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c_u7hiaqu7wczn | Carathéodory function | Summary | Carathéodory_function | In mathematical analysis, a Carathéodory function (or Carathéodory integrand) is a multivariable function that allows us to solve the following problem effectively: A composition of two Lebesgue-measurable functions does not have to be Lebesgue-measurable as well. Nevertheless, a composition of a measurable function wi... |
c_oadm9fbx2agx | Hermitian function | Summary | Hermitian_symmetry | In mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign: f ∗ ( x ) = f ( − x ) {\displaystyle f^{*}(x)=f(-x)} (where the ∗ {\displaystyle ^{*}} indicates the complex conjugate) for all x {\displ... |
c_xredb32kysli | Young measure | Summary | Young_measure | In mathematical analysis, a Young measure is a parameterized measure that is associated with certain subsequences of a given bounded sequence of measurable functions. They are a quantification of the oscillation effect of the sequence in the limit. Young measures have applications in the calculus of variations, especia... |
c_rfp9djq2gl6t | Bounded domain | Summary | Closed_region | In mathematical analysis, a domain or region is a non-empty connected open set in a topological space, in particular any non-empty connected open subset of the real coordinate space Rn or the complex coordinate space Cn. A connected open subset of coordinate space is frequently used for the domain of a function, but in... |
c_zcmu3kyq6mo2 | Equicontinuous linear maps | Summary | Equicontinuity | In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions. Equicontinuity appears in the fo... |
c_q1gxm1yb22n4 | Equicontinuous linear maps | Summary | Equicontinuity | As a corollary, a sequence in C(X) is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous a-priori). In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions fn on either metric space or locally compact spa... |
c_e23xy11ykvkr | Function of bounded variation | Summary | Bv_space | In mathematical analysis, a function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means tha... |
c_hcetedjy7667 | Function of bounded variation | Summary | Bv_space | Another characterization states that the functions of bounded variation on a compact interval are exactly those f which can be written as a difference g − h, where both g and h are bounded monotone. In particular, a BV function may have discontinuities, but at most countably many. |
c_partjb2f3dxj | Function of bounded variation | Summary | Bv_space | In the case of several variables, a function f defined on an open subset Ω of R n {\displaystyle \mathbb {R} ^{n}} is said to have bounded variation if its distributional derivative is a vector-valued finite Radon measure. One of the most important aspects of functions of bounded variation is that they form an algebra ... |
c_wavpa0728pz8 | Metric differential | Summary | Metric_differential | In mathematical analysis, a metric differential is a generalization of a derivative for a Lipschitz continuous function defined on a Euclidean space and taking values in an arbitrary metric space. With this definition of a derivative, one can generalize Rademacher's theorem to metric space-valued Lipschitz functions. |
c_8i8ndhegcru4 | Complete metric space | Summary | Completion_(topology) | In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, becau... |
c_erkw62454c48 | Modulus of continuity | Summary | Modulus_of_continuity | In mathematical analysis, a modulus of continuity is a function ω: → used to measure quantitatively the uniform continuity of functions. So, a function f: I → R admits ω as a modulus of continuity if and only if | f ( x ) − f ( y ) | ≤ ω ( | x − y | ) , {\displaystyle |f(x)-f(y)|\leq \omega (|x-y|),} for all x and y ... |
c_2t80ei1tlszw | Modulus of continuity | Summary | Modulus_of_continuity | In general, the role of ω is to fix some explicit functional dependence of ε on δ in the (ε, δ) definition of uniform continuity. The same notions generalize naturally to functions between metric spaces. Moreover, a suitable local version of these notions allows to describe quantitatively the continuity at a point in t... |
c_eoemg8bybg99 | Modulus of continuity | Summary | Modulus_of_continuity | A special role is played by concave moduli of continuity, especially in connection with extension properties, and with approximation of uniformly continuous functions. For a function between metric spaces, it is equivalent to admit a modulus of continuity that is either concave, or subadditive, or uniformly continuous,... |
c_23dpnyu2x98v | Measure zero | Summary | Null_set | In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set should not be confused with the empty set as defined in set theory. |
c_hrzud5fhbjje | Measure zero | Summary | Null_set | Although the empty set has Lebesgue measure zero, there are also non-empty sets which are null. For example, any non-empty countable set of real numbers has Lebesgue measure zero and therefore is null. More generally, on a given measure space M = ( X , Σ , μ ) {\displaystyle M=(X,\Sigma ,\mu )} a null set is a set S ∈ ... |
c_xaprntlu9in1 | Positive invariant set | Summary | Positive_invariant_set | In mathematical analysis, a positively (or positive) invariant set is a set with the following properties: Suppose x ˙ = f ( x ) {\displaystyle {\dot {x}}=f(x)} is a dynamical system, x ( t , x 0 ) {\displaystyle x(t,x_{0})} is a trajectory, and x 0 {\displaystyle x_{0}} is the initial point. Let O := { x ∈ R n ∣ φ ( x... |
c_ooyjnr72ousd | Space-filling curves | Summary | Space-filling_curve | In mathematical analysis, a space-filling curve is a curve whose range reaches every point in a higher dimensional region, typically the unit square (or more generally an n-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are ... |
c_w1gi2tk9j741 | Strong measure zero set | Summary | Strong_measure_zero_set | In mathematical analysis, a strong measure zero set is a subset A of the real line with the following property: for every sequence (εn) of positive reals there exists a sequence (In) of intervals such that |In| < εn for all n and A is contained in the union of the In. (Here |In| denotes the length of the interval In.) ... |
c_quealn13bm48 | Strong measure zero set | Summary | Strong_measure_zero_set | The Cantor set is an example of an uncountable set of Lebesgue measure 0 which is not of strong measure zero.Borel's conjecture states that every strong measure zero set is countable. It is now known that this statement is independent of ZFC (the Zermelo–Fraenkel axioms of set theory, which is the standard axiom system... |
c_rcwego6totmb | Strong measure zero set | Summary | Strong_measure_zero_set | Sierpiński proved in 1928 that the continuum hypothesis (which is now also known to be independent of ZFC) implies the existence of uncountable strong measure zero sets. In 1976 Laver used a method of forcing to construct a model of ZFC in which Borel's conjecture holds. These two results together establish the indepen... |
c_36k604pzsyof | Strong measure zero set | Summary | Strong_measure_zero_set | The following characterization of strong measure zero sets was proved in 1973: A set A ⊆ R has strong measure zero if and only if A + M ≠ R for every meagre set M ⊆ R.This result establishes a connection to the notion of strongly meagre set, defined as follows: A set M ⊆ R is strongly meagre if and only if A + M ≠ R fo... |
c_9sfkypp2vtrv | Thin set (analysis) | Summary | Thin_set_(analysis) | In mathematical analysis, a thin set is a subset of n-dimensional complex space Cn with the property that each point has a neighbourhood on which some non-zero holomorphic function vanishes. Since the set on which a holomorphic function vanishes is closed and has empty interior (by the Identity theorem), a thin set is ... |
c_u4ia7o5wmg8v | Osgood curve | Summary | Osgood_curve | In mathematical analysis, an Osgood curve is a non-self-intersecting curve that has positive area. Despite its area, it is not possible for such a curve to cover any two-dimensional region, distinguishing them from space-filling curves. Osgood curves are named after William Fogg Osgood. |
c_e7y52l3t9oj3 | Improper integrals | Summary | Improper_integrals | In mathematical analysis, an improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context of Riemann integrals (or, equivalently, Darboux integrals), this typically involves unboundedness, either of the set over which the in... |
c_5biduppnpbal | Improper integrals | Summary | Improper_integrals | If a regular definite integral (which may retronymically be called a proper integral) is worked out as if it is improper, the same answer will result. In the simplest case of a real-valued function of a single variable integrated in the sense of Riemann (or Darboux) over a single interval, improper integrals may be in ... |
c_uodmgwqx9ucm | Improper integrals | Summary | Improper_integrals | Such an integral is sometimes described as being of the "first" type or kind if the integrand otherwise satisfies the assumptions of integration. Integrals in the fourth form that are improper because f ( x ) {\displaystyle f(x)} has a vertical asymptote somewhere on the interval {\displaystyle } may be described as b... |
c_8d2p4nr0s263 | Improper integrals | Summary | Improper_integrals | For example, in case 1, if f ( x ) {\displaystyle f(x)} is continuous on the entire interval [ a , ∞ ) {\displaystyle [a,\infty )} , then ∫ a ∞ f ( x ) d x = lim b → ∞ ∫ a b f ( x ) d x . {\displaystyle \int _{a}^{\infty }f(x)\,dx=\lim _{b\to \infty }\int _{a}^{b}f(x)\,dx.} The limit on the right is taken to be the def... |
c_dwopjlyfnd50 | Improper integrals | Summary | Improper_integrals | If f ( x ) {\displaystyle f(x)} is only continuous on ( a , ∞ ) {\displaystyle (a,\infty )} and not at a {\displaystyle a} itself, then typically this is rewritten as ∫ a ∞ f ( x ) d x = lim t → a + ∫ t c f ( x ) d x + lim b → ∞ ∫ c b f ( x ) d x , {\displaystyle \int _{a}^{\infty }f(x)\,dx=\lim _{t\to a^{+}}\int _{t}^... |
c_3a4wb1p7xudi | Improper integrals | Summary | Improper_integrals | Alternatively, an iterated limit could be used or a single limit based on the Cauchy principal value. If f ( x ) {\displaystyle f(x)} is continuous on [ a , d ) {\displaystyle [a,d)} and ( d , ∞ ) {\displaystyle (d,\infty )} , with a discontinuity of any kind at d {\displaystyle d} , then ∫ a ∞ f ( x ) d x = lim t → d ... |
c_4tj6joi6y79v | Improper integrals | Summary | Improper_integrals | The function f ( x ) {\displaystyle f(x)} can have more discontinuities, in which case even more limits would be required (or a more complicated principal value expression). Cases 2–4 are handled similarly. See the examples below. Improper integrals can also be evaluated in the context of complex numbers, in higher dim... |
c_1qhv2i8119o5 | Function of a real variable | Summary | Function_of_a_real_variable | In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers R {\displaystyle \mathbb {R} } , or a subset of R {\displaystyle \mathbb {R} } that contains an interval of positive length. Most r... |
c_nyzvymtzjlvn | Function of a real variable | Summary | Function_of_a_real_variable | However, it is often assumed to have a structure of R {\displaystyle \mathbb {R} } -vector space over the reals. That is, the codomain may be a Euclidean space, a coordinate vector, the set of matrices of real numbers of a given size, or an R {\displaystyle \mathbb {R} } -algebra, such as the complex numbers or the qua... |
c_3zqp8y9qwl7s | Function of a real variable | Summary | Function_of_a_real_variable | If the codomain has a structure of R {\displaystyle \mathbb {R} } -algebra, the same is true for the functions. The image of a function of a real variable is a curve in the codomain. In this context, a function that defines curve is called a parametric equation of the curve. When the codomain of a function of a real va... |
c_ff60itbs94n5 | Continuous functions on a compact Hausdorff space | Summary | Continuous_functions_on_a_compact_Hausdorff_space | In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space X {\displaystyle X} with values in the real or complex numbers. This space, denoted by C ( X ) , {\displaystyle {\mathcal {C}}(X),} is a vector space with respect ... |
c_39slakibjw8g | Continuous functions on a compact Hausdorff space | Summary | Continuous_functions_on_a_compact_Hausdorff_space | {\displaystyle X.} The space C ( X ) {\displaystyle {\mathcal {C}}(X)} is a Banach algebra with respect to this norm. (Rudin 1973, §11.3) |
c_vac81ky92r4q | Function (mathematics) | Function space | Functional_notation > Function space | In mathematical analysis, and more specifically in functional analysis, a function space is a set of scalar-valued or vector-valued functions, which share a specific property and form a topological vector space. For example, the real smooth functions with a compact support (that is, they are zero outside some compact s... |
c_m82za17bp2ta | Asymptotic theory | Summary | Asymptotically_equal | In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function f (n) as n becomes very large. If f(n) = n2 + 3n, then as n becomes very large, the term 3n becomes insignificant comp... |
c_292in3v00cno | Asymptotic theory | Summary | Asymptotically_equal | This is often written symbolically as f (n) ~ n2, which is read as "f(n) is asymptotic to n2". An example of an important asymptotic result is the prime number theorem. Let π(x) denote the prime-counting function (which is not directly related to the constant pi), i.e. π(x) is the number of prime numbers that are less ... |
c_28uyklxxco7m | Constructive function theory | Summary | Constructive_function_theory | In mathematical analysis, constructive function theory is a field which studies the connection between the smoothness of a function and its degree of approximation. It is closely related to approximation theory. The term was coined by Sergei Bernstein. |
c_ihdm0005xs38 | Epi-convergence | Summary | Epi-convergence | In mathematical analysis, epi-convergence is a type of convergence for real-valued and extended real-valued functions. Epi-convergence is important because it is the appropriate notion of convergence with which to approximate minimization problems in the field of mathematical optimization. The symmetric notion of hypo-... |
c_vhzl46wlpgc9 | Power series ring | Interpreting formal power series as functions | Non-commuting_formal_power_series > Interpreting formal power series as functions | In mathematical analysis, every convergent power series defines a function with values in the real or complex numbers. Formal power series over certain special rings can also be interpreted as functions, but one has to be careful with the domain and codomain. Let f = ∑ a n X n ∈ R ] , {\displaystyle f=\sum a_{n}X^{n}\... |
c_bry3orcebb9d | Power series ring | Interpreting formal power series as functions | Non-commuting_formal_power_series > Interpreting formal power series as functions | {\displaystyle f(x)=\sum _{n\geq 0}a_{n}x^{n}.} This series is guaranteed to converge in S {\displaystyle S} given the above assumptions on x {\displaystyle x} . Furthermore, we have ( f + g ) ( x ) = f ( x ) + g ( x ) {\displaystyle (f+g)(x)=f(x)+g(x)} and ( f g ) ( x ) = f ( x ) g ( x ) . |
c_7fg6shi206hk | Power series ring | Interpreting formal power series as functions | Non-commuting_formal_power_series > Interpreting formal power series as functions | {\displaystyle (fg)(x)=f(x)g(x).} Unlike in the case of bona fide functions, these formulas are not definitions but have to be proved. Since the topology on R ] {\displaystyle R]} is the ( X ) {\displaystyle (X)} -adic topology and R ] {\displaystyle R]} is complete, we can in particular apply power series to other p... |
c_p518kz5nchxm | Power series ring | Interpreting formal power series as functions | Non-commuting_formal_power_series > Interpreting formal power series as functions | {\displaystyle f\in R].} With this formalism, we can give an explicit formula for the multiplicative inverse of a power series f {\displaystyle f} whose constant coefficient a = f ( 0 ) {\displaystyle a=f(0)} is invertible in R {\displaystyle R}: f − 1 = ∑ n ≥ 0 a − n − 1 ( a − f ) n . {\displaystyle f^{-1}=\sum _{n\ge... |
c_dqwo65pb0gaf | Idempotent analysis | Summary | Idempotent_analysis | In mathematical analysis, idempotent analysis is the study of idempotent semirings, such as the tropical semiring. The lack of an additive inverse in the semiring is compensated somewhat by the idempotent rule A ⊕ A = A {\displaystyle A\oplus A=A} . == References == |
c_dl11n5u502yw | Proper convex function | Summary | Proper_convex_function | In mathematical analysis, in particular the subfields of convex analysis and optimization, a proper convex function is an extended real-valued convex function with a non-empty domain, that never takes on the value − ∞ {\displaystyle -\infty } and also is not identically equal to + ∞ . {\displaystyle +\infty .} In conve... |
c_t8pwbxg0no8w | Proper convex function | Summary | Proper_convex_function | Such a point, if it exists, is called a global minimum point of the function and its value at this point is called the global minimum (value) of the function. If the function takes − ∞ {\displaystyle -\infty } as a value then − ∞ {\displaystyle -\infty } is necessarily the global minimum value and the minimization prob... |
c_5gcduzf2mcem | Proper convex function | Summary | Proper_convex_function | Extended real-valued function for which the minimization problem is not solved by any one of these three trivial cases are exactly those that are called proper. Many (although not all) results whose hypotheses require that the function be proper add this requirement specifically to exclude these trivial cases. If the p... |
c_btf95eqtr02d | Parameter | Mathematical analysis | Parameter > Mathematical functions > Mathematical analysis | In mathematical analysis, integrals dependent on a parameter are often considered. These are of the form F ( t ) = ∫ x 0 ( t ) x 1 ( t ) f ( x ; t ) d x . {\displaystyle F(t)=\int _{x_{0}(t)}^{x_{1}(t)}f(x;t)\,dx.} In this formula, t is the argument of the function F, and on the right-hand side the parameter on which t... |
c_gtozwvpyidlh | Parameter | Mathematical analysis | Parameter > Mathematical functions > Mathematical analysis | When evaluating the integral, t is held constant, and so it is considered to be a parameter. If we are interested in the value of F for different values of t, we then consider t to be a variable. The quantity x is a dummy variable or variable of integration (confusingly, also sometimes called a parameter of integration... |
c_ht9zkdaqh9d1 | Limit inferior | The case of sequences of real numbers | Limit_supremum > The case of sequences of real numbers | In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers. Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number sys... |
c_ub11upedklwe | Generalized Fourier series | Summary | Generalized_Fourier_series | In mathematical analysis, many generalizations of Fourier series have proved to be useful. They are all special cases of decompositions over an orthonormal basis of an inner product space. Here we consider that of square-integrable functions defined on an interval of the real line, which is important, among others, for... |
c_ydxhdt2lmxax | Microlocalization functor | Summary | Microlocal_analysis | In mathematical analysis, microlocal analysis comprises techniques developed from the 1950s onwards based on Fourier transforms related to the study of variable-coefficients-linear and nonlinear partial differential equations. This includes generalized functions, pseudo-differential operators, wave front sets, Fourier ... |
c_c86zit2s654k | Wavefront set | Summary | Wave_front_set | In mathematical analysis, more precisely in microlocal analysis, the wave front (set) WF(f) characterizes the singularities of a generalized function f, not only in space, but also with respect to its Fourier transform at each point. The term "wave front" was coined by Lars Hörmander around 1970. |
c_0p92msopfslb | Nested sequence of closed intervals | The construction of the real numbers | Nested_sequences_of_intervals > The construction of the real numbers | In mathematical analysis, nested intervals provide one method of axiomatically introducing the real numbers as the completion of the rational numbers, being a necessity for discussing the concepts of continuity and differentiability. Historically, Isaac Newton's and Gottfried Wilhelm Leibniz's discovery of differential... |
c_psxpsqzage99 | Nullcline | Summary | Nullcline | In mathematical analysis, nullclines, sometimes called zero-growth isoclines, are encountered in a system of ordinary differential equations x 1 ′ = f 1 ( x 1 , … , x n ) {\displaystyle x_{1}'=f_{1}(x_{1},\ldots ,x_{n})} x 2 ′ = f 2 ( x 1 , … , x n ) {\displaystyle x_{2}'=f_{2}(x_{1},\ldots ,x_{n})} ⋮ {\displaystyle \v... |
c_9oeg4aeoqeyb | P-variation | Summary | P-variation | In mathematical analysis, p-variation is a collection of seminorms on functions from an ordered set to a metric space, indexed by a real number p ≥ 1 {\displaystyle p\geq 1} . p-variation is a measure of the regularity or smoothness of a function. Specifically, if f: I → ( M , d ) {\displaystyle f:I\to (M,d)} , where (... |
c_ds0wskt2db5j | Upper semi-continuous | Summary | Upper_semi-continuous | In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f {\displaystyle f} is upper (respectively, lower) semicontinuous at a point x 0 {\displaystyle x_{0}} if, roughly speaking, the function values ... |
c_tc3w9rdaooto | Agranovich–Dynin formula | Summary | Agranovich–Dynin_formula | In mathematical analysis, the Agranovich–Dynin formula is a formula for the index of an elliptic system of differential operators, introduced by Agranovich and Dynin (1962). |
c_nfv299d68m3p | Alexandrov theorem | Summary | Alexandrov_theorem | In mathematical analysis, the Alexandrov theorem, named after Aleksandr Danilovich Aleksandrov, states that if U is an open subset of R n {\displaystyle \mathbb {R} ^{n}} and f: U → R m {\displaystyle f\colon U\to \mathbb {R} ^{m}} is a convex function, then f {\displaystyle f} has a second derivative almost everywhere... |
c_w4sqaow8w3gp | Bessel–Clifford function | Summary | Bessel–Clifford_function | In mathematical analysis, the Bessel–Clifford function, named after Friedrich Bessel and William Kingdon Clifford, is an entire function of two complex variables that can be used to provide an alternative development of the theory of Bessel functions. If π ( x ) = 1 Π ( x ) = 1 Γ ( x + 1 ) {\displaystyle \pi (x)={\frac... |
c_sd3h4gygewrr | Bohr–Mollerup theorem | Summary | Bohr–Mollerup_theorem | In mathematical analysis, the Bohr–Mollerup theorem is a theorem proved by the Danish mathematicians Harald Bohr and Johannes Mollerup. The theorem characterizes the gamma function, defined for x > 0 by Γ ( x ) = ∫ 0 ∞ t x − 1 e − t d t {\displaystyle \Gamma (x)=\int _{0}^{\infty }t^{x-1}e^{-t}\,\mathrm {d} t} as the o... |
c_w027eor13ec2 | Brezis–Gallouët inequality | Summary | Brezis–Gallouët_inequality | In mathematical analysis, the Brezis–Gallouët inequality, named after Haïm Brezis and Thierry Gallouët, is an inequality valid in 2 spatial dimensions. It shows that a function of two variables which is sufficiently smooth is (essentially) bounded, and provides an explicit bound, which depends only logarithmically on t... |
c_kq72707aex7d | Brezis–Gallouët inequality | Summary | Brezis–Gallouët_inequality | Let Ω ⊂ R 2 {\displaystyle \Omega \subset \mathbb {R} ^{2}} be the exterior or the interior of a bounded domain with regular boundary, or R 2 {\displaystyle \mathbb {R} ^{2}} itself. Then the Brezis–Gallouët inequality states that there exists a real C {\displaystyle C} only depending on Ω {\displaystyle \Omega } such ... |
c_hl4ic40u3w3m | Cauchy index | Summary | Cauchy_index | In mathematical analysis, the Cauchy index is an integer associated to a real rational function over an interval. By the Routh–Hurwitz theorem, we have the following interpretation: the Cauchy index of r(x) = p(x)/q(x)over the real line is the difference between the number of roots of f(z) located in the right half-pla... |
c_khzni0hyuoxu | Chebyshev–Markov–Stieltjes inequalities | Summary | Chebyshev–Markov–Stieltjes_inequalities | In mathematical analysis, the Chebyshev–Markov–Stieltjes inequalities are inequalities related to the problem of moments that were formulated in the 1880s by Pafnuty Chebyshev and proved independently by Andrey Markov and (somewhat later) by Thomas Jan Stieltjes. Informally, they provide sharp bounds on a measure from ... |
c_ueebyjc1g5oc | Dirichlet kernel | Summary | Dirichlet_kernel | In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as where n is any nonnegative integer. The kernel functions are periodic with period 2 π {\displaystyle 2\pi } . The importance of the Dirichlet kernel com... |
c_bdwrt468123b | Foias constant | Summary | Foias_constant | In mathematical analysis, the Foias constant is a real number named after Ciprian Foias. It is defined in the following way: for every real number x1 > 0, there is a sequence defined by the recurrence relation x n + 1 = ( 1 + 1 x n ) n {\displaystyle x_{n+1}=\left(1+{\frac {1}{x_{n}}}\right)^{n}} for n = 1, 2, 3, .... ... |
c_44ff4kz09ovs | Haar integral | Summary | Haar's_theorem | In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1... |
c_bwerdz5lw6ba | Karamata's tauberian theorem | Summary | Hardy–Littlewood_Tauberian_theorem | In mathematical analysis, the Hardy–Littlewood Tauberian theorem is a Tauberian theorem relating the asymptotics of the partial sums of a series with the asymptotics of its Abel summation. In this form, the theorem asserts that if the sequence a n ≥ 0 {\displaystyle a_{n}\geq 0} is such that there is an asymptotic equi... |
c_lwsqgqy8nsfb | Karamata's tauberian theorem | Summary | Hardy–Littlewood_Tauberian_theorem | The theorem was proved in 1914 by G. H. Hardy and J. E. Littlewood. : 226 In 1930, Jovan Karamata gave a new and much simpler proof. : 226 |
c_bngt2fbilcwl | Hardy–Littlewood inequality | Summary | Hardy–Littlewood_inequality | In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if f {\displaystyle f} and g {\displaystyle g} are nonnegative measurable real functions vanishing at infinity that are defined on n {\displaystyle n} -dimensional Euclidean space R n {\displaysty... |
c_35yd2rasda2g | Hilbert–Schmidt theorem | Summary | Hilbert–Schmidt_theorem | In mathematical analysis, the Hilbert–Schmidt theorem, also known as the eigenfunction expansion theorem, is a fundamental result concerning compact, self-adjoint operators on Hilbert spaces. In the theory of partial differential equations, it is very useful in solving elliptic boundary value problems. |
c_9v2fbcd9isfl | Kakutani fixed-point theorem | Summary | Kakutani_fixed-point_theorem | In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixe... |
c_5ev3dhu7bq90 | Kakutani fixed-point theorem | Summary | Kakutani_fixed-point_theorem | Kakutani's theorem extends this to set-valued functions. The theorem was developed by Shizuo Kakutani in 1941, and was used by John Nash in his description of Nash equilibria. It has subsequently found widespread application in game theory and economics. |
c_p06adjfs6tc8 | Lagrange–Bürmann formula | Summary | Lagrange–Bürmann_formula | In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function. |
c_uargji9dm1ae | Minkowski inequality | Summary | Minkowski_inequality | In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. Let S {\displaystyle S} be a measure space, let 1 ≤ p < ∞ {\displaystyle 1\leq p<\infty } and let f {\displaystyle f} and g {\displaystyle g} be elements of L p ( S ) . {\displaystyle L^{p}(S).} Then f + g {\disp... |
c_d8r8tgiyg3mn | Pólya–Szegő inequality | Summary | Pólya–Szegő_inequality | In mathematical analysis, the Pólya–Szegő inequality (or Szegő inequality) states that the Sobolev energy of a function in a Sobolev space does not increase under symmetric decreasing rearrangement. The inequality is named after the mathematicians George Pólya and Gábor Szegő. |
c_x2a1jbftzt7w | Rademacher–Menchov theorem | Summary | Rademacher–Menchov_theorem | In mathematical analysis, the Rademacher–Menchov theorem, introduced by Rademacher (1922) and Menchoff (1923), gives a sufficient condition for a series of orthogonal functions on an interval to converge almost everywhere. |
c_87caax1x4hr3 | Schur test | Summary | Schur_test | In mathematical analysis, the Schur test, named after German mathematician Issai Schur, is a bound on the L 2 → L 2 {\displaystyle L^{2}\to L^{2}} operator norm of an integral operator in terms of its Schwartz kernel (see Schwartz kernel theorem). Here is one version. Let X , Y {\displaystyle X,\,Y} be two measurable s... |
c_p4mwk5hgv5dt | Schur test | Summary | Schur_test | {\displaystyle Tf(x)=\int _{Y}K(x,y)f(y)\,dy.} If there exist real functions p ( x ) > 0 {\displaystyle \,p(x)>0} and q ( y ) > 0 {\displaystyle \,q(y)>0} and numbers α , β > 0 {\displaystyle \,\alpha ,\beta >0} such that ( 1 ) ∫ Y K ( x , y ) q ( y ) d y ≤ α p ( x ) {\displaystyle (1)\qquad \int _{Y}K(x,y)q(y)\,dy\leq... |
c_p3g1n4jus90h | Szegő limit theorems | Summary | Szegő_limit_theorems | In mathematical analysis, the Szegő limit theorems describe the asymptotic behaviour of the determinants of large Toeplitz matrices. They were first proved by Gábor Szegő. |
c_w6mv7xsj3e7x | Weierstrass approximation theorem | Summary | Stone-Weierstrass_theorem | In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials,... |
c_3nub7hbohpbb | Weierstrass approximation theorem | Summary | Stone-Weierstrass_theorem | His result is known as the Stone–Weierstrass theorem. The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval , an arbitrary compact Hausdorff space X is considered, and instead of the algebra of polynomial functions, a variety of other families of... |
c_9h6j1b2sawhb | Whitney covering lemma | Summary | Whitney_covering_lemma | In mathematical analysis, the Whitney covering lemma, or Whitney decomposition, asserts the existence of a certain type of partition of an open set in a Euclidean space. Originally it was employed in the proof of Hassler Whitney's extension theorem. The lemma was subsequently applied to prove generalizations of the Cal... |
c_kvhr6u4gx5qt | Whitney covering lemma | Summary | Whitney_covering_lemma | More precisely: Whitney Covering Lemma (Grafakos 2008, Appendix J) Let Ω {\displaystyle \Omega } be an open non-empty proper subset of R n {\displaystyle \mathbb {R} ^{n}} . Then there exists a family of closed cubes { Q j } j {\displaystyle \{Q_{j}\}_{j}} such that ∪ j Q j = Ω {\displaystyle \cup _{j}Q_{j}=\Omega } an... |
c_ftnpdwzl09u4 | Whitney covering lemma | Summary | Whitney_covering_lemma | {\displaystyle {\sqrt {n}}\ell (Q_{j})\leq \mathrm {dist} (Q_{j},\Omega ^{c})\leq 4{\sqrt {n}}\ell (Q_{j}).} If the boundaries of two cubes Q j {\displaystyle Q_{j}} and Q k {\displaystyle Q_{k}} touch then 1 4 ≤ ℓ ( Q j ) ℓ ( Q k ) ≤ 4. {\displaystyle {\frac {1}{4}}\leq {\frac {\ell (Q_{j})}{\ell (Q_{k})}}\leq 4.} For... |
c_nvojqkvjva7a | Young's inequality for integral operators | Summary | Young's_inequality_for_integral_operators | In mathematical analysis, the Young's inequality for integral operators, is a bound on the L p → L q {\displaystyle L^{p}\to L^{q}} operator norm of an integral operator in terms of L r {\displaystyle L^{r}} norms of the kernel itself. |
c_ay269pz2lgye | Alternating series test | Summary | Alternating_series_test | In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit. The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz cri... |
c_0uoen8kunmnh | Characteristic variety | Summary | Characteristic_variety | In mathematical analysis, the characteristic variety of a microdifferential operator P is an algebraic variety that is the zero set of the principal symbol of P in the cotangent bundle. It is invariant under a quantized contact transformation. The notion is also defined more generally in commutative algebra. A basic th... |
c_3zzwxh5ffd2p | Mean-periodic function | Summary | Mean-periodic_function | In mathematical analysis, the concept of a mean-periodic function is a generalization of the concept of a periodic function introduced in 1935 by Jean Delsarte. Further results were made by Laurent Schwartz. |
c_5lis8a4pdqwu | Final value theorem | Summary | Final_value_theorem | In mathematical analysis, the final value theorem (FVT) is one of several similar theorems used to relate frequency domain expressions to the time domain behavior as time approaches infinity. Mathematically, if f ( t ) {\displaystyle f(t)} in continuous time has (unilateral) Laplace transform F ( s ) {\displaystyle F(s... |
c_0mblbcgchezj | Initial value theorem | Summary | Initial_value_theorem | In mathematical analysis, the initial value theorem is a theorem used to relate frequency domain expressions to the time domain behavior as time approaches zero.Let F ( s ) = ∫ 0 ∞ f ( t ) e − s t d t {\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt} be the (one-sided) Laplace transform of ƒ(t). If f {\displaysty... |
c_5ra1mmj6ygs6 | Intermediate Value Theorem | Summary | Bolzano's_theorem | In mathematical analysis, the intermediate value theorem states that if f {\displaystyle f} is a continuous function whose domain contains the interval , then it takes on any given value between f ( a ) {\displaystyle f(a)} and f ( b ) {\displaystyle f(b)} at some point within the interval. This has two important corol... |
c_2a5150d2itg7 | Local maxima | Summary | Maximum_and_minimum | In mathematical analysis, the maximum and minimum of a function are, respectively, the largest and smallest value taken by the function. Known generically as extremum, they may be defined either within a given range (the local or relative extrema) or on the entire domain (the global or absolute extrema) of a function. ... |
c_fmduebhpxx67 | Local maxima | Summary | Maximum_and_minimum | As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. In statistics, the corresponding concept is the sample maximum and minimum. |
c_o93t6a1dixi5 | Mean value theorem for divided differences | Summary | Mean_value_theorem_(divided_differences) | In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives. |
c_1x4c6t21pxjk | Rising sun lemma | Summary | Rising_sun_lemma | In mathematical analysis, the rising sun lemma is a lemma due to Frigyes Riesz, used in the proof of the Hardy–Littlewood maximal theorem. The lemma was a precursor in one dimension of the Calderón–Zygmund lemma.The lemma is stated as follows: Suppose g is a real-valued continuous function on the interval and S is the... |
c_ht0qqj2v63cb | Differentiability class | Summary | Parametric_continuity | In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called differentiability class. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also... |
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