Baire's category theorem and trigonometric series
type: Item
instance of: scholarly article
title: Baire's category theorem and trigonometric series
zbMATH Open document ID: 0961.42001
DOI: 10.1007/BF02791536
author: Jean-Pierre Kahane
published in: Journal d'Analyse Mathématique
publication date: 2001-05-10T00:00:00Z
review text: This is an expository article. The subject is Baire's theorem and its applications to Fourier series, general trigonometric series, Taylor series, Dirichlet series, and thin sets in harmonic analysis. Baire's theorem says that in a complete metric space \(X\), any countable intersection of dense open sets \(\{G_n: n= 1,2,\dots\}\) is dense. Its proof is so easy that it has been qualified by T. Körner as a ``profound triviality''. On the other hand, difficult constructions can be replaced by a convenient choice of the space \(X\) and the dense open sets \(G_n\). Instead of exhibiting one object \(x\) in \(X\) with a given property \(P\), one proves that \(P\) holds on a countable intersection of open dense subsets of \(X\). In that case, one says that quasi every element \(x\) in \(X\) has property \(P\). The applications of Baire's theorem are surprisingly abundant in various areas. We try to illustrate them by presenting a specimen from each of the first four chapters. Theorem 1.5. Quasi all Fourier-Lebesgue series diverge everywhere. Corollary 2.8. Quasi all trigonometric series with coefficients tending to \(0\) are universal in the sense of Menshov, that is, every measurable function of \(\mathbb{T}\) can be approximated by some subsequence of the partial sums of the trigonometric series in question almost everywhere on \(\mathbb{T}\). Denote by \(H(D)\) an appropriate Fréchet space of all analytic functions in the open unit disc \(D\). Proposition 3.2. For quasi all \(F\in H(D)\), \(F(\Delta)= \mathbb{C}\), where \(\Delta\) is an open set in \(D\) such that the closure \(\overline\Delta\) contains at least a point on the boundary \(|z|= 1\). Following V. Nestoridis, a series \(S\):\(= \sum^\infty_{n= 0} c_nz^n\), convergent in \(D\) (that is, a Taylor series which represents an analytic function in \(D\)), is called \(H(D)\)-universal if, given any compact set \(K\) such that \(K\cap D= \emptyset\), and any function \(g(z)\) continuous on \(K\) and analytic ins…
reviewed by: Ferenc Móricz
Mathematics Subject Classification ID: 42-02
Mathematics Subject Classification ID: 42A20
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 43A46
Mathematics Subject Classification ID: 54E52
zbMATH DE Number: 1506553
zbMATH Keywords: Baire's theorem
zbMATH Keywords: Fourier series
zbMATH Keywords: trigonometric series
zbMATH Keywords: Taylor series
zbMATH Keywords: Dirichlet series
zbMATH Keywords: thin sets
zbMATH Keywords: harmonic analysis
MaRDI profile type: Publication
cites work: Q4892644
cites work: On the convergence in category of trigonometric series
cites work: Q5815582
cites work: On convergence and growth of partial sums of Fourier series
cites work: An Almost Everywhere Divergent Fourier Series of the Class L (log+ log+ L )1−ε
cites work: Approximation by overconvergence of a power series
cites work: Some remarks on universal functions and Taylor series
cites work: On the Range of Universal Functions
cites work: On the existence of O-universal functions
cites work: Universal families and hypercyclic operators
cites work: Topics in approximation theory
cites work: Fourier transforms on perfect sets
cites work: Q5542909
cites work: Q5587572
cites work: Q5590907
cites work: Q5665415
cites work: Sur le théorème de superposition de Kolmogorov
cites work: Sur le treizieme probleme de Hilbert, le theoreme de superposition de Kolmogorov et les sommes algebriques d'arcs croissants
cites work: Sur les ensembles de divergence des séries trigonométriques
cites work: Ordre, convergence et sommabilité de produits de séries de Dirichlet. (Order, convergence and summability of products of Dirichlet series.)
cites work: Q5719528
cites work: Q5556050
cites work: A functional method for linear sets
cites work: Thin sets, differentiable functions, and the category method
cites work: Q3263814
cites work: On divergence of trigonometric Fourier series everywhere
cites work: Q3126394
cites work: Q4767743
cites work: Kahane's Helson curve
cites work: Q5638512
cites work: Holomorphic monsters
cites work: Q5767507
cites work: On the growth of universal functions
cites work: Universality of Taylor series as a generic property of holomorphic functions
cites work: On various types of universal taylor series
cites work: Growth of coefficients of universal Taylor series and comparison of two classes of functions
cites work: Q5835542
cites work: Q5801448
cites work: Universal Taylor series
cites work: Q4270264
cites work: Q3874454
cites work: Propriétés Presque Sûres et Quasi-Sûres Des Séries de Dirichletet et des Produits D'Euler
cites work: Q5806092
cites work: An inequality of Paley and convergence a.e. of Walsh-Fourier series
cites work: On representation of functions by means of superpositions and related topics
description: scientific article; zbMATH DE number 1506553