(hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε) :
dist ((f ⋆[L, μ] g : G → F) x₀) (∫ t, L (f t) z₀ ∂μ) ≤ ‖L‖ * ∫ x, ‖f x‖ ∂μ * ε | begin
have hfg : convolution_exists_at f g x₀ L μ,
{ refine bdd_above.convolution_exists_at L _ metric.is_open_ball.measurable_set
(subset_trans _ hf) hif.integrable_on hmg,
swap, { refine λ t, mt (λ ht : f t = 0, _), simp_rw [ht, L.map_zero₂] },
rw [bdd_above_def],
refine ⟨‖z₀‖ + ε, _⟩,
rintro ... | lemma | dist_convolution_le' | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"bdd_above.convolution_exists_at",
"bdd_above_def",
"convolution_def",
"convolution_exists_at",
"dist_self",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"subset_trans",
"zero_mul"
] | Approximate `(f ⋆ g) x₀` if the support of the `f` is bounded within a ball, and `g` is near
`g x₀` on a ball with the same radius around `x₀`. See `dist_convolution_le` for a special case.
We can simplify the second argument of `dist` further if we add some extra type-classes on `E`
and `𝕜` or if `L` is scalar multi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_convolution_le {f : G → ℝ} {x₀ : G} {R ε : ℝ} {z₀ : E'}
(hε : 0 ≤ ε)
(hf : support f ⊆ ball (0 : G) R)
(hnf : ∀ x, 0 ≤ f x)
(hintf : ∫ x, f x ∂μ = 1)
(hmg : ae_strongly_measurable g μ)
(hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε) :
dist ((f ⋆[lsmul ℝ ℝ, μ] g : G → E') x₀) z₀ ≤ ε | begin
have hif : integrable f μ,
{ by_contra hif, exact zero_ne_one ((integral_undef hif).symm.trans hintf) },
convert (dist_convolution_le' _ hε hif hf hmg hg).trans _,
{ simp_rw [lsmul_apply, integral_smul_const, hintf, one_smul] },
{ simp_rw [real.norm_of_nonneg (hnf _), hintf, mul_one],
exact (mul_le_... | lemma | dist_convolution_le | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"by_contra",
"dist_convolution_le'",
"integral_smul_const",
"mul_le_mul_of_nonneg_right",
"mul_one",
"one_mul",
"one_smul",
"real.norm_of_nonneg",
"zero_ne_one"
] | Approximate `f ⋆ g` if the support of the `f` is bounded within a ball, and `g` is near `g x₀`
on a ball with the same radius around `x₀`.
This is a special case of `dist_convolution_le'` where `L` is `(•)`, `f` has integral 1 and `f` is
nonnegative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convolution_tendsto_right
{ι} {g : ι → G → E'} {l : filter ι} {x₀ : G} {z₀ : E'}
{φ : ι → G → ℝ} {k : ι → G}
(hnφ : ∀ᶠ i in l, ∀ x, 0 ≤ φ i x)
(hiφ : ∀ᶠ i in l, ∫ x, φ i x ∂μ = 1) -- todo: we could weaken this to "the integral tends to 1"
(hφ : tendsto (λ n, support (φ n)) l (𝓝 0).small_sets)
(hmg : ∀ᶠ i i... | begin
simp_rw [tendsto_small_sets_iff] at hφ,
rw [metric.tendsto_nhds] at hcg ⊢,
simp_rw [metric.eventually_prod_nhds_iff] at hcg,
intros ε hε,
have h2ε : 0 < ε / 3 := div_pos hε (by norm_num),
obtain ⟨p, hp, δ, hδ, hgδ⟩ := hcg _ h2ε,
dsimp only [uncurry] at hgδ,
have h2k := hk.eventually (ball_mem_nhds... | lemma | convolution_tendsto_right | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"add_halves",
"dist_convolution_le",
"dist_triangle",
"dist_triangle_right",
"div_pos",
"filter",
"half_pos",
"metric.eventually_prod_nhds_iff",
"metric.tendsto_nhds"
] | `(φ i ⋆ g i) (k i)` tends to `z₀` as `i` tends to some filter `l` if
* `φ` is a sequence of nonnegative functions with integral `1` as `i` tends to `l`;
* The support of `φ` tends to small neighborhoods around `(0 : G)` as `i` tends to `l`;
* `g i` is `mu`-a.e. strongly measurable as `i` tends to `l`;
* `g i x` tends t... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convolution_eq_right {x₀ : G}
(hg : ∀ x ∈ ball x₀ φ.R, g x = g x₀) : (φ ⋆[lsmul ℝ ℝ, μ] g : G → E') x₀ = integral μ φ • g x₀ | by simp_rw [convolution_eq_right' _ φ.support_eq.subset hg, lsmul_apply, integral_smul_const] | lemma | cont_diff_bump.convolution_eq_right | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_eq_right'",
"integral_smul_const"
] | If `φ` is a bump function, compute `(φ ⋆ g) x₀` if `g` is constant on `metric.ball x₀ φ.R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_convolution_eq_right {x₀ : G}
(hg : ∀ x ∈ ball x₀ φ.R, g x = g x₀) : (φ.normed μ ⋆[lsmul ℝ ℝ, μ] g : G → E') x₀ = g x₀ | by { simp_rw [convolution_eq_right' _ φ.support_normed_eq.subset hg, lsmul_apply],
exact integral_normed_smul φ μ (g x₀) } | lemma | cont_diff_bump.normed_convolution_eq_right | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_eq_right'"
] | If `φ` is a normed bump function, compute `φ ⋆ g` if `g` is constant on `metric.ball x₀ φ.R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dist_normed_convolution_le {x₀ : G} {ε : ℝ}
(hmg : ae_strongly_measurable g μ)
(hg : ∀ x ∈ ball x₀ φ.R, dist (g x) (g x₀) ≤ ε) :
dist ((φ.normed μ ⋆[lsmul ℝ ℝ, μ] g : G → E') x₀) (g x₀) ≤ ε | dist_convolution_le (by simp_rw [← dist_self (g x₀), hg x₀ (mem_ball_self φ.R_pos)])
φ.support_normed_eq.subset φ.nonneg_normed φ.integral_normed hmg hg | lemma | cont_diff_bump.dist_normed_convolution_le | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"dist_convolution_le",
"dist_self"
] | If `φ` is a normed bump function, approximate `(φ ⋆ g) x₀` if `g` is near `g x₀` on a ball with
radius `φ.R` around `x₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convolution_tendsto_right {ι} {φ : ι → cont_diff_bump (0 : G)}
{g : ι → G → E'} {k : ι → G} {x₀ : G} {z₀ : E'} {l : filter ι}
(hφ : tendsto (λ i, (φ i).R) l (𝓝 0))
(hig : ∀ᶠ i in l, ae_strongly_measurable (g i) μ)
(hcg : tendsto (uncurry g) (l ×ᶠ 𝓝 x₀) (𝓝 z₀))
(hk : tendsto k l (𝓝 x₀)) :
tendsto (λ i, (... | convolution_tendsto_right (eventually_of_forall $ λ i, (φ i).nonneg_normed)
(eventually_of_forall $ λ i, (φ i).integral_normed)
(tendsto_support_normed_small_sets hφ) hig hcg hk | lemma | cont_diff_bump.convolution_tendsto_right | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"cont_diff_bump",
"convolution_tendsto_right",
"filter"
] | `(φ i ⋆ g i) (k i)` tends to `z₀` as `i` tends to some filter `l` if
* `φ` is a sequence of normed bump functions such that `(φ i).R` tends to `0` as `i` tends to `l`;
* `g i` is `mu`-a.e. strongly measurable as `i` tends to `l`;
* `g i x` tends to `z₀` as `(i, x)` tends to `l ×ᶠ 𝓝 x₀`;
* `k i` tends to `x₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convolution_tendsto_right_of_continuous {ι} {φ : ι → cont_diff_bump (0 : G)}
{l : filter ι} (hφ : tendsto (λ i, (φ i).R) l (𝓝 0))
(hg : continuous g) (x₀ : G) :
tendsto (λ i, ((λ x, (φ i).normed μ x) ⋆[lsmul ℝ ℝ, μ] g : G → E') x₀) l (𝓝 (g x₀)) | convolution_tendsto_right hφ (eventually_of_forall $ λ _, hg.ae_strongly_measurable)
((hg.tendsto x₀).comp tendsto_snd) tendsto_const_nhds | lemma | cont_diff_bump.convolution_tendsto_right_of_continuous | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"cont_diff_bump",
"continuous",
"convolution_tendsto_right",
"filter",
"tendsto_const_nhds"
] | Special case of `cont_diff_bump.convolution_tendsto_right` where `g` is continuous,
and the limit is taken only in the first function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_convolution
[has_measurable_add₂ G] [has_measurable_neg G]
[normed_space ℝ E] [normed_space ℝ E']
[complete_space E] [complete_space E']
(hf : integrable f ν) (hg : integrable g μ) :
∫ x, (f ⋆[L, ν] g) x ∂μ = L (∫ x, f x ∂ν) (∫ x, g x ∂μ) | begin
refine (integral_integral_swap (by apply hf.convolution_integrand L hg)).trans _,
simp_rw [integral_comp_comm _ (hg.comp_sub_right _), integral_sub_right_eq_self],
exact (L.flip (∫ x, g x ∂μ)).integral_comp_comm hf
end | lemma | integral_convolution | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"complete_space",
"has_measurable_add₂",
"has_measurable_neg",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convolution_assoc' (hL : ∀ (x : E) (y : E') (z : E''), L₂ (L x y) z = L₃ x (L₄ y z))
{x₀ : G}
(hfg : ∀ᵐ y ∂μ, convolution_exists_at f g y L ν)
(hgk : ∀ᵐ x ∂ν, convolution_exists_at g k x L₄ μ)
(hi : integrable (uncurry (λ x y, (L₃ (f y)) ((L₄ (g (x - y))) (k (x₀ - x))))) (μ.prod ν)) :
((f ⋆[L, ν] g) ⋆[L₂, μ] ... | calc ((f ⋆[L, ν] g) ⋆[L₂, μ] k) x₀
= ∫ t, L₂ (∫ s, L (f s) (g (t - s)) ∂ν) (k (x₀ - t)) ∂μ : rfl
... = ∫ t, ∫ s, L₂ (L (f s) (g (t - s))) (k (x₀ - t)) ∂ν ∂μ :
integral_congr_ae (hfg.mono $ λ t ht, ((L₂.flip (k (x₀ - t))).integral_comp_comm ht).symm)
... = ∫ t, ∫ s, L₃ (f s) (L₄ (g (t - s)) (k (x₀ - t))... | lemma | convolution_assoc' | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_exists_at"
] | Convolution is associative. This has a weak but inconvenient integrability condition.
See also `convolution_assoc`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convolution_assoc (hL : ∀ (x : E) (y : E') (z : E''), L₂ (L x y) z = L₃ x (L₄ y z))
{x₀ : G}
(hf : ae_strongly_measurable f ν)
(hg : ae_strongly_measurable g μ)
(hk : ae_strongly_measurable k μ)
(hfg : ∀ᵐ y ∂μ, convolution_exists_at f g y L ν)
(hgk : ∀ᵐ x ∂ν, convolution_exists_at (λ x, ‖g x‖) (λ x, ‖k x‖) ... | begin
refine convolution_assoc' L L₂ L₃ L₄ hL hfg (hgk.mono $ λ x hx, hx.of_norm L₄ hg hk) _,
/- the following is similar to `integrable.convolution_integrand` -/
have h_meas : ae_strongly_measurable
(uncurry (λ x y, L₃ (f y) (L₄ (g x) (k (x₀ - y - x))))) (μ.prod ν),
{ refine L₃.ae_strongly_measurable_com... | lemma | convolution_assoc | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"convolution_assoc'",
"convolution_def",
"convolution_exists_at",
"function.comp_apply",
"map_eq",
"measurable_fst",
"measurable_snd",
"mul_assoc",
"mul_le_mul_of_nonneg_left",
"mul_mul_mul_comm",
"real.norm_of_nonneg"
] | Convolution is associative. This requires that
* all maps are a.e. strongly measurable w.r.t one of the measures
* `f ⋆[L, ν] g` exists almost everywhere
* `‖g‖ ⋆[μ] ‖k‖` exists almost everywhere
* `‖f‖ ⋆[ν] (‖g‖ ⋆[μ] ‖k‖)` exists at `x₀` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convolution_precompR_apply {g : G → E'' →L[𝕜] E'}
(hf : locally_integrable f μ) (hcg : has_compact_support g) (hg : continuous g)
(x₀ : G) (x : E'') : (f ⋆[L.precompR E'', μ] g) x₀ x = (f ⋆[L, μ] (λ a, g a x)) x₀ | begin
have := hcg.convolution_exists_right (L.precompR E'' : _) hf hg x₀,
simp_rw [convolution_def, continuous_linear_map.integral_apply this],
refl,
end | lemma | convolution_precompR_apply | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"continuous",
"continuous_linear_map.integral_apply",
"convolution_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support.has_fderiv_at_convolution_right
(hcg : has_compact_support g) (hf : locally_integrable f μ) (hg : cont_diff 𝕜 1 g) (x₀ : G) :
has_fderiv_at (f ⋆[L, μ] g) ((f ⋆[L.precompR G, μ] fderiv 𝕜 g) x₀) x₀ | begin
rcases hcg.eq_zero_or_finite_dimensional 𝕜 hg.continuous with rfl|fin_dim,
{ have : fderiv 𝕜 (0 : G → E') = 0, from fderiv_const (0 : E'),
simp only [this, convolution_zero, pi.zero_apply],
exact has_fderiv_at_const (0 : F) x₀ },
resetI,
haveI : proper_space G, from finite_dimensional.proper_is_... | lemma | has_compact_support.has_fderiv_at_convolution_right | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"cont_diff",
"convolution_zero",
"fderiv",
"fderiv_const",
"finite_dimensional.proper_is_R_or_C",
"has_fderiv_at",
"has_fderiv_at.comp",
"has_fderiv_at_const",
"has_fderiv_at_id",
"has_fderiv_at_integral_of_dominated_of_fderiv_le",
"is_compact",
"le_rfl",
"proper_space",
"zero_lt_one"
] | Compute the total derivative of `f ⋆ g` if `g` is `C^1` with compact support and `f` is locally
integrable. To write down the total derivative as a convolution, we use
`continuous_linear_map.precompR`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_compact_support.has_fderiv_at_convolution_left [is_neg_invariant μ]
(hcf : has_compact_support f) (hf : cont_diff 𝕜 1 f) (hg : locally_integrable g μ) (x₀ : G) :
has_fderiv_at (f ⋆[L, μ] g) ((fderiv 𝕜 f ⋆[L.precompL G, μ] g) x₀) x₀ | begin
simp only [← convolution_flip] {single_pass := tt},
exact hcf.has_fderiv_at_convolution_right L.flip hg hf x₀,
end | lemma | has_compact_support.has_fderiv_at_convolution_left | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"cont_diff",
"convolution_flip",
"fderiv",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support.has_deriv_at_convolution_right
(hf : locally_integrable f₀ μ) (hcg : has_compact_support g₀) (hg : cont_diff 𝕜 1 g₀)
(x₀ : 𝕜) :
has_deriv_at (f₀ ⋆[L, μ] g₀) ((f₀ ⋆[L, μ] deriv g₀) x₀) x₀ | begin
convert (hcg.has_fderiv_at_convolution_right L hf hg x₀).has_deriv_at,
rw [convolution_precompR_apply L hf (hcg.fderiv 𝕜) (hg.continuous_fderiv le_rfl)],
refl,
end | lemma | has_compact_support.has_deriv_at_convolution_right | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"cont_diff",
"convolution_precompR_apply",
"deriv",
"has_deriv_at",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support.has_deriv_at_convolution_left [is_neg_invariant μ]
(hcf : has_compact_support f₀) (hf : cont_diff 𝕜 1 f₀)
(hg : locally_integrable g₀ μ) (x₀ : 𝕜) :
has_deriv_at (f₀ ⋆[L, μ] g₀) ((deriv f₀ ⋆[L, μ] g₀) x₀) x₀ | begin
simp only [← convolution_flip] {single_pass := tt},
exact hcf.has_deriv_at_convolution_right L.flip hg hf x₀,
end | lemma | has_compact_support.has_deriv_at_convolution_left | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"cont_diff",
"convolution_flip",
"deriv",
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_convolution_right_with_param
{g : P → G → E'} {s : set P} {k : set G} (hs : is_open s) (hk : is_compact k)
(hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : locally_integrable f μ) (hg : cont_diff_on 𝕜 1 ↿g (s ×ˢ univ))
(q₀ : P × G) (hq₀ : q₀.1 ∈ s) :
has_fderiv_at (λ (q : P × G), (f ⋆[L, μ] g ... | begin
let g' := fderiv 𝕜 ↿g,
have A : ∀ p ∈ s, continuous (g p),
{ assume p hp,
apply hg.continuous_on.comp_continuous (continuous_const.prod_mk continuous_id') (λ x, _),
simpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true] using hp },
have A' : ∀ (q : P × G), q.1 ∈ s → s ×ˢ univ ∈ 𝓝 q,
{ assume... | lemma | has_fderiv_at_convolution_right_with_param | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"bound",
"closure_minimal",
"cont_diff_on",
"continuous",
"continuous_id'",
"continuous_linear_map.id",
"continuous_on",
"convolution_integrand_bound_right_of_le_of_subset",
"differentiable_at",
"dist_self",
"exists_eq_right",
"fderiv",
"has_compact_support.convolution_exists_right",
"has_... | The derivative of the convolution `f * g` is given by `f * Dg`, when `f` is locally integrable
and `g` is `C^1` and compactly supported. Version where `g` depends on an additional parameter in an
open subset `s` of a parameter space `P` (and the compact support `k` is independent of the
parameter in `s`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_convolution_right_with_param_aux
{G : Type uP} {E' : Type uP} {F : Type uP} {P : Type uP}
[normed_add_comm_group E'] [normed_add_comm_group F]
[normed_space 𝕜 E'] [normed_space ℝ F] [normed_space 𝕜 F] [complete_space F]
[measurable_space G] {μ : measure G} [normed_add_comm_group G] [borel_space G... | begin
/- We have a formula for the derivation of `f * g`, which is of the same form, thanks to
`has_fderiv_at_convolution_right_with_param`. Therefore, we can prove the result by induction on
`n` (but for this we need the spaces at the different steps of the induction to live in the same
universe, which is why ... | lemma | cont_diff_on_convolution_right_with_param_aux | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"borel_space",
"complete_space",
"cont_diff_on",
"cont_diff_on_succ_iff_fderiv_of_open",
"cont_diff_on_top",
"cont_diff_on_zero",
"continuous_on_convolution_right_with_param",
"differentiable_at.differentiable_within_at",
"enat.nat_induction",
"fderiv",
"has_fderiv_at",
"has_fderiv_at_convolut... | The convolution `f * g` is `C^n` when `f` is locally integrable and `g` is `C^n` and compactly
supported. Version where `g` depends on an additional parameter in an open subset `s` of a
parameter space `P` (and the compact support `k` is independent of the parameter in `s`).
In this version, all the types belong to the... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_convolution_right_with_param
{f : G → E} {n : ℕ∞} (L : E →L[𝕜] E' →L[𝕜] F) {g : P → G → E'}
{s : set P} {k : set G} (hs : is_open s) (hk : is_compact k)
(hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : locally_integrable f μ) (hg : cont_diff_on 𝕜 n ↿g (s ×ˢ univ)) :
cont_diff_on 𝕜 n (λ (q : ... | begin
/- The result is known when all the universes are the same, from
`cont_diff_on_convolution_right_with_param_aux`. We reduce to this situation by pushing
everything through `ulift` continuous linear equivalences. -/
let eG : Type (max uG uE' uF uP) := ulift G,
borelize eG,
let eE' : Type (max uE' uG uF... | lemma | cont_diff_on_convolution_right_with_param | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"closed_embedding.integral_map",
"cont_diff",
"cont_diff_on",
"cont_diff_on_convolution_right_with_param_aux",
"continuous_linear_equiv.apply_symm_apply",
"continuous_linear_equiv.arrow_congr",
"continuous_linear_equiv.coe_coe",
"continuous_linear_equiv.coe_to_homeomorph",
"continuous_linear_equiv.c... | The convolution `f * g` is `C^n` when `f` is locally integrable and `g` is `C^n` and compactly
supported. Version where `g` depends on an additional parameter in an open subset `s` of a
parameter space `P` (and the compact support `k` is independent of the parameter in `s`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_convolution_right_with_param_comp
{n : ℕ∞} (L : E →L[𝕜] E' →L[𝕜] F)
{s : set P} {v : P → G} (hv : cont_diff_on 𝕜 n v s)
{f : G → E} {g : P → G → E'} {k : set G} (hs : is_open s) (hk : is_compact k)
(hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : locally_integrable f μ) (hg : cont_diff_on 𝕜 ... | begin
apply (cont_diff_on_convolution_right_with_param L hs hk hgs hf hg).comp
(cont_diff_on_id.prod hv),
assume x hx,
simp only [hx, mem_preimage, prod_mk_mem_set_prod_eq, mem_univ, and_self, id.def],
end | lemma | cont_diff_on_convolution_right_with_param_comp | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"cont_diff_on",
"cont_diff_on_convolution_right_with_param",
"is_compact",
"is_open"
] | The convolution `f * g` is `C^n` when `f` is locally integrable and `g` is `C^n` and compactly
supported. Version where `g` depends on an additional parameter in an open subset `s` of a
parameter space `P` (and the compact support `k` is independent of the parameter in `s`),
given in terms of composition with an additi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_convolution_left_with_param [μ.is_add_left_invariant] [μ.is_neg_invariant]
(L : E' →L[𝕜] E →L[𝕜] F) {f : G → E} {n : ℕ∞}
{g : P → G → E'} {s : set P} {k : set G} (hs : is_open s) (hk : is_compact k)
(hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : locally_integrable f μ) (hg : cont_diff_on 𝕜 n ... | by simpa only [convolution_flip]
using cont_diff_on_convolution_right_with_param L.flip hs hk hgs hf hg | lemma | cont_diff_on_convolution_left_with_param | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"cont_diff_on",
"cont_diff_on_convolution_right_with_param",
"convolution_flip",
"is_compact",
"is_open"
] | The convolution `g * f` is `C^n` when `f` is locally integrable and `g` is `C^n` and compactly
supported. Version where `g` depends on an additional parameter in an open subset `s` of a
parameter space `P` (and the compact support `k` is independent of the parameter in `s`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_convolution_left_with_param_comp [μ.is_add_left_invariant] [μ.is_neg_invariant]
(L : E' →L[𝕜] E →L[𝕜] F) {s : set P} {n : ℕ∞} {v : P → G} (hv : cont_diff_on 𝕜 n v s)
{f : G → E} {g : P → G → E'} {k : set G} (hs : is_open s) (hk : is_compact k)
(hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : lo... | begin
apply (cont_diff_on_convolution_left_with_param L hs hk hgs hf hg).comp (cont_diff_on_id.prod hv),
assume x hx,
simp only [hx, mem_preimage, prod_mk_mem_set_prod_eq, mem_univ, and_self, id.def],
end | lemma | cont_diff_on_convolution_left_with_param_comp | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"cont_diff_on",
"cont_diff_on_convolution_left_with_param",
"is_compact",
"is_open"
] | The convolution `g * f` is `C^n` when `f` is locally integrable and `g` is `C^n` and compactly
supported. Version where `g` depends on an additional parameter in an open subset `s` of a
parameter space `P` (and the compact support `k` is independent of the parameter in `s`),
given in terms of composition with additiona... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_compact_support.cont_diff_convolution_right {n : ℕ∞}
(hcg : has_compact_support g) (hf : locally_integrable f μ) (hg : cont_diff 𝕜 n g) :
cont_diff 𝕜 n (f ⋆[L, μ] g) | begin
rcases exists_compact_iff_has_compact_support.2 hcg with ⟨k, hk, h'k⟩,
rw ← cont_diff_on_univ,
exact cont_diff_on_convolution_right_with_param_comp L cont_diff_on_id is_open_univ hk
(λ p x hp hx, h'k x hx) hf (hg.comp cont_diff_snd).cont_diff_on,
end | lemma | has_compact_support.cont_diff_convolution_right | analysis | src/analysis/convolution.lean | [
"analysis.calculus.bump_function_inner",
"analysis.calculus.parametric_integral",
"measure_theory.constructions.prod.integral",
"measure_theory.function.locally_integrable",
"measure_theory.group.integration",
"measure_theory.group.prod",
"measure_theory.integral.interval_integral"
] | [
"cont_diff",
"cont_diff_on",
"cont_diff_on_convolution_right_with_param_comp",
"cont_diff_on_id",
"cont_diff_on_univ",
"cont_diff_snd",
"is_open_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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