Teorema de Nachbin

Nachbin's theorem This article is about Nachbin's theorem on complex analysis. For Nachbin's theorem on approximation theory, see Stone–Weierstrass theorem § Nachbin's theorem.

Na matemática, in the area of complex analysis, Teorema de Nachbin (named after Leopoldo Nachbin) is commonly used to establish a bound on the growth rates for an analytic function. This article provides a brief review of growth rates, including the idea of a function of exponential type. Classification of growth rates based on type help provide a finer tool than big O or Landau notation, since a number of theorems about the analytic structure of the bounded function and its integral transforms can be stated. Em particular, Nachbin's theorem may be used to give the domain of convergence of the generalized Borel transform, dado abaixo.

Conteúdo 1 Exponential type 2 Ψ type 3 Borel transform 4 Nachbin resummation 5 Fréchet space 6 Veja também 7 References Exponential type Main article: Exponential type A function f(z) defined on the complex plane is said to be of exponential type if there exist constants M and α such that {estilo de exibição |f(re^{ittheta })|leq Me^{alpha r}} in the limit of {displaystyle rto infty } . Aqui, the complex variable z was written as {displaystyle z=re^{ittheta }} to emphasize that the limit must hold in all directions θ. Letting α stand for the infimum of all such α, one then says that the function f is of exponential type α.

Por exemplo, deixar {estilo de exibição f(z)=sin(pi z)} . Then one says that {pecado de estilo de exibição(pi z)} is of exponential type π, since π is the smallest number that bounds the growth of {pecado de estilo de exibição(pi z)} along the imaginary axis. Então, for this example, Carlson's theorem cannot apply, as it requires functions of exponential type less than π.

Ψ type Bounding may be defined for other functions besides the exponential function. No geral, uma função {displaystyle Psi (t)} is a comparison function if it has a series {displaystyle Psi (t)=soma _{n=0}^{infty }Psi _{n}t^{n}} com {displaystyle Psi _{n}>0} for all n, e {displaystyle lim _{até o infinito }{fratura {Psi _{n+1}}{Psi _{n}}}=0.} Comparison functions are necessarily entire, which follows from the ratio test. Se {displaystyle Psi (t)} is such a comparison function, one then says that f is of Ψ-type if there exist constants M and τ such that {estilo de exibição à esquerda|abandonou(re^{ittheta }certo)certo|leq MPsi (tau r)} Como {displaystyle rto infty } . If τ is the infimum of all such τ one says that f is of Ψ-type τ.

Nachbin's theorem states that a function f(z) with the series {estilo de exibição f(z)=soma _{n=0}^{infty }f_{n}z^{n}} is of Ψ-type τ if and only if {displaystyle limsup _{até o infinito }deixei|{fratura {f_{n}}{Psi _{n}}}certo|^{1/n}=tau .} Borel transform Nachbin's theorem has immediate applications in Cauchy theorem-like situations, and for integral transforms. Por exemplo, the generalized Borel transform is given by {estilo de exibição F(W)=soma _{n=0}^{infty }{fratura {f_{n}}{Psi _{n}w^{n+1}}}.} If f is of Ψ-type τ, then the exterior of the domain of convergence of {estilo de exibição F(W)} , and all of its singular points, are contained within the disk {estilo de exibição |W|leq tau .} Além disso, um tem {estilo de exibição f(z)={fratura {1}{2pi eu}}pomada _{gama }psi (zw)F(W),dw} where the contour of integration γ encircles the disk {estilo de exibição |W|leq tau } . This generalizes the usual Borel transform for exponential type, Onde {displaystyle Psi (t)=e^{t}} . The integral form for the generalized Borel transform follows as well. Deixar {alfa de estilo de exibição (t)} be a function whose first derivative is bounded on the interval {estilo de exibição [0,infty )} , de modo a {estilo de exibição {fratura {1}{Psi _{n}}}=int_{0}^{infty }t^{n},dalpha (t)} Onde {displaystyle dalpha (t)=alpha ^{melhor }(t),dt} . Then the integral form of the generalized Borel transform is {estilo de exibição F(W)={fratura {1}{W}}int_{0}^{infty }abandonou({fratura {t}{W}}certo),dalpha (t).} The ordinary Borel transform is regained by setting {alfa de estilo de exibição (t)=e^{-t}} . Note that the integral form of the Borel transform is just the Laplace transform.

Nachbin resummation Nachbin resummation (generalized Borel transform) can be used to sum divergent series that escape to the usual Borel summation or even to solve (assintoticamente) integral equations of the form: {estilo de exibição g(s)=sint_{0}^{infty }K(st)f(t),dt} where f(t) may or may not be of exponential growth and the kernel K(você) has a Mellin transform. The solution can be obtained as {estilo de exibição f(x)=soma _{n=0}^{infty }{fratura {uma_{n}}{M(n+1)}}x^{n}} com {estilo de exibição g(s)=soma _{n=0}^{infty }uma_{n}s^{-n}} e M(n) is the Mellin transform of K(você). An example of this is the Gram series {estilo de exibição pi (x)approx 1+sum _{n=1}^{infty }{fratura {log ^{n}(x)}{ncdot n!zeta (n+1)}}.} in some cases as an extra condition we require {estilo de exibição int _{0}^{infty }K(t)t^{n},dt} to be finite for {displaystyle n=0,1,2,3,...} and different from 0.

Fréchet space Collections of functions of exponential type {estilo de exibição tau } can form a complete uniform space, namely a Fréchet space, by the topology induced by the countable family of norms {estilo de exibição |f|_{n}=sup_{zin mathbb {C} }exp esquerda[-deixei(sim +{fratura {1}{n}}certo)|z|certo]|f(z)|.} See also Divergent series Borel summation Euler summation Cesàro summation Lambert summation Mittag-Leffler summation Phragmén–Lindelöf principle Abelian and tauberian theorems Van Wijngaarden transformation References L. Nachbin, "An extension of the notion of integral functions of the finite exponential type", Anais Acad. Brasil. Ciencias. 16 (1944) 143-147. Ralph P. Boas, Jr. e R. Creighton Buck, Polynomial Expansions of Analytic Functions (Second Printing Corrected), (1964) Academic Press Inc., Publishers New York, Springer-Verlag, Berlim. Library of Congress Card Number 63-23263. (Provides a statement and proof of Nachbin's theorem, as well as a general review of this topic.) A.F. Leont'ev (2001) [1994], "Function of exponential type", Enciclopédia de Matemática, EMS Press A.F. Leont'ev (2001) [1994], "Borel transform", Enciclopédia de Matemática, Categorias de Imprensa EMS: Integral transformsTheorems in complex analysisSummability methods