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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|---|---|---|---|---|---|---|---|---|---|---|
convex_on_zpow : ∀ m : ℤ, convex_on ℝ (Ioi 0) (λ x : ℝ, x^m) | | (n : ℕ) :=
begin
simp_rw zpow_coe_nat,
exact (convex_on_pow n).subset Ioi_subset_Ici_self (convex_Ioi _)
end
| -[1+ n] :=
begin
simp_rw zpow_neg_succ_of_nat,
refine ⟨convex_Ioi _, _⟩,
rintros a (ha : 0 < a) b (hb : 0 < b) μ ν hμ hν h,
have ha' : 0 < a ^ (n + 1) := by positivity,
have hb' : 0 < b ^ (n + ... | lemma | convex_on_zpow | analysis.convex.specific_functions | src/analysis/convex/specific_functions/basic.lean | [
"analysis.convex.slope",
"analysis.special_functions.pow.real",
"tactic.linear_combination"
] | [
"convex_Ioi",
"convex_on",
"convex_on_pow",
"div_le_div_iff",
"mul_le_mul_of_nonneg_right",
"mul_pow",
"pow_le_pow_of_le_left",
"zpow_coe_nat",
"zpow_neg_succ_of_nat"
] | `x^m`, `m : ℤ` is convex on `(0, +∞)` for all `m`.
We give an elementary proof rather than using the second derivative test, since this lemma is
needed early in the analysis library. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_concave_on_log_Ioi : strict_concave_on ℝ (Ioi 0) log | begin
apply strict_concave_on_of_slope_strict_anti_adjacent (convex_Ioi (0:ℝ)),
rintros x y z (hx : 0 < x) (hz : 0 < z) hxy hyz,
have hy : 0 < y := hx.trans hxy,
transitivity y⁻¹,
{ have h : 0 < z - y := by linarith,
rw div_lt_iff h,
have hyz' : 0 < z / y := by positivity,
have hyz'' : z / y ≠ 1,
... | lemma | strict_concave_on_log_Ioi | analysis.convex.specific_functions | src/analysis/convex/specific_functions/basic.lean | [
"analysis.convex.slope",
"analysis.special_functions.pow.real",
"tactic.linear_combination"
] | [
"convex_Ioi",
"div_eq_one_iff_eq",
"div_lt_iff",
"lt_div_iff",
"ring",
"strict_concave_on",
"strict_concave_on_of_slope_strict_anti_adjacent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_add_mul_self_lt_rpow_one_add {s : ℝ} (hs : -1 ≤ s) (hs' : s ≠ 0) {p : ℝ} (hp : 1 < p) :
1 + p * s < (1 + s) ^ p | begin
rcases eq_or_lt_of_le hs with rfl | hs,
{ have : p ≠ 0 := by positivity,
simpa [zero_rpow this], },
have hs1 : 0 < 1 + s := by linarith,
cases le_or_lt (1 + p * s) 0 with hs2 hs2,
{ exact hs2.trans_lt (rpow_pos_of_pos hs1 _) },
rw [rpow_def_of_pos hs1, ← exp_log hs2],
apply exp_strict_mono,
ha... | lemma | one_add_mul_self_lt_rpow_one_add | analysis.convex.specific_functions | src/analysis/convex/specific_functions/basic.lean | [
"analysis.convex.slope",
"analysis.special_functions.pow.real",
"tactic.linear_combination"
] | [
"div_lt_div_right",
"div_lt_div_right_of_neg",
"div_lt_iff",
"eq_or_lt_of_le",
"zero_lt_one"
] | **Bernoulli's inequality** for real exponents, strict version: for `1 < p` and `-1 ≤ s`, with
`s ≠ 0`, we have `1 + p * s < (1 + s) ^ p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_add_mul_self_le_rpow_one_add {s : ℝ} (hs : -1 ≤ s) {p : ℝ} (hp : 1 ≤ p) :
1 + p * s ≤ (1 + s) ^ p | begin
rcases eq_or_lt_of_le hp with rfl | hp,
{ simp },
by_cases hs' : s = 0,
{ simp [hs'] },
exact (one_add_mul_self_lt_rpow_one_add hs hs' hp).le,
end | lemma | one_add_mul_self_le_rpow_one_add | analysis.convex.specific_functions | src/analysis/convex/specific_functions/basic.lean | [
"analysis.convex.slope",
"analysis.special_functions.pow.real",
"tactic.linear_combination"
] | [
"eq_or_lt_of_le",
"one_add_mul_self_lt_rpow_one_add"
] | **Bernoulli's inequality** for real exponents, non-strict version: for `1 ≤ p` and `-1 ≤ s`
we have `1 + p * s ≤ (1 + s) ^ p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_on_rpow {p : ℝ} (hp : 1 < p) : strict_convex_on ℝ (Ici 0) (λ x : ℝ, x^p) | begin
apply strict_convex_on_of_slope_strict_mono_adjacent (convex_Ici (0:ℝ)),
rintros x y z (hx : 0 ≤ x) (hz : 0 ≤ z) hxy hyz,
have hy : 0 < y := by linarith,
have hy' : 0 < y ^ p := rpow_pos_of_pos hy _,
have H1 : y ^ ((p - 1) + 1) = y ^ (p - 1) * y := rpow_add_one hy.ne' _,
ring_nf at H1,
transitivity ... | lemma | strict_convex_on_rpow | analysis.convex.specific_functions | src/analysis/convex/specific_functions/basic.lean | [
"analysis.convex.slope",
"analysis.special_functions.pow.real",
"tactic.linear_combination"
] | [
"convex_Ici",
"div_lt_div_right",
"div_lt_iff",
"div_lt_one",
"lt_div_iff",
"one_add_mul_self_lt_rpow_one_add",
"one_lt_div",
"strict_convex_on",
"strict_convex_on_of_slope_strict_mono_adjacent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on_rpow {p : ℝ} (hp : 1 ≤ p) : convex_on ℝ (Ici 0) (λ x : ℝ, x^p) | begin
rcases eq_or_lt_of_le hp with rfl | hp,
{ simpa using convex_on_id (convex_Ici _), },
exact (strict_convex_on_rpow hp).convex_on,
end | lemma | convex_on_rpow | analysis.convex.specific_functions | src/analysis/convex/specific_functions/basic.lean | [
"analysis.convex.slope",
"analysis.special_functions.pow.real",
"tactic.linear_combination"
] | [
"convex_Ici",
"convex_on",
"convex_on_id",
"eq_or_lt_of_le",
"strict_convex_on_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_concave_on_log_Iio : strict_concave_on ℝ (Iio 0) log | begin
refine ⟨convex_Iio _, _⟩,
rintros x (hx : x < 0) y (hy : y < 0) hxy a b ha hb hab,
have hx' : 0 < -x := by linarith,
have hy' : 0 < -y := by linarith,
have hxy' : - x ≠ - y := by contrapose! hxy; linarith,
calc a • log x + b • log y = a • log (-x) + b • log (-y) : by simp_rw [log_neg_eq_log]
... < l... | lemma | strict_concave_on_log_Iio | analysis.convex.specific_functions | src/analysis/convex/specific_functions/basic.lean | [
"analysis.convex.slope",
"analysis.special_functions.pow.real",
"tactic.linear_combination"
] | [
"algebra.id.smul_eq_mul",
"ring",
"strict_concave_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_on_pow {n : ℕ} (hn : 2 ≤ n) : strict_convex_on ℝ (Ici 0) (λ x : ℝ, x^n) | begin
apply strict_mono_on.strict_convex_on_of_deriv (convex_Ici _) (continuous_on_pow _),
rw [deriv_pow', interior_Ici],
exact λ x (hx : 0 < x) y hy hxy, mul_lt_mul_of_pos_left (pow_lt_pow_of_lt_left hxy hx.le $
nat.sub_pos_of_lt hn) (nat.cast_pos.2 $ zero_lt_two.trans_le hn),
end | lemma | strict_convex_on_pow | analysis.convex.specific_functions | src/analysis/convex/specific_functions/deriv.lean | [
"analysis.calculus.deriv.zpow",
"analysis.special_functions.pow.deriv",
"analysis.special_functions.sqrt"
] | [
"continuous_on_pow",
"convex_Ici",
"deriv_pow'",
"interior_Ici",
"mul_lt_mul_of_pos_left",
"pow_lt_pow_of_lt_left",
"strict_convex_on",
"strict_mono_on.strict_convex_on_of_deriv"
] | `x^n`, `n : ℕ` is strictly convex on `[0, +∞)` for all `n` greater than `2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
even.strict_convex_on_pow {n : ℕ} (hn : even n) (h : n ≠ 0) :
strict_convex_on ℝ set.univ (λ x : ℝ, x^n) | begin
apply strict_mono.strict_convex_on_univ_of_deriv (continuous_pow n),
rw deriv_pow',
replace h := nat.pos_of_ne_zero h,
exact strict_mono.const_mul (odd.strict_mono_pow $ nat.even.sub_odd h hn $ nat.odd_iff.2 rfl)
(nat.cast_pos.2 h),
end | lemma | even.strict_convex_on_pow | analysis.convex.specific_functions | src/analysis/convex/specific_functions/deriv.lean | [
"analysis.calculus.deriv.zpow",
"analysis.special_functions.pow.deriv",
"analysis.special_functions.sqrt"
] | [
"continuous_pow",
"deriv_pow'",
"nat.even.sub_odd",
"odd.strict_mono_pow",
"strict_convex_on",
"strict_mono.const_mul",
"strict_mono.strict_convex_on_univ_of_deriv"
] | `x^n`, `n : ℕ` is strictly convex on the whole real line whenever `n ≠ 0` is even. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finset.prod_nonneg_of_card_nonpos_even
{α β : Type*} [linear_ordered_comm_ring β]
{f : α → β} [decidable_pred (λ x, f x ≤ 0)]
{s : finset α} (h0 : even (s.filter (λ x, f x ≤ 0)).card) :
0 ≤ ∏ x in s, f x | calc 0 ≤ (∏ x in s, ((if f x ≤ 0 then (-1:β) else 1) * f x)) :
finset.prod_nonneg (λ x _, by
{ split_ifs with hx hx, by simp [hx], simp at hx ⊢, exact le_of_lt hx })
... = _ : by rw [finset.prod_mul_distrib, finset.prod_ite, finset.prod_const_one,
mul_one, finset.prod_const, neg_one_pow_eq_pow_mod_two, nat.even... | lemma | finset.prod_nonneg_of_card_nonpos_even | analysis.convex.specific_functions | src/analysis/convex/specific_functions/deriv.lean | [
"analysis.calculus.deriv.zpow",
"analysis.special_functions.pow.deriv",
"analysis.special_functions.sqrt"
] | [
"finset",
"finset.prod_const",
"finset.prod_const_one",
"finset.prod_ite",
"finset.prod_mul_distrib",
"finset.prod_nonneg",
"linear_ordered_comm_ring",
"mul_one",
"neg_one_pow_eq_pow_mod_two",
"one_mul",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_prod_range_nonneg (m : ℤ) (n : ℕ) (hn : even n) :
0 ≤ ∏ k in finset.range n, (m - k) | begin
rcases hn with ⟨n, rfl⟩,
induction n with n ihn, { simp },
rw ← two_mul at ihn,
rw [← two_mul, nat.succ_eq_add_one, mul_add, mul_one, bit0, ← add_assoc, finset.prod_range_succ,
finset.prod_range_succ, mul_assoc],
refine mul_nonneg ihn _, generalize : (1 + 1) * n = k,
cases le_or_lt m k with hmk hm... | lemma | int_prod_range_nonneg | analysis.convex.specific_functions | src/analysis/convex/specific_functions/deriv.lean | [
"analysis.calculus.deriv.zpow",
"analysis.special_functions.pow.deriv",
"analysis.special_functions.sqrt"
] | [
"finset.prod_range_succ",
"finset.range",
"lt_add_one",
"mul_assoc",
"mul_nonneg_of_nonpos_of_nonpos",
"mul_one",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_prod_range_pos {m : ℤ} {n : ℕ} (hn : even n) (hm : m ∉ Ico (0 : ℤ) n) :
0 < ∏ k in finset.range n, (m - k) | begin
refine (int_prod_range_nonneg m n hn).lt_of_ne (λ h, hm _),
rw [eq_comm, finset.prod_eq_zero_iff] at h,
obtain ⟨a, ha, h⟩ := h,
rw sub_eq_zero.1 h,
exact ⟨int.coe_zero_le _, int.coe_nat_lt.2 $ finset.mem_range.1 ha⟩,
end | lemma | int_prod_range_pos | analysis.convex.specific_functions | src/analysis/convex/specific_functions/deriv.lean | [
"analysis.calculus.deriv.zpow",
"analysis.special_functions.pow.deriv",
"analysis.special_functions.sqrt"
] | [
"finset.prod_eq_zero_iff",
"finset.range",
"int_prod_range_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_convex_on_zpow {m : ℤ} (hm₀ : m ≠ 0) (hm₁ : m ≠ 1) :
strict_convex_on ℝ (Ioi 0) (λ x : ℝ, x^m) | begin
apply strict_convex_on_of_deriv2_pos' (convex_Ioi 0),
{ exact (continuous_on_zpow₀ m).mono (λ x hx, ne_of_gt hx) },
intros x hx,
rw iter_deriv_zpow,
refine mul_pos _ (zpow_pos_of_pos hx _),
exact_mod_cast int_prod_range_pos (even_bit0 1) (λ hm, _),
norm_cast at hm,
rw ← finset.coe_Ico at hm,
fin... | lemma | strict_convex_on_zpow | analysis.convex.specific_functions | src/analysis/convex/specific_functions/deriv.lean | [
"analysis.calculus.deriv.zpow",
"analysis.special_functions.pow.deriv",
"analysis.special_functions.sqrt"
] | [
"continuous_on_zpow₀",
"convex_Ioi",
"even_bit0",
"finset.coe_Ico",
"int_prod_range_pos",
"iter_deriv_zpow",
"strict_convex_on",
"strict_convex_on_of_deriv2_pos'",
"zpow_pos_of_pos"
] | `x^m`, `m : ℤ` is convex on `(0, +∞)` for all `m` except `0` and `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_sqrt_mul_log {x : ℝ} (hx : x ≠ 0) :
has_deriv_at (λ x, sqrt x * log x) ((2 + log x) / (2 * sqrt x)) x | begin
convert (has_deriv_at_sqrt hx).mul (has_deriv_at_log hx),
rw [add_div, div_mul_right (sqrt x) two_ne_zero, ←div_eq_mul_inv, sqrt_div_self',
add_comm, div_eq_mul_one_div, mul_comm],
end | lemma | has_deriv_at_sqrt_mul_log | analysis.convex.specific_functions | src/analysis/convex/specific_functions/deriv.lean | [
"analysis.calculus.deriv.zpow",
"analysis.special_functions.pow.deriv",
"analysis.special_functions.sqrt"
] | [
"add_div",
"div_eq_mul_one_div",
"div_mul_right",
"has_deriv_at",
"mul_comm",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_sqrt_mul_log (x : ℝ) : deriv (λ x, sqrt x * log x) x = (2 + log x) / (2 * sqrt x) | begin
cases lt_or_le 0 x with hx hx,
{ exact (has_deriv_at_sqrt_mul_log hx.ne').deriv },
{ rw [sqrt_eq_zero_of_nonpos hx, mul_zero, div_zero],
refine has_deriv_within_at.deriv_eq_zero _ (unique_diff_on_Iic 0 x hx),
refine (has_deriv_within_at_const x _ 0).congr_of_mem (λ x hx, _) hx,
rw [sqrt_eq_zero_... | lemma | deriv_sqrt_mul_log | analysis.convex.specific_functions | src/analysis/convex/specific_functions/deriv.lean | [
"analysis.calculus.deriv.zpow",
"analysis.special_functions.pow.deriv",
"analysis.special_functions.sqrt"
] | [
"deriv",
"div_zero",
"has_deriv_at_sqrt_mul_log",
"has_deriv_within_at.deriv_eq_zero",
"has_deriv_within_at_const",
"mul_zero",
"unique_diff_on_Iic",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_sqrt_mul_log' : deriv (λ x, sqrt x * log x) = λ x, (2 + log x) / (2 * sqrt x) | funext deriv_sqrt_mul_log | lemma | deriv_sqrt_mul_log' | analysis.convex.specific_functions | src/analysis/convex/specific_functions/deriv.lean | [
"analysis.calculus.deriv.zpow",
"analysis.special_functions.pow.deriv",
"analysis.special_functions.sqrt"
] | [
"deriv",
"deriv_sqrt_mul_log"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv2_sqrt_mul_log (x : ℝ) :
deriv^[2] (λ x, sqrt x * log x) x = -log x / (4 * sqrt x ^ 3) | begin
simp only [nat.iterate, deriv_sqrt_mul_log'],
cases le_or_lt x 0 with hx hx,
{ rw [sqrt_eq_zero_of_nonpos hx, zero_pow zero_lt_three, mul_zero, div_zero],
refine has_deriv_within_at.deriv_eq_zero _ (unique_diff_on_Iic 0 x hx),
refine (has_deriv_within_at_const _ _ 0).congr_of_mem (λ x hx, _) hx,
... | lemma | deriv2_sqrt_mul_log | analysis.convex.specific_functions | src/analysis/convex/specific_functions/deriv.lean | [
"analysis.calculus.deriv.zpow",
"analysis.special_functions.pow.deriv",
"analysis.special_functions.sqrt"
] | [
"deriv",
"deriv_sqrt_mul_log'",
"div_zero",
"has_deriv_within_at.deriv_eq_zero",
"has_deriv_within_at_const",
"mul_ne_zero",
"mul_zero",
"ring",
"two_ne_zero",
"unique_diff_on_Iic",
"zero_lt_three",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_concave_on_sqrt_mul_log_Ioi : strict_concave_on ℝ (set.Ioi 1) (λ x, sqrt x * log x) | begin
apply strict_concave_on_of_deriv2_neg' (convex_Ioi 1) _ (λ x hx, _),
{ exact continuous_sqrt.continuous_on.mul
(continuous_on_log.mono (λ x hx, ne_of_gt (zero_lt_one.trans hx))) },
{ rw [deriv2_sqrt_mul_log x],
exact div_neg_of_neg_of_pos (neg_neg_of_pos (log_pos hx))
(mul_pos four_pos (pow_... | lemma | strict_concave_on_sqrt_mul_log_Ioi | analysis.convex.specific_functions | src/analysis/convex/specific_functions/deriv.lean | [
"analysis.calculus.deriv.zpow",
"analysis.special_functions.pow.deriv",
"analysis.special_functions.sqrt"
] | [
"convex_Ioi",
"deriv2_sqrt_mul_log",
"div_neg_of_neg_of_pos",
"pow_pos",
"set.Ioi",
"strict_concave_on",
"strict_concave_on_of_deriv2_neg'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_concave_on_sin_Icc : strict_concave_on ℝ (Icc 0 π) sin | begin
apply strict_concave_on_of_deriv2_neg (convex_Icc _ _) continuous_on_sin (λ x hx, _),
rw interior_Icc at hx,
simp [sin_pos_of_mem_Ioo hx],
end | lemma | strict_concave_on_sin_Icc | analysis.convex.specific_functions | src/analysis/convex/specific_functions/deriv.lean | [
"analysis.calculus.deriv.zpow",
"analysis.special_functions.pow.deriv",
"analysis.special_functions.sqrt"
] | [
"convex_Icc",
"interior_Icc",
"strict_concave_on",
"strict_concave_on_of_deriv2_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_concave_on_cos_Icc : strict_concave_on ℝ (Icc (-(π/2)) (π/2)) cos | begin
apply strict_concave_on_of_deriv2_neg (convex_Icc _ _) continuous_on_cos (λ x hx, _),
rw interior_Icc at hx,
simp [cos_pos_of_mem_Ioo hx],
end | lemma | strict_concave_on_cos_Icc | analysis.convex.specific_functions | src/analysis/convex/specific_functions/deriv.lean | [
"analysis.calculus.deriv.zpow",
"analysis.special_functions.pow.deriv",
"analysis.special_functions.sqrt"
] | [
"convex_Icc",
"interior_Icc",
"strict_concave_on",
"strict_concave_on_of_deriv2_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
scaled_exp_map_periodic :
function.periodic (λ x, exp_map_circle (2 * π / T * x)) T | begin
-- The case T = 0 is not interesting, but it is true, so we prove it to save hypotheses
rcases eq_or_ne T 0 with rfl | hT,
{ intro x, simp },
{ intro x, simp_rw mul_add, rw [div_mul_cancel _ hT, periodic_exp_map_circle] }
end | lemma | add_circle.scaled_exp_map_periodic | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"div_mul_cancel",
"eq_or_ne",
"exp_map_circle",
"function.periodic",
"periodic_exp_map_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_circle : add_circle T → circle | (@scaled_exp_map_periodic T).lift | def | add_circle.to_circle | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"circle",
"lift"
] | The canonical map `λ x, exp (2 π i x / T)` from `ℝ / ℤ • T` to the unit circle in `ℂ`.
If `T = 0` we understand this as the constant function 1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_circle_add (x : add_circle T) (y : add_circle T) :
to_circle (x + y) = to_circle x * to_circle y | begin
induction x using quotient_add_group.induction_on',
induction y using quotient_add_group.induction_on',
simp_rw [←quotient_add_group.coe_add, to_circle, function.periodic.lift_coe,
mul_add, exp_map_circle_add],
end | lemma | add_circle.to_circle_add | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"exp_map_circle_add",
"function.periodic.lift_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_to_circle : continuous (@to_circle T) | continuous_coinduced_dom.mpr (exp_map_circle.continuous.comp $ continuous_const.mul continuous_id') | lemma | add_circle.continuous_to_circle | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"continuous",
"continuous_id'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_to_circle (hT : T ≠ 0) : function.injective (@to_circle T) | begin
intros a b h,
induction a using quotient_add_group.induction_on',
induction b using quotient_add_group.induction_on',
simp_rw [to_circle, function.periodic.lift_coe] at h,
obtain ⟨m, hm⟩ := exp_map_circle_eq_exp_map_circle.mp h.symm,
simp_rw [quotient_add_group.eq, add_subgroup.mem_zmultiples_iff, zsm... | lemma | add_circle.injective_to_circle | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"function.periodic.lift_coe",
"mul_right_inj'",
"zsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
haar_add_circle : measure (add_circle T) | add_haar_measure ⊤ | def | add_circle.haar_add_circle | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle"
] | Haar measure on the additive circle, normalised to have total measure 1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
volume_eq_smul_haar_add_circle :
(volume : measure (add_circle T)) = ennreal.of_real T • haar_add_circle | rfl | lemma | add_circle.volume_eq_smul_haar_add_circle | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"ennreal.of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier (n : ℤ) : C(add_circle T, ℂ) | { to_fun := λ x, to_circle (n • x),
continuous_to_fun := continuous_induced_dom.comp $ continuous_to_circle.comp $ continuous_zsmul _} | def | fourier | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle"
] | The family of exponential monomials `λ x, exp (2 π i n x / T)`, parametrized by `n : ℤ` and
considered as bundled continuous maps from `ℝ / ℤ • T` to `ℂ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_apply {n : ℤ} {x : add_circle T} : fourier n x = to_circle (n • x) | rfl | lemma | fourier_apply | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"fourier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier_coe_apply {n : ℤ} {x : ℝ} :
fourier n (x : add_circle T) = complex.exp (2 * π * complex.I * n * x / T) | begin
rw [fourier_apply, ←quotient_add_group.coe_zsmul, to_circle, function.periodic.lift_coe,
exp_map_circle_apply, complex.of_real_mul, complex.of_real_div, complex.of_real_mul,
zsmul_eq_mul, complex.of_real_mul, complex.of_real_int_cast, complex.of_real_bit0,
complex.of_real_one],
congr' 1, ring,
end | lemma | fourier_coe_apply | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"complex.I",
"complex.exp",
"complex.of_real_bit0",
"complex.of_real_div",
"complex.of_real_int_cast",
"complex.of_real_mul",
"complex.of_real_one",
"exp_map_circle_apply",
"fourier",
"fourier_apply",
"function.periodic.lift_coe",
"ring",
"zsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier_zero {x : add_circle T} : fourier 0 x = 1 | begin
induction x using quotient_add_group.induction_on',
simp only [fourier_coe_apply, algebra_map.coe_zero, mul_zero, zero_mul,
zero_div, complex.exp_zero],
end | lemma | fourier_zero | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"algebra_map.coe_zero",
"complex.exp_zero",
"fourier",
"fourier_coe_apply",
"mul_zero",
"zero_div",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier_eval_zero (n : ℤ) : fourier n (0 : add_circle T) = 1 | by rw [←quotient_add_group.coe_zero, fourier_coe_apply, complex.of_real_zero,
mul_zero, zero_div, complex.exp_zero] | lemma | fourier_eval_zero | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"complex.exp_zero",
"complex.of_real_zero",
"fourier",
"fourier_coe_apply",
"mul_zero",
"zero_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier_one {x : add_circle T} : fourier 1 x = to_circle x | by rw [fourier_apply, one_zsmul] | lemma | fourier_one | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"fourier",
"fourier_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier_neg {n : ℤ} {x : add_circle T} : fourier (-n) x = conj (fourier n x) | begin
induction x using quotient_add_group.induction_on',
simp_rw [fourier_apply, to_circle, ←quotient_add_group.coe_zsmul,
function.periodic.lift_coe, ←coe_inv_circle_eq_conj, ←exp_map_circle_neg, neg_smul, mul_neg],
end | lemma | fourier_neg | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"fourier",
"fourier_apply",
"function.periodic.lift_coe",
"mul_neg",
"neg_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier_add {m n : ℤ} {x : add_circle T} :
fourier (m + n) x = fourier m x * fourier n x | by simp_rw [fourier_apply, add_zsmul, to_circle_add, coe_mul_unit_sphere] | lemma | fourier_add | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"coe_mul_unit_sphere",
"fourier",
"fourier_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier_norm [fact (0 < T)] (n : ℤ) : ‖@fourier T n‖ = 1 | begin
rw continuous_map.norm_eq_supr_norm,
have : ∀ (x : add_circle T), ‖fourier n x‖ = 1 := λ x, abs_coe_circle _,
simp_rw this,
exact @csupr_const _ _ _ has_zero.nonempty _,
end | lemma | fourier_norm | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"abs_coe_circle",
"add_circle",
"continuous_map.norm_eq_supr_norm",
"csupr_const",
"fact",
"fourier",
"has_zero.nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier_add_half_inv_index {n : ℤ} (hn : n ≠ 0) (hT : 0 < T) (x : add_circle T) :
fourier n (x + ((T / 2 / n) : ℝ)) = - fourier n x | begin
rw [fourier_apply, zsmul_add, ←quotient_add_group.coe_zsmul, to_circle_add, coe_mul_unit_sphere],
have : (n : ℂ) ≠ 0 := by simpa using hn,
have : ((@to_circle T ((n • (T / 2 / n)) : ℝ)) : ℂ) = -1,
{ rw [zsmul_eq_mul, to_circle, function.periodic.lift_coe, exp_map_circle_apply],
replace hT := complex.o... | lemma | fourier_add_half_inv_index | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"coe_mul_unit_sphere",
"complex.exp_pi_mul_I",
"exp_map_circle_apply",
"fourier",
"fourier_apply",
"function.periodic.lift_coe",
"ring",
"zsmul_eq_mul"
] | For `n ≠ 0`, a translation by `T / 2 / n` negates the function `fourier n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_subalgebra : subalgebra ℂ C(add_circle T, ℂ) | algebra.adjoin ℂ (range fourier) | def | fourier_subalgebra | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"algebra.adjoin",
"fourier",
"subalgebra"
] | The subalgebra of `C(add_circle T, ℂ)` generated by `fourier n` for `n ∈ ℤ` . | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_subalgebra_coe : (@fourier_subalgebra T).to_submodule = span ℂ (range fourier) | begin
apply adjoin_eq_span_of_subset,
refine subset.trans _ submodule.subset_span,
intros x hx,
apply submonoid.closure_induction hx (λ _, id) ⟨0, _⟩,
{ rintros _ _ ⟨m, rfl⟩ ⟨n, rfl⟩,
refine ⟨m + n, _⟩,
ext1 z,
exact fourier_add },
{ ext1 z, exact fourier_zero }
end | lemma | fourier_subalgebra_coe | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"fourier",
"fourier_add",
"fourier_subalgebra",
"fourier_zero",
"submodule.subset_span",
"submonoid.closure_induction"
] | The subalgebra of `C(add_circle T, ℂ)` generated by `fourier n` for `n ∈ ℤ` is in fact the
linear span of these functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_subalgebra_conj_invariant :
conj_invariant_subalgebra ((@fourier_subalgebra T).restrict_scalars ℝ) | begin
apply subalgebra_conj_invariant,
rintros _ ⟨n, rfl⟩,
exact ⟨-n, ext (λ _, fourier_neg)⟩
end | lemma | fourier_subalgebra_conj_invariant | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"fourier_neg",
"fourier_subalgebra",
"restrict_scalars"
] | The subalgebra of `C(add_circle T, ℂ)` generated by `fourier n` for `n ∈ ℤ` is invariant under
complex conjugation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_subalgebra_separates_points : (@fourier_subalgebra T).separates_points | begin
intros x y hxy,
refine ⟨_, ⟨fourier 1, subset_adjoin ⟨1, rfl⟩, rfl⟩, _⟩,
dsimp only, rw [fourier_one, fourier_one],
contrapose! hxy,
rw subtype.coe_inj at hxy,
exact injective_to_circle hT.elim.ne' hxy,
end | lemma | fourier_subalgebra_separates_points | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"fourier_one",
"fourier_subalgebra",
"subtype.coe_inj"
] | The subalgebra of `C(add_circle T, ℂ)` generated by `fourier n` for `n ∈ ℤ`
separates points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_subalgebra_closure_eq_top : (@fourier_subalgebra T).topological_closure = ⊤ | continuous_map.subalgebra_is_R_or_C_topological_closure_eq_top_of_separates_points
fourier_subalgebra fourier_subalgebra_separates_points fourier_subalgebra_conj_invariant | lemma | fourier_subalgebra_closure_eq_top | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"continuous_map.subalgebra_is_R_or_C_topological_closure_eq_top_of_separates_points",
"fourier_subalgebra",
"fourier_subalgebra_conj_invariant",
"fourier_subalgebra_separates_points"
] | The subalgebra of `C(add_circle T, ℂ)` generated by `fourier n` for `n ∈ ℤ` is dense. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span_fourier_closure_eq_top : (span ℂ (range $ @fourier T)).topological_closure = ⊤ | begin
rw ← fourier_subalgebra_coe,
exact congr_arg subalgebra.to_submodule fourier_subalgebra_closure_eq_top,
end | lemma | span_fourier_closure_eq_top | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"fourier",
"fourier_subalgebra_closure_eq_top",
"fourier_subalgebra_coe",
"subalgebra.to_submodule"
] | The linear span of the monomials `fourier n` is dense in `C(add_circle T, ℂ)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_Lp (p : ℝ≥0∞) [fact (1 ≤ p)] (n : ℤ) : Lp ℂ p (@haar_add_circle T hT) | to_Lp p haar_add_circle ℂ (fourier n) | abbreviation | fourier_Lp | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"fact",
"fourier"
] | The family of monomials `fourier n`, parametrized by `n : ℤ` and considered as
elements of the `Lp` space of functions `add_circle T → ℂ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fn_fourier_Lp (p : ℝ≥0∞) [fact (1 ≤ p)] (n : ℤ) :
(@fourier_Lp T hT p _ n) =ᵐ[haar_add_circle] fourier n | coe_fn_to_Lp haar_add_circle (fourier n) | lemma | coe_fn_fourier_Lp | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"fact",
"fourier",
"fourier_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_fourier_Lp_closure_eq_top {p : ℝ≥0∞} [fact (1 ≤ p)] (hp : p ≠ ∞) :
(span ℂ (range (@fourier_Lp T _ p _))).topological_closure = ⊤ | begin
convert (continuous_map.to_Lp_dense_range ℂ (@haar_add_circle T hT) hp ℂ
).topological_closure_map_submodule (span_fourier_closure_eq_top),
rw [map_span, range_comp],
simp only [continuous_linear_map.coe_coe],
end | lemma | span_fourier_Lp_closure_eq_top | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"continuous_linear_map.coe_coe",
"continuous_map.to_Lp_dense_range",
"fact",
"fourier_Lp",
"span_fourier_closure_eq_top"
] | For each `1 ≤ p < ∞`, the linear span of the monomials `fourier n` is dense in
`Lp ℂ p haar_circle`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthonormal_fourier : orthonormal ℂ (@fourier_Lp T _ 2 _) | begin
rw orthonormal_iff_ite,
intros i j,
rw continuous_map.inner_to_Lp (@haar_add_circle T hT) (fourier i) (fourier j),
simp_rw [←fourier_neg, ←fourier_add],
split_ifs,
{ simp_rw [h, neg_add_self],
have : ⇑(@fourier T 0) = (λ x, 1 : (add_circle T) → ℂ),
{ ext1, exact fourier_zero },
rw [this, i... | lemma | orthonormal_fourier | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"complex.of_real_one",
"complex.real_smul",
"ennreal.one_to_real",
"fourier",
"fourier_Lp",
"fourier_add_half_inv_index",
"fourier_zero",
"mul_one",
"orthonormal",
"orthonormal_iff_ite"
] | The monomials `fourier n` are an orthonormal set with respect to normalised Haar measure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_coeff (f : add_circle T → E) (n : ℤ) : E | ∫ (t : add_circle T), fourier (-n) t • f t ∂ haar_add_circle | def | fourier_coeff | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"fourier"
] | The `n`-th Fourier coefficient of a function `add_circle T → E`, for `E` a complete normed
`ℂ`-vector space, defined as the integral over `add_circle T` of `fourier (-n) t • f t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_coeff_eq_interval_integral (f : add_circle T → E) (n : ℤ) (a : ℝ) :
fourier_coeff f n = (1 / T) • ∫ x in a .. a + T, @fourier T (-n) x • f x | begin
have : ∀ (x : ℝ), @fourier T (-n) x • f x = (λ (z : add_circle T), @fourier T (-n) z • f z) x,
{ intro x, refl, },
simp_rw this,
rw [fourier_coeff, add_circle.interval_integral_preimage T a,
volume_eq_smul_haar_add_circle, integral_smul_measure, ennreal.to_real_of_real hT.out.le,
←smul_assoc, smul... | lemma | fourier_coeff_eq_interval_integral | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"add_circle.interval_integral_preimage",
"ennreal.to_real_of_real",
"fourier",
"fourier_coeff",
"one_div_mul_cancel",
"one_smul",
"smul_eq_mul"
] | The Fourier coefficients of a function on `add_circle T` can be computed as an integral
over `[a, a + T]`, for any real `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_coeff.const_smul (f : add_circle T → E) (c : ℂ) (n : ℤ) :
fourier_coeff (c • f) n = c • fourier_coeff f n | by simp_rw [fourier_coeff, pi.smul_apply, ←smul_assoc, smul_eq_mul, mul_comm, ←smul_eq_mul,
smul_assoc, integral_smul] | lemma | fourier_coeff.const_smul | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"fourier_coeff",
"mul_comm",
"pi.smul_apply",
"smul_assoc",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier_coeff.const_mul (f : add_circle T → ℂ) (c : ℂ) (n : ℤ) :
fourier_coeff (λ x, c * f x) n = c * fourier_coeff f n | fourier_coeff.const_smul f c n | lemma | fourier_coeff.const_mul | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"fourier_coeff",
"fourier_coeff.const_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier_coeff_on {a b : ℝ} (hab : a < b) (f : ℝ → E) (n : ℤ) : E | begin
haveI := fact.mk (by linarith : 0 < b - a),
exact fourier_coeff (add_circle.lift_Ioc (b - a) a f) n
end | def | fourier_coeff_on | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle.lift_Ioc",
"fourier_coeff"
] | For a function on `ℝ`, the Fourier coefficients of `f` on `[a, b]` are defined as the
Fourier coefficients of the unique periodic function agreeing with `f` on `Ioc a b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_coeff_on_eq_integral {a b : ℝ} (f : ℝ → E) (n : ℤ) (hab : a < b) :
fourier_coeff_on hab f n =
(1 / (b - a)) • ∫ x in a ..b, fourier (-n) (x : add_circle (b - a)) • f x | begin
rw [fourier_coeff_on, fourier_coeff_eq_interval_integral _ _ a],
congr' 1,
rw [add_sub, add_sub_cancel'],
simp_rw interval_integral.integral_of_le hab.le,
refine set_integral_congr measurable_set_Ioc (λ x hx, _),
dsimp only,
rwa [lift_Ioc_coe_apply],
rwa [add_sub, add_sub_cancel'],
end | lemma | fourier_coeff_on_eq_integral | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"fourier",
"fourier_coeff_eq_interval_integral",
"fourier_coeff_on",
"interval_integral.integral_of_le",
"measurable_set_Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier_coeff_on.const_smul {a b : ℝ} (f : ℝ → E) (c : ℂ) (n : ℤ) (hab : a < b) :
fourier_coeff_on hab (c • f) n = c • fourier_coeff_on hab f n | by apply fourier_coeff.const_smul | lemma | fourier_coeff_on.const_smul | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"fourier_coeff.const_smul",
"fourier_coeff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier_coeff_on.const_mul {a b : ℝ} (f : ℝ → ℂ) (c : ℂ) (n : ℤ) (hab : a < b) :
fourier_coeff_on hab (λ x, c * f x) n = c * fourier_coeff_on hab f n | fourier_coeff_on.const_smul _ _ _ _ | lemma | fourier_coeff_on.const_mul | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"fourier_coeff_on",
"fourier_coeff_on.const_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier_coeff_lift_Ioc_eq {a : ℝ} (f : ℝ → ℂ) (n : ℤ) :
fourier_coeff (add_circle.lift_Ioc T a f) n =
fourier_coeff_on (lt_add_of_pos_right a hT.out) f n | begin
rw [fourier_coeff_on_eq_integral, fourier_coeff_eq_interval_integral, add_sub_cancel' a T],
congr' 1,
refine interval_integral.integral_congr_ae (ae_of_all _ (λ x hx, _)),
rw lift_Ioc_coe_apply,
rwa uIoc_of_le (lt_add_of_pos_right a hT.out).le at hx,
end | lemma | fourier_coeff_lift_Ioc_eq | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle.lift_Ioc",
"fourier_coeff",
"fourier_coeff_eq_interval_integral",
"fourier_coeff_on",
"fourier_coeff_on_eq_integral",
"interval_integral.integral_congr_ae"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier_coeff_lift_Ico_eq {a : ℝ} (f : ℝ → ℂ) (n : ℤ) :
fourier_coeff (add_circle.lift_Ico T a f) n =
fourier_coeff_on (lt_add_of_pos_right a hT.out) f n | begin
rw [fourier_coeff_on_eq_integral, fourier_coeff_eq_interval_integral _ _ a, add_sub_cancel' a T],
congr' 1,
simp_rw [interval_integral.integral_of_le (lt_add_of_pos_right a hT.out).le,
integral_Ioc_eq_integral_Ioo],
refine set_integral_congr measurable_set_Ioo (λ x hx, _),
dsimp only,
rw lift_Ico_... | lemma | fourier_coeff_lift_Ico_eq | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle.lift_Ico",
"fourier_coeff",
"fourier_coeff_eq_interval_integral",
"fourier_coeff_on",
"fourier_coeff_on_eq_integral",
"interval_integral.integral_of_le",
"measurable_set_Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier_basis : hilbert_basis ℤ ℂ (Lp ℂ 2 $ @haar_add_circle T hT) | hilbert_basis.mk orthonormal_fourier (span_fourier_Lp_closure_eq_top (by norm_num)).ge | def | fourier_basis | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"hilbert_basis",
"hilbert_basis.mk",
"orthonormal_fourier",
"span_fourier_Lp_closure_eq_top"
] | We define `fourier_basis` to be a `ℤ`-indexed Hilbert basis for `Lp ℂ 2 haar_add_circle`,
which by definition is an isometric isomorphism from `Lp ℂ 2 haar_add_circle` to `ℓ²(ℤ, ℂ)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fourier_basis : ⇑(@fourier_basis _ hT) = fourier_Lp 2 | hilbert_basis.coe_mk _ _ | lemma | coe_fourier_basis | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"fourier_Lp",
"fourier_basis",
"hilbert_basis.coe_mk"
] | The elements of the Hilbert basis `fourier_basis` are the functions `fourier_Lp 2`, i.e. the
monomials `fourier n` on the circle considered as elements of `L²`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_basis_repr (f : Lp ℂ 2 $ @haar_add_circle T hT) (i : ℤ) :
fourier_basis.repr f i = fourier_coeff f i | begin
transitivity ∫ (t : add_circle T),
conj (((@fourier_Lp T hT 2 _ i) : add_circle T → ℂ) t) * f t ∂ haar_add_circle,
{ simp [fourier_basis.repr_apply_apply f i, measure_theory.L2.inner_def] },
{ apply integral_congr_ae,
filter_upwards [coe_fn_fourier_Lp 2 i] with _ ht,
rw [ht, ←fourier_neg, smul_e... | lemma | fourier_basis_repr | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"coe_fn_fourier_Lp",
"fourier_Lp",
"fourier_coeff",
"measure_theory.L2.inner_def",
"smul_eq_mul"
] | Under the isometric isomorphism `fourier_basis` from `Lp ℂ 2 haar_circle` to `ℓ²(ℤ, ℂ)`, the
`i`-th coefficient is `fourier_coeff f i`, i.e., the integral over `add_circle T` of
`λ t, fourier (-i) t * f t` with respect to the Haar measure of total mass 1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum_fourier_series_L2 (f : Lp ℂ 2 $ @haar_add_circle T hT) :
has_sum (λ i, fourier_coeff f i • fourier_Lp 2 i) f | by { simp_rw ←fourier_basis_repr, simpa using hilbert_basis.has_sum_repr fourier_basis f } | lemma | has_sum_fourier_series_L2 | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"fourier_Lp",
"fourier_basis",
"fourier_coeff",
"has_sum",
"hilbert_basis.has_sum_repr"
] | The Fourier series of an `L2` function `f` sums to `f`, in the `L²` space of `add_circle T`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsum_sq_fourier_coeff (f : Lp ℂ 2 $ @haar_add_circle T hT) :
∑' i : ℤ, ‖fourier_coeff f i‖ ^ 2 = ∫ (t : add_circle T), ‖f t‖ ^ 2 ∂ haar_add_circle | begin
simp_rw ←fourier_basis_repr,
have H₁ : ‖fourier_basis.repr f‖ ^ 2 = ∑' i, ‖fourier_basis.repr f i‖ ^ 2,
{ exact_mod_cast lp.norm_rpow_eq_tsum _ (fourier_basis.repr f),
norm_num },
have H₂ : ‖fourier_basis.repr f‖ ^ 2 = ‖f‖ ^ 2 := by simp,
have H₃ := congr_arg is_R_or_C.re (@L2.inner_def (add_circle ... | lemma | tsum_sq_fourier_coeff | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"integral_re",
"lp.norm_rpow_eq_tsum"
] | **Parseval's identity**: for an `L²` function `f` on `add_circle T`, the sum of the squared
norms of the Fourier coefficients equals the `L²` norm of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_coeff_to_Lp (n : ℤ) :
fourier_coeff (to_Lp 2 haar_add_circle ℂ f) n = fourier_coeff f n | integral_congr_ae (filter.eventually_eq.mul
(filter.eventually_of_forall (by tauto))
(continuous_map.coe_fn_to_ae_eq_fun haar_add_circle f)) | lemma | fourier_coeff_to_Lp | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"continuous_map.coe_fn_to_ae_eq_fun",
"filter.eventually_eq.mul",
"filter.eventually_of_forall",
"fourier_coeff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_fourier_series_of_summable (h : summable (fourier_coeff f)) :
has_sum (λ i, fourier_coeff f i • fourier i) f | begin
have sum_L2 := has_sum_fourier_series_L2 (to_Lp 2 haar_add_circle ℂ f),
simp_rw fourier_coeff_to_Lp at sum_L2,
refine continuous_map.has_sum_of_has_sum_Lp (summable_of_summable_norm _) sum_L2,
simp_rw [norm_smul, fourier_norm, mul_one, summable_norm_iff],
exact h,
end | lemma | has_sum_fourier_series_of_summable | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"continuous_map.has_sum_of_has_sum_Lp",
"fourier",
"fourier_coeff",
"fourier_coeff_to_Lp",
"fourier_norm",
"has_sum",
"has_sum_fourier_series_L2",
"mul_one",
"norm_smul",
"summable",
"summable_norm_iff",
"summable_of_summable_norm"
] | If the sequence of Fourier coefficients of `f` is summable, then the Fourier series converges
uniformly to `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pointwise_sum_fourier_series_of_summable
(h : summable (fourier_coeff f)) (x : add_circle T) :
has_sum (λ i, fourier_coeff f i • fourier i x) (f x) | (continuous_map.eval_clm ℂ x).has_sum (has_sum_fourier_series_of_summable h) | lemma | has_pointwise_sum_fourier_series_of_summable | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"continuous_map.eval_clm",
"fourier",
"fourier_coeff",
"has_sum",
"has_sum_fourier_series_of_summable",
"summable"
] | If the sequence of Fourier coefficients of `f` is summable, then the Fourier series of `f`
converges everywhere pointwise to `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_fourier (n : ℤ) (x : ℝ) : has_deriv_at (λ y:ℝ, fourier n (y : add_circle T))
(2 * π * I * n / T * fourier n (x : add_circle T)) x | begin
simp_rw [fourier_coe_apply],
refine (_ : has_deriv_at (λ y, exp (2 * π * I * n * y / T)) _ _).comp_of_real,
rw (λ α β, by ring : ∀ (α β : ℂ), α * exp β = exp β * α),
refine (has_deriv_at_exp _).comp x _,
convert has_deriv_at_mul_const (2 * ↑π * I * ↑n / T),
ext1 y, ring,
end | lemma | has_deriv_at_fourier | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"exp",
"fourier",
"fourier_coe_apply",
"has_deriv_at",
"has_deriv_at_exp",
"has_deriv_at_mul_const",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_fourier_neg (n : ℤ) (x : ℝ) :
has_deriv_at (λ y:ℝ, fourier (-n) (y : add_circle T))
(-2 * π * I * n / T * fourier (-n) (x : add_circle T)) x | by simpa using has_deriv_at_fourier T (-n) x | lemma | has_deriv_at_fourier_neg | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"fourier",
"has_deriv_at",
"has_deriv_at_fourier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_antideriv_at_fourier_neg (hT : fact (0 < T)) {n : ℤ} (hn : n ≠ 0) (x : ℝ) :
has_deriv_at (λ (y : ℝ), (T : ℂ) / (-2 * π * I * n) * fourier (-n) (y : add_circle T))
(fourier (-n) (x : add_circle T)) x | begin
convert (has_deriv_at_fourier_neg T n x).div_const (-2 * π * I * n / T) using 1,
{ ext1 y, rw div_div_eq_mul_div, ring, },
{ rw mul_div_cancel_left,
simp only [ne.def, div_eq_zero_iff, neg_eq_zero, mul_eq_zero, bit0_eq_zero, one_ne_zero,
of_real_eq_zero, false_or, int.cast_eq_zero, not_or_distrib]... | lemma | has_antideriv_at_fourier_neg | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"bit0_eq_zero",
"div_div_eq_mul_div",
"div_eq_zero_iff",
"fact",
"fourier",
"has_deriv_at",
"has_deriv_at_fourier_neg",
"int.cast_eq_zero",
"mul_div_cancel_left",
"mul_eq_zero",
"not_or_distrib",
"one_ne_zero",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier_coeff_on_of_has_deriv_at {a b : ℝ} (hab : a < b) {f f' : ℝ → ℂ} {n : ℤ}
(hn : n ≠ 0) (hf : ∀ x, x ∈ [a, b] → has_deriv_at f (f' x) x)
(hf' : interval_integrable f' volume a b) :
fourier_coeff_on hab f n =
1 / (-2 * π * I * n) * (fourier (-n) (a : add_circle (b - a)) * (f b - f a)
- (b - a) * fourier... | begin
rw ←of_real_sub,
have hT : fact (0 < b - a) := ⟨by linarith⟩,
simp_rw [fourier_coeff_on_eq_integral, smul_eq_mul, real_smul, of_real_div, of_real_one],
conv { for (fourier _ _ * _) [1, 2, 3] { rw mul_comm } },
rw integral_mul_deriv_eq_deriv_mul hf (λ x hx, has_antideriv_at_fourier_neg hT hn x) hf'
(... | lemma | fourier_coeff_on_of_has_deriv_at | analysis.fourier | src/analysis/fourier/add_circle.lean | [
"analysis.special_functions.exp_deriv",
"analysis.special_functions.complex.circle",
"analysis.inner_product_space.l2_space",
"measure_theory.function.continuous_map_dense",
"measure_theory.function.l2_space",
"measure_theory.group.integration",
"measure_theory.integral.periodic",
"topology.continuous... | [
"add_circle",
"add_circle.continuous_mk'",
"div_eq_iff",
"fact",
"fourier",
"fourier_coeff_on",
"fourier_coeff_on_eq_integral",
"has_antideriv_at_fourier_neg",
"has_deriv_at",
"interval_integrable",
"interval_integral",
"mul_comm",
"mul_div",
"mul_one",
"ring",
"smul_eq_mul"
] | Express Fourier coefficients of `f` on an interval in terms of those of its derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_integral
(e : (multiplicative 𝕜) →* 𝕊) (μ : measure V) (L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜)
(f : V → E) (w : W) : E | ∫ v, e [-L v w] • f v ∂μ | def | vector_fourier.fourier_integral | analysis.fourier | src/analysis/fourier/fourier_transform.lean | [
"analysis.complex.circle",
"measure_theory.group.integration",
"measure_theory.measure.haar.of_basis"
] | [
"multiplicative"
] | The Fourier transform integral for `f : V → E`, with respect to a bilinear form `L : V × W → 𝕜`
and an additive character `e`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_integral_smul_const
(e : (multiplicative 𝕜) →* 𝕊) (μ : measure V) (L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜)
(f : V → E) (r : ℂ) :
fourier_integral e μ L (r • f) = r • (fourier_integral e μ L f) | begin
ext1 w,
simp only [pi.smul_apply, fourier_integral, smul_comm _ r, integral_smul],
end | lemma | vector_fourier.fourier_integral_smul_const | analysis.fourier | src/analysis/fourier/fourier_transform.lean | [
"analysis.complex.circle",
"measure_theory.group.integration",
"measure_theory.measure.haar.of_basis"
] | [
"multiplicative",
"pi.smul_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_fourier_integral_le_integral_norm (e : (multiplicative 𝕜) →* 𝕊) (μ : measure V)
(L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E) (w : W) :
‖fourier_integral e μ L f w‖ ≤ ∫ (v : V), ‖f v‖ ∂μ | begin
refine (norm_integral_le_integral_norm _).trans (le_of_eq _),
simp_rw [norm_smul, complex.norm_eq_abs, abs_coe_circle, one_mul],
end | lemma | vector_fourier.norm_fourier_integral_le_integral_norm | analysis.fourier | src/analysis/fourier/fourier_transform.lean | [
"analysis.complex.circle",
"measure_theory.group.integration",
"measure_theory.measure.haar.of_basis"
] | [
"abs_coe_circle",
"complex.norm_eq_abs",
"multiplicative",
"norm_smul",
"one_mul"
] | The uniform norm of the Fourier integral of `f` is bounded by the `L¹` norm of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_integral_comp_add_right [has_measurable_add V]
(e : (multiplicative 𝕜) →* 𝕊) (μ : measure V) [μ.is_add_right_invariant]
(L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E) (v₀ : V) :
fourier_integral e μ L (f ∘ (λ v, v + v₀)) = λ w, e [L v₀ w] • fourier_integral e μ L f w | begin
ext1 w,
dsimp only [fourier_integral, function.comp_apply],
conv in (L _) { rw ←add_sub_cancel v v₀ },
rw integral_add_right_eq_self (λ (v : V), e [-L (v - v₀) w] • f v),
swap, apply_instance,
dsimp only,
rw ←integral_smul,
congr' 1 with v,
rw [←smul_assoc, smul_eq_mul, ←submonoid.coe_mul, ←e.ma... | lemma | vector_fourier.fourier_integral_comp_add_right | analysis.fourier | src/analysis/fourier/fourier_transform.lean | [
"analysis.complex.circle",
"measure_theory.group.integration",
"measure_theory.measure.haar.of_basis"
] | [
"function.comp_apply",
"has_measurable_add",
"linear_map.map_sub",
"multiplicative",
"smul_eq_mul"
] | The Fourier integral converts right-translation into scalar multiplication by a phase factor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_integral_convergent_iff (he : continuous e) (hL : continuous (λ p : V × W, L p.1 p.2))
{f : V → E} (w : W) :
integrable f μ ↔ integrable (λ (v : V), (e [-L v w]) • f v) μ | begin
-- first prove one-way implication
have aux : ∀ {g : V → E} (hg : integrable g μ) (x : W),
integrable (λ (v : V), (e [-L v x]) • g v) μ,
{ intros g hg x,
have c : continuous (λ v, e[-L v x]),
{ refine (continuous_induced_rng.mp he).comp (continuous_of_add.comp (continuous.neg _)),
exact hL... | lemma | vector_fourier.fourier_integral_convergent_iff | analysis.fourier | src/analysis/fourier/fourier_transform.lean | [
"analysis.complex.circle",
"measure_theory.group.integration",
"measure_theory.measure.haar.of_basis"
] | [
"abs_coe_circle",
"aux",
"complex.norm_eq_abs",
"continuous",
"linear_map.map_neg",
"monoid_hom.map_one",
"norm_smul",
"of_add_zero",
"one_mul",
"one_smul",
"smul_eq_mul",
"submonoid.coe_one"
] | For any `w`, the Fourier integral is convergent iff `f` is integrable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_integral_add
(he : continuous e) (hL : continuous (λ p : V × W, L p.1 p.2))
{f g : V → E} (hf : integrable f μ) (hg : integrable g μ) :
(fourier_integral e μ L f) + (fourier_integral e μ L g) = fourier_integral e μ L (f + g) | begin
ext1 w,
dsimp only [pi.add_apply, fourier_integral],
simp_rw smul_add,
rw integral_add,
{ exact (fourier_integral_convergent_iff he hL w).mp hf },
{ exact (fourier_integral_convergent_iff he hL w).mp hg },
end | lemma | vector_fourier.fourier_integral_add | analysis.fourier | src/analysis/fourier/fourier_transform.lean | [
"analysis.complex.circle",
"measure_theory.group.integration",
"measure_theory.measure.haar.of_basis"
] | [
"continuous",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier_integral_continuous [topological_space.first_countable_topology W]
(he : continuous e) (hL : continuous (λ p : V × W, L p.1 p.2))
{f : V → E} (hf : integrable f μ) :
continuous (fourier_integral e μ L f) | begin
apply continuous_of_dominated,
{ exact λ w, ((fourier_integral_convergent_iff he hL w).mp hf).1 },
{ refine λ w, ae_of_all _ (λ v, _),
{ exact λ v, ‖f v‖ },
{ rw [norm_smul, complex.norm_eq_abs, abs_coe_circle, one_mul] } },
{ exact hf.norm },
{ rw continuous_induced_rng at he,
refine ae_of_... | lemma | vector_fourier.fourier_integral_continuous | analysis.fourier | src/analysis/fourier/fourier_transform.lean | [
"analysis.complex.circle",
"measure_theory.group.integration",
"measure_theory.measure.haar.of_basis"
] | [
"abs_coe_circle",
"complex.norm_eq_abs",
"continuous",
"continuous_const",
"continuous_induced_rng",
"norm_smul",
"one_mul",
"topological_space.first_countable_topology"
] | The Fourier integral of an `L^1` function is a continuous function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_integral
(e : (multiplicative 𝕜) →* 𝕊) (μ : measure 𝕜) (f : 𝕜 → E) (w : 𝕜) : E | vector_fourier.fourier_integral e μ (linear_map.mul 𝕜 𝕜) f w | def | fourier.fourier_integral | analysis.fourier | src/analysis/fourier/fourier_transform.lean | [
"analysis.complex.circle",
"measure_theory.group.integration",
"measure_theory.measure.haar.of_basis"
] | [
"linear_map.mul",
"multiplicative",
"vector_fourier.fourier_integral"
] | The Fourier transform integral for `f : 𝕜 → E`, with respect to the measure `μ` and additive
character `e`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_integral_def (e : (multiplicative 𝕜) →* 𝕊) (μ : measure 𝕜) (f : 𝕜 → E) (w : 𝕜) :
fourier_integral e μ f w = ∫ (v : 𝕜), e[-(v * w)] • f v ∂μ | rfl | lemma | fourier.fourier_integral_def | analysis.fourier | src/analysis/fourier/fourier_transform.lean | [
"analysis.complex.circle",
"measure_theory.group.integration",
"measure_theory.measure.haar.of_basis"
] | [
"multiplicative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier_integral_smul_const
(e : (multiplicative 𝕜) →* 𝕊) (μ : measure 𝕜) (f : 𝕜 → E) (r : ℂ) :
fourier_integral e μ (r • f) = r • (fourier_integral e μ f) | vector_fourier.fourier_integral_smul_const _ _ _ _ _ | lemma | fourier.fourier_integral_smul_const | analysis.fourier | src/analysis/fourier/fourier_transform.lean | [
"analysis.complex.circle",
"measure_theory.group.integration",
"measure_theory.measure.haar.of_basis"
] | [
"multiplicative",
"vector_fourier.fourier_integral_smul_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_fourier_integral_le_integral_norm
(e : (multiplicative 𝕜) →* 𝕊) (μ : measure 𝕜) (f : 𝕜 → E) (w : 𝕜) :
‖fourier_integral e μ f w‖ ≤ ∫ x : 𝕜, ‖f x‖ ∂μ | vector_fourier.norm_fourier_integral_le_integral_norm _ _ _ _ _ | lemma | fourier.norm_fourier_integral_le_integral_norm | analysis.fourier | src/analysis/fourier/fourier_transform.lean | [
"analysis.complex.circle",
"measure_theory.group.integration",
"measure_theory.measure.haar.of_basis"
] | [
"multiplicative",
"vector_fourier.norm_fourier_integral_le_integral_norm"
] | The uniform norm of the Fourier transform of `f` is bounded by the `L¹` norm of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_integral_comp_add_right [has_measurable_add 𝕜]
(e : (multiplicative 𝕜) →* 𝕊) (μ : measure 𝕜) [μ.is_add_right_invariant] (f : 𝕜 → E) (v₀ : 𝕜) :
fourier_integral e μ (f ∘ (λ v, v + v₀)) = λ w, e [v₀ * w] • fourier_integral e μ f w | vector_fourier.fourier_integral_comp_add_right _ _ _ _ _ | lemma | fourier.fourier_integral_comp_add_right | analysis.fourier | src/analysis/fourier/fourier_transform.lean | [
"analysis.complex.circle",
"measure_theory.group.integration",
"measure_theory.measure.haar.of_basis"
] | [
"has_measurable_add",
"multiplicative",
"vector_fourier.fourier_integral_comp_add_right"
] | The Fourier transform converts right-translation into scalar multiplication by a phase factor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_char : (multiplicative ℝ) →* 𝕊 | { to_fun := λ z, exp_map_circle (2 * π * z.to_add),
map_one' := by rw [to_add_one, mul_zero, exp_map_circle_zero],
map_mul' := λ x y, by rw [to_add_mul, mul_add, exp_map_circle_add] } | def | real.fourier_char | analysis.fourier | src/analysis/fourier/fourier_transform.lean | [
"analysis.complex.circle",
"measure_theory.group.integration",
"measure_theory.measure.haar.of_basis"
] | [
"exp_map_circle",
"exp_map_circle_add",
"exp_map_circle_zero",
"mul_zero",
"multiplicative",
"to_add_mul",
"to_add_one"
] | The standard additive character of `ℝ`, given by `λ x, exp (2 * π * x * I)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_char_apply (x : ℝ) :
real.fourier_char [x] = complex.exp (↑(2 * π * x) * complex.I) | by refl | lemma | real.fourier_char_apply | analysis.fourier | src/analysis/fourier/fourier_transform.lean | [
"analysis.complex.circle",
"measure_theory.group.integration",
"measure_theory.measure.haar.of_basis"
] | [
"complex.I",
"complex.exp",
"real.fourier_char"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_fourier_char : continuous real.fourier_char | (map_continuous exp_map_circle).comp (continuous_const.mul continuous_to_add) | lemma | real.continuous_fourier_char | analysis.fourier | src/analysis/fourier/fourier_transform.lean | [
"analysis.complex.circle",
"measure_theory.group.integration",
"measure_theory.measure.haar.of_basis"
] | [
"continuous",
"continuous_to_add",
"exp_map_circle",
"real.fourier_char"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vector_fourier_integral_eq_integral_exp_smul
{V : Type*} [add_comm_group V] [module ℝ V] [measurable_space V]
{W : Type*} [add_comm_group W] [module ℝ W]
(L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ) (μ : measure V) (f : V → E) (w : W) :
vector_fourier.fourier_integral fourier_char μ L f w
= ∫ (v : V), complex.exp (↑(-2 * π * L v ... | by simp_rw [vector_fourier.fourier_integral, real.fourier_char_apply, mul_neg, neg_mul] | lemma | real.vector_fourier_integral_eq_integral_exp_smul | analysis.fourier | src/analysis/fourier/fourier_transform.lean | [
"analysis.complex.circle",
"measure_theory.group.integration",
"measure_theory.measure.haar.of_basis"
] | [
"add_comm_group",
"complex.I",
"complex.exp",
"measurable_space",
"module",
"mul_neg",
"neg_mul",
"real.fourier_char_apply",
"vector_fourier.fourier_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier_integral (f : ℝ → E) (w : ℝ) | fourier.fourier_integral fourier_char volume f w | def | real.fourier_integral | analysis.fourier | src/analysis/fourier/fourier_transform.lean | [
"analysis.complex.circle",
"measure_theory.group.integration",
"measure_theory.measure.haar.of_basis"
] | [
"fourier.fourier_integral"
] | The Fourier integral for `f : ℝ → E`, with respect to the standard additive character and
measure on `ℝ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_integral_def (f : ℝ → E) (w : ℝ) :
fourier_integral f w = ∫ (v : ℝ), fourier_char [-(v * w)] • f v | rfl | lemma | real.fourier_integral_def | analysis.fourier | src/analysis/fourier/fourier_transform.lean | [
"analysis.complex.circle",
"measure_theory.group.integration",
"measure_theory.measure.haar.of_basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fourier_integral_eq_integral_exp_smul
{E : Type*} [normed_add_comm_group E] [complete_space E] [normed_space ℂ E]
(f : ℝ → E) (w : ℝ) :
𝓕 f w = ∫ (v : ℝ), complex.exp (↑(-2 * π * v * w) * complex.I) • f v | by simp_rw [fourier_integral_def, real.fourier_char_apply, mul_neg, neg_mul, mul_assoc] | lemma | real.fourier_integral_eq_integral_exp_smul | analysis.fourier | src/analysis/fourier/fourier_transform.lean | [
"analysis.complex.circle",
"measure_theory.group.integration",
"measure_theory.measure.haar.of_basis"
] | [
"complete_space",
"complex.I",
"complex.exp",
"mul_assoc",
"mul_neg",
"neg_mul",
"normed_add_comm_group",
"normed_space",
"real.fourier_char_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.fourier_coeff_tsum_comp_add {f : C(ℝ, ℂ)}
(hf : ∀ (K : compacts ℝ), summable (λ n : ℤ, ‖(f.comp (continuous_map.add_right n)).restrict K‖))
(m : ℤ) :
fourier_coeff (periodic.lift $ f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m | begin
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(ℝ, ℂ) := (fourier (-m)).comp ⟨(coe : ℝ → unit_add_circle), continuous_quotient_mk⟩,
have neK : ∀ (K... | lemma | real.fourier_coeff_tsum_comp_add | analysis.fourier | src/analysis/fourier/poisson_summation.lean | [
"analysis.fourier.add_circle",
"analysis.fourier.fourier_transform",
"analysis.p_series",
"analysis.schwartz_space",
"measure_theory.measure.lebesgue.integral"
] | [
"abs_coe_circle",
"add_circle.coe_add_period",
"continuous_map.comp_apply",
"div_one",
"fourier",
"fourier_coe_apply",
"fourier_coeff",
"fourier_coeff_eq_interval_integral",
"interval_integral.tsum_interval_integral_eq_of_summable_norm",
"mul_one",
"norm_mul",
"one_mul",
"one_smul",
"pi.mu... | The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
`∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.tsum_eq_tsum_fourier_integral {f : C(ℝ, ℂ)}
(h_norm : ∀ (K : compacts ℝ),
summable (λ n : ℤ, ‖(f.comp $ continuous_map.add_right n).restrict K‖))
(h_sum : summable (λ n : ℤ, 𝓕 f n)) :
∑' (n : ℤ), f n = ∑' (n : ℤ), 𝓕 f n | begin
let F : C(unit_add_circle, ℂ) := ⟨(f.periodic_tsum_comp_add_zsmul 1).lift,
continuous_coinduced_dom.mpr (map_continuous _)⟩,
have : summable (fourier_coeff F),
{ convert h_sum,
exact funext (λ n, real.fourier_coeff_tsum_comp_add h_norm n) },
convert (has_pointwise_sum_fourier_series_of_summable th... | theorem | real.tsum_eq_tsum_fourier_integral | analysis.fourier | src/analysis/fourier/poisson_summation.lean | [
"analysis.fourier.add_circle",
"analysis.fourier.fourier_transform",
"analysis.p_series",
"analysis.schwartz_space",
"measure_theory.measure.lebesgue.integral"
] | [
"fourier_coeff",
"fourier_eval_zero",
"has_pointwise_sum_fourier_series_of_summable",
"has_sum",
"lift",
"mul_one",
"real.fourier_coeff_tsum_comp_add",
"smul_eq_mul",
"summable",
"unit_add_circle",
"zsmul_one"
] | **Poisson's summation formula**, most general form. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O_norm_Icc_restrict_at_top {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : is_O at_top f (λ x : ℝ, |x| ^ (-b))) (R S : ℝ) :
is_O at_top (λ x : ℝ, ‖f.restrict (Icc (x + R) (x + S))‖) (λ x : ℝ, |x| ^ (-b)) | begin
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved using `async` rather than being formulated as a separate lemma.
have claim : ∀ (x : ℝ), max 0 (-2 * R) < x →
∀ (y... | lemma | is_O_norm_Icc_restrict_at_top | analysis.fourier | src/analysis/fourier/poisson_summation.lean | [
"analysis.fourier.add_circle",
"analysis.fourier.fourier_transform",
"analysis.p_series",
"analysis.schwartz_space",
"measure_theory.measure.lebesgue.integral"
] | [
"abs_of_nonneg",
"abs_of_pos",
"continuous_map.norm_le",
"ge_iff_le",
"inv_le_inv",
"inv_le_inv_of_le",
"max_le_iff",
"max_lt_iff",
"mul_assoc",
"mul_le_mul_left",
"neg_mul",
"norm_norm",
"one_half_pos",
"two_mul"
] | If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O_norm_Icc_restrict_at_bot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : is_O at_bot f (λ x : ℝ, |x| ^ (-b))) (R S : ℝ) :
is_O at_bot (λ x : ℝ, ‖f.restrict (Icc (x + R) (x + S))‖) (λ x : ℝ, |x| ^ (-b)) | begin
have h1 : is_O at_top (f.comp (continuous_map.mk _ continuous_neg)) (λ x : ℝ, |x| ^ (-b)),
{ convert hf.comp_tendsto tendsto_neg_at_top_at_bot,
ext1 x, simp only [function.comp_app, abs_neg] },
have h2 := (is_O_norm_Icc_restrict_at_top hb h1 (-S) (-R)).comp_tendsto tendsto_neg_at_bot_at_top,
have : ((... | lemma | is_O_norm_Icc_restrict_at_bot | analysis.fourier | src/analysis/fourier/poisson_summation.lean | [
"analysis.fourier.add_circle",
"analysis.fourier.fourier_transform",
"analysis.p_series",
"analysis.schwartz_space",
"measure_theory.measure.lebesgue.integral"
] | [
"abs_neg",
"continuous_map.coe_mk",
"continuous_map.comp_apply",
"continuous_map.norm_coe_le_norm",
"continuous_map.norm_le",
"continuous_map.restrict_apply_mk",
"is_O_norm_Icc_restrict_at_top",
"norm_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : is_O (cocompact ℝ) f (λ x : ℝ, |x| ^ (-b))) (K : compacts ℝ) :
is_O (cocompact ℝ) (λ x, ‖(f.comp (continuous_map.add_right x)).restrict K‖) (λ x, |x| ^ (-b)) | begin
obtain ⟨r, hr⟩ := K.is_compact.bounded.subset_ball 0,
rw [closed_ball_eq_Icc, zero_add, zero_sub] at hr,
have : ∀ (x : ℝ), ‖(f.comp (continuous_map.add_right x)).restrict K‖ ≤
‖f.restrict (Icc (x - r) (x + r))‖,
{ intro x,
rw continuous_map.norm_le _ (norm_nonneg _),
rintro ⟨y, hy⟩,
refine... | lemma | is_O_norm_restrict_cocompact | analysis.fourier | src/analysis/fourier/poisson_summation.lean | [
"analysis.fourier.add_circle",
"analysis.fourier.fourier_transform",
"analysis.p_series",
"analysis.schwartz_space",
"measure_theory.measure.lebesgue.integral"
] | [
"continuous_map.comp_apply",
"continuous_map.norm_coe_le_norm",
"continuous_map.norm_le",
"continuous_map.restrict_apply",
"is_O_norm_Icc_restrict_at_bot",
"is_O_norm_Icc_restrict_at_top",
"norm_norm",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.tsum_eq_tsum_fourier_integral_of_rpow_decay_of_summable {f : ℝ → ℂ} (hc : continuous f)
{b : ℝ} (hb : 1 < b) (hf : is_O (cocompact ℝ) f (λ x : ℝ, |x| ^ (-b)))
(hFf : summable (λ n : ℤ, 𝓕 f n)) :
∑' (n : ℤ), f n = ∑' (n : ℤ), 𝓕 f n | real.tsum_eq_tsum_fourier_integral
(λ K, summable_of_is_O (real.summable_abs_int_rpow hb)
((is_O_norm_restrict_cocompact (continuous_map.mk _ hc)
(zero_lt_one.trans hb) hf K).comp_tendsto int.tendsto_coe_cofinite)) hFf | lemma | real.tsum_eq_tsum_fourier_integral_of_rpow_decay_of_summable | analysis.fourier | src/analysis/fourier/poisson_summation.lean | [
"analysis.fourier.add_circle",
"analysis.fourier.fourier_transform",
"analysis.p_series",
"analysis.schwartz_space",
"measure_theory.measure.lebesgue.integral"
] | [
"continuous",
"int.tendsto_coe_cofinite",
"is_O_norm_restrict_cocompact",
"real.summable_abs_int_rpow",
"real.tsum_eq_tsum_fourier_integral",
"summable",
"summable_of_is_O"
] | **Poisson's summation formula**, assuming that `f` decays as
`|x| ^ (-b)` for some `1 < b` and its Fourier transform is summable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.tsum_eq_tsum_fourier_integral_of_rpow_decay {f : ℝ → ℂ} (hc : continuous f)
{b : ℝ} (hb : 1 < b) (hf : is_O (cocompact ℝ) f (λ x : ℝ, |x| ^ (-b)))
(hFf : is_O (cocompact ℝ) (𝓕 f) (λ x : ℝ, |x| ^ (-b))) :
∑' (n : ℤ), f n = ∑' (n : ℤ), 𝓕 f n | real.tsum_eq_tsum_fourier_integral_of_rpow_decay_of_summable hc hb hf
(summable_of_is_O (real.summable_abs_int_rpow hb) (hFf.comp_tendsto int.tendsto_coe_cofinite)) | lemma | real.tsum_eq_tsum_fourier_integral_of_rpow_decay | analysis.fourier | src/analysis/fourier/poisson_summation.lean | [
"analysis.fourier.add_circle",
"analysis.fourier.fourier_transform",
"analysis.p_series",
"analysis.schwartz_space",
"measure_theory.measure.lebesgue.integral"
] | [
"continuous",
"int.tendsto_coe_cofinite",
"real.summable_abs_int_rpow",
"real.tsum_eq_tsum_fourier_integral_of_rpow_decay_of_summable",
"summable_of_is_O"
] | **Poisson's summation formula**, assuming that both `f` and its Fourier transform decay as
`|x| ^ (-b)` for some `1 < b`. (This is the one-dimensional case of Corollary VII.2.6 of Stein and
Weiss, *Introduction to Fourier analysis on Euclidean spaces*.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
schwartz_map.tsum_eq_tsum_fourier_integral
(f g : schwartz_map ℝ ℂ) (hfg : 𝓕 f = g) :
∑' (n : ℤ), f n = ∑' (n : ℤ), g n | begin
-- We know that Schwartz functions are `O(‖x ^ (-b)‖)` for *every* `b`; for this argument we take
-- `b = 2` and work with that.
simp_rw ←hfg,
exact real.tsum_eq_tsum_fourier_integral_of_rpow_decay f.continuous one_lt_two
(f.is_O_cocompact_rpow (-2)) (by simpa only [hfg] using g.is_O_cocompact_rpow (-... | lemma | schwartz_map.tsum_eq_tsum_fourier_integral | analysis.fourier | src/analysis/fourier/poisson_summation.lean | [
"analysis.fourier.add_circle",
"analysis.fourier.fourier_transform",
"analysis.p_series",
"analysis.schwartz_space",
"measure_theory.measure.lebesgue.integral"
] | [
"one_lt_two",
"real.tsum_eq_tsum_fourier_integral_of_rpow_decay",
"schwartz_map"
] | **Poisson's summation formula** for Schwartz functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_integrand_integrable (w : V) :
integrable f ↔ integrable (λ v : V, e [-⟪v, w⟫] • f v) | begin
have hL : continuous (λ p : V × V, bilin_form_of_real_inner.to_lin p.1 p.2) := continuous_inner,
rw vector_fourier.fourier_integral_convergent_iff real.continuous_fourier_char hL w,
{ simp only [bilin_form.to_lin_apply, bilin_form_of_real_inner_apply] },
{ apply_instance },
end | lemma | fourier_integrand_integrable | analysis.fourier | src/analysis/fourier/riemann_lebesgue_lemma.lean | [
"analysis.fourier.fourier_transform",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.function.continuous_map_dense",
"measure_theory.group.integration",
"measure_theory.integral.set_integral",
"measure_theory.measure.haar.normed_space",
"topology.me... | [
"bilin_form.to_lin_apply",
"continuous",
"continuous_inner",
"real.continuous_fourier_char",
"vector_fourier.fourier_integral_convergent_iff"
] | The integrand in the Riemann-Lebesgue lemma for `f` is integrable iff `f` is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_integral_half_period_translate {w : V} (hw : w ≠ 0) :
∫ (v : V), e [-⟪v, w⟫] • f (v + i w) = -∫ (v : V), e [-⟪v, w⟫] • f v | begin
have hiw : ⟪i w, w⟫ = 1 / 2,
{ rw [inner_smul_left, inner_self_eq_norm_sq_to_K, is_R_or_C.coe_real_eq_id, id.def,
is_R_or_C.conj_to_real, ←div_div, div_mul_cancel],
rwa [ne.def, sq_eq_zero_iff, norm_eq_zero] },
have : (λ v : V, e [-⟪v, w⟫] • f (v + i w)) = (λ v : V, (λ x : V, -e[-⟪x, w⟫] • f x) (v... | lemma | fourier_integral_half_period_translate | analysis.fourier | src/analysis/fourier/riemann_lebesgue_lemma.lean | [
"analysis.fourier.fourier_transform",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.function.continuous_map_dense",
"measure_theory.group.integration",
"measure_theory.integral.set_integral",
"measure_theory.measure.haar.normed_space",
"topology.me... | [
"div_mul_cancel",
"exp_add",
"exp_neg",
"inner_add_left",
"inner_self_eq_norm_sq_to_K",
"inner_smul_left",
"inv_neg",
"inv_one",
"is_R_or_C.coe_real_eq_id",
"is_R_or_C.conj_to_real",
"mul_neg_one",
"neg_mul",
"neg_smul",
"norm_eq_zero",
"real.fourier_char_apply",
"ring",
"sq_eq_zero_... | Shifting `f` by `(1 / (2 * ‖w‖ ^ 2)) • w` negates the integral in the Riemann-Lebesgue lemma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fourier_integral_eq_half_sub_half_period_translate {w : V} (hw : w ≠ 0) (hf : integrable f) :
∫ v : V, e[-⟪v, w⟫] • f v = (1 / (2 : ℂ)) • ∫ v : V, e[-⟪v, w⟫] • (f v - f (v + i w)) | begin
simp_rw [smul_sub],
rw [integral_sub, fourier_integral_half_period_translate hw, sub_eq_add_neg, neg_neg,
←two_smul ℂ _, ←@smul_assoc _ _ _ _ _ _ (is_scalar_tower.left ℂ), smul_eq_mul],
norm_num,
exacts [(fourier_integrand_integrable w).mp hf,
(fourier_integrand_integrable w).mp (hf.comp_add_right... | lemma | fourier_integral_eq_half_sub_half_period_translate | analysis.fourier | src/analysis/fourier/riemann_lebesgue_lemma.lean | [
"analysis.fourier.fourier_transform",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.function.continuous_map_dense",
"measure_theory.group.integration",
"measure_theory.integral.set_integral",
"measure_theory.measure.haar.normed_space",
"topology.me... | [
"fourier_integral_half_period_translate",
"fourier_integrand_integrable",
"is_scalar_tower.left",
"smul_assoc",
"smul_eq_mul",
"smul_sub"
] | Rewrite the Fourier integral in a form that allows us to use uniform continuity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support
(hf1 : continuous f) (hf2 : has_compact_support f) :
tendsto (λ w : V, ∫ v : V, e[-⟪v, w⟫] • f v) (cocompact V) (𝓝 0) | begin
refine normed_add_comm_group.tendsto_nhds_zero.mpr (λ ε hε, _),
suffices : ∃ (T : ℝ), ∀ (w : V), T ≤ ‖w‖ → ‖∫ (v : V), e[-⟪v, w⟫] • f v‖ < ε,
{ simp_rw [←comap_dist_left_at_top_eq_cocompact (0 : V), eventually_comap, eventually_at_top,
dist_eq_norm', sub_zero],
exact let ⟨T, hT⟩ := this in ⟨T, (λ ... | lemma | tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support | analysis.fourier | src/analysis/fourier/riemann_lebesgue_lemma.lean | [
"analysis.fourier.fourier_transform",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.euclidean_dist",
"measure_theory.function.continuous_map_dense",
"measure_theory.group.integration",
"measure_theory.integral.set_integral",
"measure_theory.measure.haar.normed_space",
"topology.me... | [
"abs_coe_circle",
"complex.abs_of_nonneg",
"continuous",
"continuous.ae_strongly_measurable",
"continuous.comp",
"continuous_const",
"div_div",
"div_le_one",
"div_lt_iff",
"div_mul_cancel",
"div_mul_eq_mul_div",
"div_pos",
"ennreal",
"ennreal.coe_add",
"ennreal.coe_lt_top",
"ennreal.co... | Riemann-Lebesgue Lemma for continuous and compactly-supported functions: the integral
`∫ v, exp (-2 * π * ⟪w, v⟫ * I) • f v` tends to 0 wrt `cocompact V`. Note that this is primarily
of interest as a preparatory step for the more general result
`tendsto_integral_exp_inner_smul_cocompact` in which `f` can be arbitrary. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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