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Trudinger's theorem
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The inequality is a limiting case of Sobolev imbedding and can be stated as the following theorem: Let Ω {\displaystyle \Omega } be a bounded domain in R n {\displaystyle \mathbb {R} ^{n}} satisfying the cone condition. Let m p = n {\displaystyle mp=n} and p > 1 {\displaystyle p>1} . Set A ( t ) = exp ( t n / ( n − m ) ) − 1.
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Trudinger's theorem
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{\displaystyle A(t)=\exp \left(t^{n/(n-m)}\right)-1.} Then there exists the embedding W m , p ( Ω ) ↪ L A ( Ω ) {\displaystyle W^{m,p}(\Omega )\hookrightarrow L_{A}(\Omega )} where L A ( Ω ) = { u ∈ M f ( Ω ): ‖ u ‖ A , Ω = inf { k > 0: ∫ Ω A ( | u ( x ) | k ) d x ≤ 1 } < ∞ } . {\displaystyle L_{A}(\Omega )=\left\{u\in M_{f}(\Omega ):\|u\|_{A,\Omega }=\inf\{k>0:\int _{\Omega }A\left({\frac {|u(x)|}{k}}\right)~dx\leq 1\}<\infty \right\}.} The space L A ( Ω ) {\displaystyle L_{A}(\Omega )} is an example of an Orlicz space.
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Wiener's Tauberian theorem
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In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in L 1 {\displaystyle L^{1}} or L 2 {\displaystyle L^{2}} can be approximated by linear combinations of translations of a given function.Informally, if the Fourier transform of a function f {\displaystyle f} vanishes on a certain set Z {\displaystyle Z} , the Fourier transform of any linear combination of translations of f {\displaystyle f} also vanishes on Z {\displaystyle Z} . Therefore, the linear combinations of translations of f {\displaystyle f} cannot approximate a function whose Fourier transform does not vanish on Z {\displaystyle Z} . Wiener's theorems make this precise, stating that linear combinations of translations of f {\displaystyle f} are dense if and only if the zero set of the Fourier transform of f {\displaystyle f} is empty (in the case of L 1 {\displaystyle L^{1}} ) or of Lebesgue measure zero (in the case of L 2 {\displaystyle L^{2}} ). Gelfand reformulated Wiener's theorem in terms of commutative C*-algebras, when it states that the spectrum of the L 1 {\displaystyle L^{1}} group ring L 1 ( R ) {\displaystyle L^{1}(\mathbb {R} )} of the group R {\displaystyle \mathbb {R} } of real numbers is the dual group of R {\displaystyle \mathbb {R} } . A similar result is true when R {\displaystyle \mathbb {R} } is replaced by any locally compact abelian group.
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Zorich's theorem
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In mathematical analysis, Zorich's theorem was proved by Vladimir A. Zorich in 1967. The result was conjectured by M. A. Lavrentev in 1938.
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Banach limit
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In mathematical analysis, a Banach limit is a continuous linear functional ϕ: ℓ ∞ → C {\displaystyle \phi :\ell ^{\infty }\to \mathbb {C} } defined on the Banach space ℓ ∞ {\displaystyle \ell ^{\infty }} of all bounded complex-valued sequences such that for all sequences x = ( x n ) {\displaystyle x=(x_{n})} , y = ( y n ) {\displaystyle y=(y_{n})} in ℓ ∞ {\displaystyle \ell ^{\infty }} , and complex numbers α {\displaystyle \alpha }: ϕ ( α x + y ) = α ϕ ( x ) + ϕ ( y ) {\displaystyle \phi (\alpha x+y)=\alpha \phi (x)+\phi (y)} (linearity); if x n ≥ 0 {\displaystyle x_{n}\geq 0} for all n ∈ N {\displaystyle n\in \mathbb {N} } , then ϕ ( x ) ≥ 0 {\displaystyle \phi (x)\geq 0} (positivity); ϕ ( x ) = ϕ ( S x ) {\displaystyle \phi (x)=\phi (Sx)} , where S {\displaystyle S} is the shift operator defined by ( S x ) n = x n + 1 {\displaystyle (Sx)_{n}=x_{n+1}} (shift-invariance); if x {\displaystyle x} is a convergent sequence, then ϕ ( x ) = lim x {\displaystyle \phi (x)=\lim x} .Hence, ϕ {\displaystyle \phi } is an extension of the continuous functional lim: c → C {\displaystyle \lim :c\to \mathbb {C} } where c ⊂ ℓ ∞ {\displaystyle c\subset \ell ^{\infty }} is the complex vector space of all sequences which converge to a (usual) limit in C {\displaystyle \mathbb {C} } . In other words, a Banach limit extends the usual limits, is linear, shift-invariant and positive. However, there exist sequences for which the values of two Banach limits do not agree. We say that the Banach limit is not uniquely determined in this case.
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Banach limit
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As a consequence of the above properties, a real-valued Banach limit also satisfies: lim inf n → ∞ x n ≤ ϕ ( x ) ≤ lim sup n → ∞ x n . {\displaystyle \liminf _{n\to \infty }x_{n}\leq \phi (x)\leq \limsup _{n\to \infty }x_{n}.} The existence of Banach limits is usually proved using the Hahn–Banach theorem (analyst's approach), or using ultrafilters (this approach is more frequent in set-theoretical expositions). These proofs necessarily use the axiom of choice (so called non-effective proof).
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Besicovitch covering theorem
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In mathematical analysis, a Besicovitch cover, named after Abram Samoilovitch Besicovitch, is an open cover of a subset E of the Euclidean space RN by balls such that each point of E is the center of some ball in the cover. The Besicovitch covering theorem asserts that there exists a constant cN depending only on the dimension N with the following property: Given any Besicovitch cover F of a bounded set E, there are cN subcollections of balls A1 = {Bn1}, …, AcN = {BncN} contained in F such that each collection Ai consists of disjoint balls, and E ⊆ ⋃ i = 1 c N ⋃ B ∈ A i B . {\displaystyle E\subseteq \bigcup _{i=1}^{c_{N}}\bigcup _{B\in A_{i}}B.}
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Besicovitch covering theorem
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Let G denote the subcollection of F consisting of all balls from the cN disjoint families A1,...,AcN. The less precise following statement is clearly true: every point x ∈ RN belongs to at most cN different balls from the subcollection G, and G remains a cover for E (every point y ∈ E belongs to at least one ball from the subcollection G). This property gives actually an equivalent form for the theorem (except for the value of the constant). There exists a constant bN depending only on the dimension N with the following property: Given any Besicovitch cover F of a bounded set E, there is a subcollection G of F such that G is a cover of the set E and every point x ∈ E belongs to at most bN different balls from the subcover G.In other words, the function SG equal to the sum of the indicator functions of the balls in G is larger than 1E and bounded on RN by the constant bN, 1 E ≤ S G := ∑ B ∈ G 1 B ≤ b N . {\displaystyle \mathbf {1} _{E}\leq S_{\mathbf {G} }:=\sum _{B\in \mathbf {G} }\mathbf {1} _{B}\leq b_{N}.}
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Carathéodory function
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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 with a continuous function is indeed Lebesgue-measurable, but in many situations, continuity is a too restrictive assumption. Carathéodory functions are more general than continuous functions, but still allow a composition with Lebesgue-measurable function to be measurable. Carathéodory functions play a significant role in calculus of variation, and it is named after the Greek mathematician Constantin Carathéodory.
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Hermitian function
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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 {\displaystyle x} in the domain of f {\displaystyle f} . In physics, this property is referred to as PT symmetry. This definition extends also to functions of two or more variables, e.g., in the case that f {\displaystyle f} is a function of two variables it is Hermitian if f ∗ ( x 1 , x 2 ) = f ( − x 1 , − x 2 ) {\displaystyle f^{*}(x_{1},x_{2})=f(-x_{1},-x_{2})} for all pairs ( x 1 , x 2 ) {\displaystyle (x_{1},x_{2})} in the domain of f {\displaystyle f} . From this definition it follows immediately that: f {\displaystyle f} is a Hermitian function if and only if the real part of f {\displaystyle f} is an even function, the imaginary part of f {\displaystyle f} is an odd function.
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Young measure
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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, especially models from material science, and the study of nonlinear partial differential equations, as well as in various optimization (or optimal control problems). They are named after Laurence Chisholm Young who invented them, already in 1937 in one dimension (curves) and later in higher dimensions in 1942.
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Bounded domain
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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 general, functions may be defined on sets that are not topological spaces. The basic idea of a connected subset of a space dates from the 19th century, but precise definitions vary slightly from generation to generation, author to author, and edition to edition, as concepts developed and terms were translated between German, French, and English works. In English, some authors use the term domain, some use the term region, some use both terms interchangeably, and some define the two terms slightly differently; some avoid ambiguity by sticking with a phrase such as non-empty connected open subset.
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Equicontinuous linear maps
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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 formulation of Ascoli's theorem, which states that a subset of C(X), the space of continuous functions on a compact Hausdorff space X, is compact if and only if it is closed, pointwise bounded and equicontinuous.
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Equicontinuous linear maps
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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 space is continuous. If, in addition, fn are holomorphic, then the limit is also holomorphic. The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous.
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Function of bounded variation
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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 that the distance along the direction of the y-axis, neglecting the contribution of motion along x-axis, traveled by a point moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function (which is a hypersurface in this case), but can be every intersection of the graph itself with a hyperplane (in the case of functions of two variables, a plane) parallel to a fixed x-axis and to the y-axis. Functions of bounded variation are precisely those with respect to which one may find Riemann–Stieltjes integrals of all continuous functions.
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Function of bounded variation
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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.
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Function of bounded variation
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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 of discontinuous functions whose first derivative exists almost everywhere: due to this fact, they can and frequently are used to define generalized solutions of nonlinear problems involving functionals, ordinary and partial differential equations in mathematics, physics and engineering. We have the following chains of inclusions for continuous functions over a closed, bounded interval of the real line: Continuously differentiable ⊆ Lipschitz continuous ⊆ absolutely continuous ⊆ continuous and bounded variation ⊆ differentiable almost everywhere
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Metric differential
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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.
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Complete metric space
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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, because e.g. 2 {\displaystyle {\sqrt {2}}} is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the completion of a given space, as explained below.
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Modulus of continuity
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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 in the domain of f. Since moduli of continuity are required to be infinitesimal at 0, a function turns out to be uniformly continuous if and only if it admits a modulus of continuity. Moreover, relevance to the notion is given by the fact that sets of functions sharing the same modulus of continuity are exactly equicontinuous families. For instance, the modulus ω(t) := kt describes the k-Lipschitz functions, the moduli ω(t) := ktα describe the Hölder continuity, the modulus ω(t) := kt(|log t|+1) describes the almost Lipschitz class, and so on.
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Modulus of continuity
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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 terms of moduli of continuity.
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Modulus of continuity
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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, or sublinear (in the sense of growth). Actually, the existence of such special moduli of continuity for a uniformly continuous function is always ensured whenever the domain is either a compact, or a convex subset of a normed space. However, a uniformly continuous function on a general metric space admits a concave modulus of continuity if and only if the ratios d Y ( f ( x ) , f ( x ′ ) ) d X ( x , x ′ ) {\displaystyle {\frac {d_{Y}(f(x),f(x'))}{d_{X}(x,x')}}} are uniformly bounded for all pairs (x, x′) bounded away from the diagonal of X x X. The functions with the latter property constitute a special subclass of the uniformly continuous functions, that in the following we refer to as the special uniformly continuous functions. Real-valued special uniformly continuous functions on the metric space X can also be characterized as the set of all functions that are restrictions to X of uniformly continuous functions over any normed space isometrically containing X. Also, it can be characterized as the uniform closure of the Lipschitz functions on X.
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Measure zero
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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.
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Positive invariant set
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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 ) = 0 } {\displaystyle {\mathcal {O}}:=\left\lbrace x\in \mathbb {R} ^{n}\mid \varphi (x)=0\right\rbrace } where φ {\displaystyle \varphi } is a real-valued function. The set O {\displaystyle {\mathcal {O}}} is said to be positively invariant if x 0 ∈ O {\displaystyle x_{0}\in {\mathcal {O}}} implies that x ( t , x 0 ) ∈ O ∀ t ≥ 0 {\displaystyle x(t,x_{0})\in {\mathcal {O}}\ \forall \ t\geq 0} In other words, once a trajectory of the system enters O {\displaystyle {\mathcal {O}}} , it will never leave it again.
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Strong measure zero set
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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 assumed in mathematics). This means that Borel's conjecture can neither be proven nor disproven in ZFC (assuming ZFC is consistent).
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Strong measure zero set
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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 independence of Borel's conjecture.
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Thin set (analysis)
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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 nowhere dense, and the closure of a thin set is also thin. The fine topology was introduced in 1940 by Henri Cartan to aid in the study of thin sets.
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Improper integrals
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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 integral is taken or of the integrand (the function being integrated), or both. It may also involve bounded but not closed sets or bounded but not continuous functions. While an improper integral is typically written symbolically just like a standard definite integral, it actually represents a limit of a definite integral or a sum of such limits; thus improper integrals are said to converge or diverge.
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Improper integrals
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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 any of the following forms: ∫ a ∞ f ( x ) d x {\displaystyle \int _{a}^{\infty }f(x)\,dx} ∫ − ∞ b f ( x ) d x {\displaystyle \int _{-\infty }^{b}f(x)\,dx} ∫ − ∞ ∞ f ( x ) d x {\displaystyle \int _{-\infty }^{\infty }f(x)\,dx} ∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx} , where f ( x ) {\displaystyle f(x)} is undefined or discontinuous somewhere on {\displaystyle } The first three forms are improper because the integrals are taken over an unbounded interval. (They may be improper for other reasons, as well, as explained below.)
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Improper integrals
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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 being of the "second" type or kind. Integrals that combine aspects of both types are sometimes described as being of the "third" type or kind.In each case above, the improper integral must be rewritten using one or more limits, depending on what is causing the integral to be improper.
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Improper integrals
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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 definition of the integral notation on the left.
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Improper integrals
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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}f(x)\,dx+\lim _{b\to \infty }\int _{c}^{b}f(x)\,dx,} for any choice of c > a {\displaystyle c>a} . Here both limits must converge to a finite value for the improper integral to be said to converge. This requirement avoids the ambiguous case of adding positive and negative infinities (i.e., the " ∞ − ∞ {\displaystyle \infty -\infty } " indeterminate form).
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Improper integrals
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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 − ∫ a t f ( x ) d x + lim u → d + ∫ u c f ( x ) d x + lim b → ∞ ∫ c b f ( x ) d x , {\displaystyle \int _{a}^{\infty }f(x)\,dx=\lim _{t\to d^{-}}\int _{a}^{t}f(x)\,dx+\lim _{u\to d^{+}}\int _{u}^{c}f(x)\,dx+\lim _{b\to \infty }\int _{c}^{b}f(x)\,dx,} for any choice of c > d {\displaystyle c>d} . The previous remarks about indeterminate forms, iterated limits, and the Cauchy principal value also apply here.
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Improper integrals
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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 dimensions, and in other theoretical frameworks such as Lebesgue integration or Henstock–Kurzweil integration. Integrals that are considered improper in one framework may not be in others.
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Function of a real variable
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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 real functions that are considered and studied are differentiable in some interval. The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers. Nevertheless, the codomain of a function of a real variable may be any set.
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Function of a real variable
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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 quaternions. The structure R {\displaystyle \mathbb {R} } -vector space of the codomain induces a structure of R {\displaystyle \mathbb {R} } -vector space on the functions.
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Function of a real variable
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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 variable is a finite-dimensional vector space, the function may be viewed as a sequence of real functions. This is often used in applications.
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Continuous functions on a compact Hausdorff space
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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 to the pointwise addition of functions and scalar multiplication by constants. It is, moreover, a normed space with norm defined by the uniform norm. The uniform norm defines the topology of uniform convergence of functions on X .
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Continuous functions on a compact Hausdorff space
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{\displaystyle X.} The space C ( X ) {\displaystyle {\mathcal {C}}(X)} is a Banach algebra with respect to this norm. (Rudin 1973, §11.3)
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Function (mathematics)
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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 set) form a function space that is at the basis of the theory of distributions. Function spaces play a fundamental role in advanced mathematical analysis, by allowing the use of their algebraic and topological properties for studying properties of functions. For example, all theorems of existence and uniqueness of solutions of ordinary or partial differential equations result of the study of function spaces.
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Asymptotic theory
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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 compared to n2. The function f(n) is said to be "asymptotically equivalent to n2, as n → ∞".
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Asymptotic theory
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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 than or equal to x. Then the theorem states that Asymptotic analysis is commonly used in computer science as part of the analysis of algorithms and is often expressed there in terms of big O notation.
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Constructive function theory
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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.
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Power series ring
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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}\in R],} and suppose S {\displaystyle S} is a commutative associative algebra over R {\displaystyle R} , I {\displaystyle I} is an ideal in S {\displaystyle S} such that the I-adic topology on S {\displaystyle S} is complete, and x {\displaystyle x} is an element of I {\displaystyle I} . Define: f ( x ) = ∑ n ≥ 0 a n x n .
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Power series ring
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{\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 ) .
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Power series ring
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{\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 power series, provided that the arguments don't have constant coefficients (so that they belong to the ideal ( X ) {\displaystyle (X)} ): f ( 0 ) {\displaystyle f(0)} , f ( X 2 − X ) {\displaystyle f(X^{2}-X)} and f ( ( 1 − X ) − 1 − 1 ) {\displaystyle f((1-X)^{-1}-1)} are all well defined for any formal power series f ∈ R ] .
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Power series ring
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{\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\geq 0}a^{-n-1}(a-f)^{n}.} If the formal power series g {\displaystyle g} with g ( 0 ) = 0 {\displaystyle g(0)=0} is given implicitly by the equation f ( g ) = X {\displaystyle f(g)=X} where f {\displaystyle f} is a known power series with f ( 0 ) = 0 {\displaystyle f(0)=0} , then the coefficients of g {\displaystyle g} can be explicitly computed using the Lagrange inversion formula.
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Proper convex function
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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 convex analysis and variational analysis, a point (in the domain) at which some given function f {\displaystyle f} is minimized is typically sought, where f {\displaystyle f} is valued in the extended real number line = R ∪ { ± ∞ } . {\displaystyle =\mathbb {R} \cup \{\pm \infty \}.}
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Proper convex function
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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 problem can be answered; this is ultimately the reason why the definition of "proper" requires that the function never take − ∞ {\displaystyle -\infty } as a value. Assuming this, if the function's domain is empty or if the function is identically equal to + ∞ {\displaystyle +\infty } then the minimization problem once again has an immediate answer.
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Proper convex function
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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 problem is instead a maximization problem (which would be clearly indicated, such as by the function being concave rather than convex) then the definition of "proper" is defined in an analogous (albeit technically different) manner but with the same goal: to exclude cases where the maximization problem can be answered immediately. Specifically, a concave function g {\displaystyle g} is called proper if its negation − g , {\displaystyle -g,} which is a convex function, is proper in the sense defined above.
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Parameter
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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 the integral depends.
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Parameter
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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).
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Limit inferior
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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 system: we add the positive and negative infinities to the real line to give the complete totally ordered set , which is a complete lattice.
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Generalized Fourier series
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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 interpolation theory.
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Microlocalization functor
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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 integral operators, oscillatory integral operators, and paradifferential operators. The term microlocal implies localisation not only with respect to location in the space, but also with respect to cotangent space directions at a given point. This gains in importance on manifolds of dimension greater than one.
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Wavefront set
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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.
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Nested sequence of closed intervals
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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 and integral calculus from the late 1600s has posed a huge challenge for mathematicians trying to prove their methods rigorously; despite their success in physics, engineering and other sciences. The axiomatic description of nested intervals (or an equivalent axiom) has become an important foundation for the modern understanding of calculus. In the context of this article, R {\displaystyle \mathbb {R} } in conjunction with + {\displaystyle +} and ⋅ {\displaystyle \cdot } is an Archimedean ordered field, meaning the axioms of order and the Archimedean property hold.
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Nullcline
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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 \vdots } x n ′ = f n ( x 1 , … , x n ) {\displaystyle x_{n}'=f_{n}(x_{1},\ldots ,x_{n})} where x ′ {\displaystyle x'} here represents a derivative of x {\displaystyle x} with respect to another parameter, such as time t {\displaystyle t} . The j {\displaystyle j} 'th nullcline is the geometric shape for which x j ′ = 0 {\displaystyle x_{j}'=0} . The equilibrium points of the system are located where all of the nullclines intersect. In a two-dimensional linear system, the nullclines can be represented by two lines on a two-dimensional plot; in a general two-dimensional system they are arbitrary curves.
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P-variation
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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 ( M , d ) {\displaystyle (M,d)} is a metric space and I a totally ordered set, its p-variation is ‖ f ‖ p -var = ( sup D ∑ t k ∈ D d ( f ( t k ) , f ( t k − 1 ) ) p ) 1 / p {\displaystyle \|f\|_{p{\text{-var}}}=\left(\sup _{D}\sum _{t_{k}\in D}d(f(t_{k}),f(t_{k-1}))^{p}\right)^{1/p}} where D ranges over all finite partitions of the interval I. The p variation of a function decreases with p. If f has finite p-variation and g is an α-Hölder continuous function, then g ∘ f {\displaystyle g\circ f} has finite p α {\displaystyle {\frac {p}{\alpha }}} -variation. The case when p is one is called total variation, and functions with a finite 1-variation are called bounded variation functions.
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Upper semi-continuous
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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 for arguments near x 0 {\displaystyle x_{0}} are not much higher (respectively, lower) than f ( x 0 ) . {\displaystyle f\left(x_{0}\right).} A function is continuous if and only if it is both upper and lower semicontinuous. If we take a continuous function and increase its value at a certain point x 0 {\displaystyle x_{0}} to f ( x 0 ) + c {\displaystyle f\left(x_{0}\right)+c} for some c > 0 {\displaystyle c>0} , then the result is upper semicontinuous; if we decrease its value to f ( x 0 ) − c {\displaystyle f\left(x_{0}\right)-c} then the result is lower semicontinuous. The notion of upper and lower semicontinuous function was first introduced and studied by René Baire in his thesis in 1899.
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Agranovich–Dynin formula
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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).
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Alexandrov theorem
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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. In this context, having a second derivative at a point means having a second-order Taylor expansion at that point with a local error smaller than any quadratic. The result is closely related to Rademacher's theorem.
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Bessel–Clifford function
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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 {1}{\Pi (x)}}={\frac {1}{\Gamma (x+1)}}} is the entire function defined by means of the reciprocal gamma function, then the Bessel–Clifford function is defined by the series C n ( z ) = ∑ k = 0 ∞ π ( k + n ) z k k ! {\displaystyle {\mathcal {C}}_{n}(z)=\sum _{k=0}^{\infty }\pi (k+n){\frac {z^{k}}{k!}}} The ratio of successive terms is z/k(n + k), which for all values of z and n tends to zero with increasing k. By the ratio test, this series converges absolutely for all z and n, and uniformly for all regions with bounded |z|, and hence the Bessel–Clifford function is an entire function of the two complex variables n and z.
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Bohr–Mollerup theorem
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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 only positive function f , with domain on the interval x > 0, that simultaneously has the following three properties: f (1) = 1, and f (x + 1) = x f (x) for x > 0 and f is logarithmically convex.A treatment of this theorem is in Artin's book The Gamma Function, which has been reprinted by the AMS in a collection of Artin's writings.The theorem was first published in a textbook on complex analysis, as Bohr and Mollerup thought it had already been proved.The theorem admits a far-reaching generalization to a wide variety of functions (that have convexity or concavity properties of any order).
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Brezis–Gallouët inequality
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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 the second derivatives. It is useful in the study of partial differential equations.
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Brezis–Gallouët inequality
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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 that, for all u ∈ H 2 ( Ω ) {\displaystyle u\in H^{2}(\Omega )} which is not a.e. equal to 0, ‖ u ‖ L ∞ ( Ω ) ≤ C ‖ u ‖ H 1 ( Ω ) ( 1 + ( log ( 1 + ‖ u ‖ H 2 ( Ω ) ‖ u ‖ H 1 ( Ω ) ) ) 1 / 2 ) .
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Cauchy index
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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-plane and those located in the left half-plane. The complex polynomial f(z) is such that f(iy) = q(y) + ip(y).We must also assume that p has degree less than the degree of q.
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Dirichlet kernel
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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 comes from its relation to Fourier series. The convolution of Dn(x) with any function f of period 2π is the nth-degree Fourier series approximation to f, i.e., we have where is the kth Fourier coefficient of f. This implies that in order to study convergence of Fourier series it is enough to study properties of the Dirichlet kernel.
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Foias constant
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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, .... The Foias constant is the unique choice α such that if x1 = α then the sequence diverges to infinity. For all other values of x1, the sequence is divergent as well, but it has two accumulation points: 1 and infinity. Numerically, it is α = 1.187452351126501 … {\displaystyle \alpha =1.187452351126501\ldots } .No closed form for the constant is known. When x1 = α then the growth rate of the sequence (xn) is given by the limit lim n → ∞ x n log n n = 1 , {\displaystyle \lim _{n\to \infty }x_{n}{\frac {\log n}{n}}=1,} where "log" denotes the natural logarithm.The same methods used in the proof of the uniqueness of the Foias constant may also be applied to other similar recursive sequences.
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Haar integral
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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 1897 under the name "invariant integral". Haar measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory.
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Karamata's tauberian theorem
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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 equivalence ∑ n = 0 ∞ a n e − n y ∼ 1 y as y ↓ 0 {\displaystyle \sum _{n=0}^{\infty }a_{n}e^{-ny}\sim {\frac {1}{y}}\ {\text{as}}\ y\downarrow 0} then there is also an asymptotic equivalence ∑ k = 0 n a k ∼ n {\displaystyle \sum _{k=0}^{n}a_{k}\sim n} as n → ∞ {\displaystyle n\to \infty } . The integral formulation of the theorem relates in an analogous manner the asymptotics of the cumulative distribution function of a function with the asymptotics of its Laplace transform.
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Karamata's tauberian theorem
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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
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Hardy–Littlewood inequality
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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 {\displaystyle \mathbb {R} ^{n}} , then ∫ R n f ( x ) g ( x ) d x ≤ ∫ R n f ∗ ( x ) g ∗ ( x ) d x {\displaystyle \int _{\mathbb {R} ^{n}}f(x)g(x)\,dx\leq \int _{\mathbb {R} ^{n}}f^{*}(x)g^{*}(x)\,dx} where f ∗ {\displaystyle f^{*}} and g ∗ {\displaystyle g^{*}} are the symmetric decreasing rearrangements of f {\displaystyle f} and g {\displaystyle g} , respectively.The decreasing rearrangement f ∗ {\displaystyle f^{*}} of f {\displaystyle f} is defined via the property that for all r > 0 {\displaystyle r>0} the two super-level sets E f ( r ) = { x ∈ X: f ( x ) > r } {\displaystyle E_{f}(r)=\left\{x\in X:f(x)>r\right\}\quad } and E f ∗ ( r ) = { x ∈ X: f ∗ ( x ) > r } {\displaystyle \quad E_{f^{*}}(r)=\left\{x\in X:f^{*}(x)>r\right\}} have the same volume ( n {\displaystyle n} -dimensional Lebesgue measure) and E f ∗ ( r ) {\displaystyle E_{f^{*}}(r)} is a ball in R n {\displaystyle \mathbb {R} ^{n}} centered at x = 0 {\displaystyle x=0} , i.e. it has maximal symmetry.
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Hilbert–Schmidt theorem
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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.
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Kakutani fixed-point theorem
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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 fixed point theorem is a generalization of the Brouwer fixed point theorem. The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces.
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Kakutani fixed-point theorem
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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.
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Lagrange–Bürmann formula
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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.
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Minkowski inequality
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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 {\displaystyle f+g} is in L p ( S ) , {\displaystyle L^{p}(S),} and we have the triangle inequality with equality for 1 < p < ∞ {\displaystyle 1
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Pólya–Szegő inequality
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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.
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Rademacher–Menchov theorem
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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.
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Schur test
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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 spaces (such as R n {\displaystyle \mathbb {R} ^{n}} ). Let T {\displaystyle \,T} be an integral operator with the non-negative Schwartz kernel K ( x , y ) {\displaystyle \,K(x,y)} , x ∈ X {\displaystyle x\in X} , y ∈ Y {\displaystyle y\in Y}: T f ( x ) = ∫ Y K ( x , y ) f ( y ) d y .
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Schur test
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{\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 \alpha p(x)} for almost all x {\displaystyle \,x} and ( 2 ) ∫ X p ( x ) K ( x , y ) d x ≤ β q ( y ) {\displaystyle (2)\qquad \int _{X}p(x)K(x,y)\,dx\leq \beta q(y)} for almost all y {\displaystyle \,y} , then T {\displaystyle \,T} extends to a continuous operator T: L 2 → L 2 {\displaystyle T:L^{2}\to L^{2}} with the operator norm ‖ T ‖ L 2 → L 2 ≤ α β . {\displaystyle \Vert T\Vert _{L^{2}\to L^{2}}\leq {\sqrt {\alpha \beta }}.} Such functions p ( x ) {\displaystyle \,p(x)} , q ( y ) {\displaystyle \,q(y)} are called the Schur test functions. In the original version, T {\displaystyle \,T} is a matrix and α = β = 1 {\displaystyle \,\alpha =\beta =1} .
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Szegő limit theorems
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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.
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Weierstrass approximation theorem
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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, this theorem has both practical and theoretical relevance, especially in polynomial interpolation. The original version of this result was established by Karl Weierstrass in 1885 using the Weierstrass transform. Marshall H. Stone considerably generalized the theorem and simplified the proof.
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Weierstrass approximation theorem
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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 continuous functions on X {\displaystyle X} are shown to suffice, as is detailed below. The Stone–Weierstrass theorem is a vital result in the study of the algebra of continuous functions on a compact Hausdorff space. Further, there is a generalization of the Stone–Weierstrass theorem to noncompact Tychonoff spaces, namely, any continuous function on a Tychonoff space is approximated uniformly on compact sets by algebras of the type appearing in the Stone–Weierstrass theorem and described below. A different generalization of Weierstrass' original theorem is Mergelyan's theorem, which generalizes it to functions defined on certain subsets of the complex plane.
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Whitney covering lemma
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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 Calderón–Zygmund decomposition. Roughly speaking, the lemma states that it is possible to decompose an open set by cubes each of whose diameters is proportional, within certain bounds, to its distance from the boundary of the open set.
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Young's inequality for integral operators
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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.
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Alternating series test
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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 criterion. The test is only sufficient, not necessary, so some convergent alternating series may fail the first part of the test.
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Characteristic variety
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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 theorem says a characteristic variety is involutive.
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Mean-periodic function
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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.
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Final value theorem
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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)} , then a final value theorem establishes conditions under which lim t → ∞ f ( t ) = lim s → 0 s F ( s ) {\displaystyle \lim _{t\to \infty }f(t)=\lim _{s\,\to \,0}{sF(s)}} Likewise, if f {\displaystyle f} in discrete time has (unilateral) Z-transform F ( z ) {\displaystyle F(z)} , then a final value theorem establishes conditions under which lim k → ∞ f = lim z → 1 ( z − 1 ) F ( z ) {\displaystyle \lim _{k\to \infty }f=\lim _{z\to 1}{(z-1)F(z)}} An Abelian final value theorem makes assumptions about the time-domain behavior of f ( t ) {\displaystyle f(t)} (or f {\displaystyle f} ) to calculate lim s → 0 s F ( s ) {\displaystyle \lim _{s\,\to \,0}{sF(s)}} . Conversely, a Tauberian final value theorem makes assumptions about the frequency-domain behaviour of F ( s ) {\displaystyle F(s)} to calculate lim t → ∞ f ( t ) {\displaystyle \lim _{t\to \infty }f(t)} (or lim k → ∞ f {\displaystyle \lim _{k\to \infty }f} ) (see Abelian and Tauberian theorems for integral transforms).
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Initial value theorem
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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 {\displaystyle f} is bounded on ( 0 , ∞ ) {\displaystyle (0,\infty )} (or if just f ( t ) = O ( e c t ) {\displaystyle f(t)=O(e^{ct})} ) and lim t → 0 + f ( t ) {\displaystyle \lim _{t\to 0^{+}}f(t)} exists then the initial value theorem says lim t → 0 f ( t ) = lim s → ∞ s F ( s ) . {\displaystyle \lim _{t\,\to \,0}f(t)=\lim _{s\to \infty }{sF(s)}.}
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Intermediate Value Theorem
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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 corollaries: If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem). The image of a continuous function over an interval is itself an interval.
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Local maxima
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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. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.
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Local maxima
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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.
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Mean value theorem for divided differences
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In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives.
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Rising sun lemma
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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 set of x in such that there exists a y∈(x,b] with g(y) > g(x). (Note that b cannot be in S, though a may be.) Define E = S ∩ (a,b).Then E is an open set, and it may be written as a countable union of disjoint intervals E = ⋃ k ( a k , b k ) {\displaystyle E=\bigcup _{k}(a_{k},b_{k})} such that g(ak) = g(bk), unless ak = a ∈ S for some k, in which case g(a) < g(bk) for that one k. Furthermore, if x ∈ (ak,bk), then g(x) < g(bk).The colorful name of the lemma comes from imagining the graph of the function g as a mountainous landscape, with the sun shining horizontally from the right. The set E consist of points that are in the shadow.
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Differentiability class
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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 possess derivatives of all orders in its domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or C ∞ {\displaystyle C^{\infty }} function).
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Spaces of test functions and distributions
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In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are usually infinitely differentiable complex-valued (or sometimes real-valued) functions on a non-empty open subset U ⊆ R n {\displaystyle U\subseteq \mathbb {R} ^{n}} that have compact support. The space of all test functions, denoted by C c ∞ ( U ) , {\displaystyle C_{c}^{\infty }(U),} is endowed with a certain topology, called the canonical LF-topology, that makes C c ∞ ( U ) {\displaystyle C_{c}^{\infty }(U)} into a complete Hausdorff locally convex TVS. The strong dual space of C c ∞ ( U ) {\displaystyle C_{c}^{\infty }(U)} is called the space of distributions on U {\displaystyle U} and is denoted by D ′ ( U ) := ( C c ∞ ( U ) ) b ′ , {\displaystyle {\mathcal {D}}^{\prime }(U):=\left(C_{c}^{\infty }(U)\right)_{b}^{\prime },} where the " b {\displaystyle b} " subscript indicates that the continuous dual space of C c ∞ ( U ) , {\displaystyle C_{c}^{\infty }(U),} denoted by ( C c ∞ ( U ) ) ′ , {\displaystyle \left(C_{c}^{\infty }(U)\right)^{\prime },} is endowed with the strong dual topology.
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Spaces of test functions and distributions
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The set of tempered distributions forms a vector subspace of the space of distributions D ′ ( U ) {\displaystyle {\mathcal {D}}^{\prime }(U)} and is thus one example of a space of distributions; there are many other spaces of distributions. There also exist other major classes of test functions that are not subsets of C c ∞ ( U ) , {\displaystyle C_{c}^{\infty }(U),} such as spaces of analytic test functions, which produce very different classes of distributions. The theory of such distributions has a different character from the previous one because there are no analytic functions with non-empty compact support. Use of analytic test functions leads to Sato's theory of hyperfunctions.
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