Relation between Fourier series and Wiener algebras

2021 ◽  
Vol 18 (1) ◽  
pp. 80-103
Author(s):  
Roald Trigub
Keyword(s):  
Fourier Series ◽  
Banach Algebras ◽  
Sufficient Conditions ◽  
Real Variable ◽  
State University ◽  
Approximation Properties ◽  
Function Classes ◽  
Almost Everywhere ◽  
Necessary And Sufficient ◽  
Complex Valued

New relations between the Banach algebras of absolutely convergent Fourier integrals of complex-valued measures of Wiener and various issues of trigonometric Fourier series (see classical monographs by A.~Zygmund [1] and N. K. Bary [2]) are described. Those bilateral interrelations allow one to derive new properties of the Fourier series from the known properties of the Wiener algebras, as well as new results to be obtained for those algebras from the known properties of Fourier series. For example, criteria, i.e. simultaneously necessary and sufficient conditions, are obtained for any trigonometric series to be a Fourier series, or the Fourier series of a function of bounded variation, and so forth. Approximation properties of various linear summability methods of Fourier series (comparison, approximation of function classes and single functions) and summability almost everywhere (often with the set indication) are considered. The presented material was reported by the author on 12.02.2021 at the Zoom-seminar on the theory of real variable functions at the Moscow State University.

On the Closure of and Integral Functions

1935 ◽  
Vol 31 (3) ◽  
pp. 335-346 ◽  
Author(s):  
Norman Levinson
Keyword(s):  
Real Variable ◽  
The Real ◽  
Almost Everywhere ◽  
Complex Valued

1. A set of functions {øn (x)} is said to be closed L over an interval (a, b) if for an f (x) belonging to Limplies that f(x) = 0 almost everywhere. Here f(x) is a complex valued function of the real variable x.

Modifications of Fourier transforms and singular continuous measures

1980 ◽  
Vol 87 (3) ◽  
pp. 383-392
Author(s):  
Alan MacLean
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Asymptotic Behaviour ◽  
Fourier Transforms ◽  
Sufficient Conditions ◽  
The Other ◽  
Absolutely Continuous ◽  
Necessary And Sufficient ◽  
Continuous Measures ◽  
Absolutely Continuous Measures

It has long been known, after Wiener (e.g. see (11), vol. 1, p. 108, (5), (8), §5·6)) that a measure μ whose Fourier transform vanishes at infinity is continuous, and generally, that μ is continuous if and only if is small ‘on the average’. Baker (1) has pursued this theme and obtained concise necessary and sufficient conditions for the continuity of μ, again expressed in terms of the rate of decrease of . On the other hand, for continuous μ, Rudin (9) points out the difficulty in obtaining criteria based solely on the asymptotic behaviour of by which one may determine whether μ has a singular component. The object of this paper is to show further that any such criteria must be complicated indeed. We shall show that the absolutely continuous measures on T = [0, 2π) whose Fourier transforms are the most well-behaved (namely, those of the form (1/2π)f(x)dx, where f has an absolutely convergent Fourier series) are such that one may modify their transforms on ‘large’ subsets of Z so that they become the transforms of singular continuous measures. Moreover, the singular continuous measures in question may be chosen so that their Fourier transforms do not vanish at infinity.

Uniform and 𝐿-Convergence of Logarithmic Means of Double Walsh–Fourier Series

2005 ◽  
Vol 12 (1) ◽  
pp. 75-88
Author(s):  
György Gát ◽  
Ushangi Goginava
Keyword(s):  
Fourier Series ◽  
Function Space ◽  
Modulus Of Continuity ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Two Dimensional ◽  
Logarithmic Means ◽  
Convergence And Divergence ◽  
Necessary And Sufficient ◽  
Series Of Functions

Abstract We discuss some convergence and divergence properties of twodimensional (Nörlund) logarithmic means of two-dimensional Walsh–Fourier series of functions both in the uniform and in the Lebesgue norm. We give necessary and sufficient conditions for the convergence regarding the modulus of continuity of the function, and also the function space.

A theorem of Stone-Weierstrass type

1966 ◽  
Vol 62 (4) ◽  
pp. 649-666 ◽  
Author(s):  
G. A. Reid
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Necessary And Sufficient Conditions ◽  
Weierstrass Theorem ◽  
Necessary And Sufficient ◽  
Complex Valued

The Stone-Weierstrass theorem gives very simple necessary and sufficient conditions for a subset A of the algebra of all real-valued continuous functions on the compact Hausdorff space X to generate a subalgebra dense in namely, this is so if and only if the functions of A strongly separate the points of X, in other words given any two distinct points of X there exists a function in A taking different values at these points, and given any point of X there exists a function in A non-zero there. In the case of the algebra of all complex-valued continuous functions on X, the same result holds provided that we consider the subalgebra generated by A together with Ā, the set of complex conjugates of the functions in A.

scholarly journals Generalization on some theorems of $ L^1 $-convergence of certain trigonometric series

2008 ◽  
Vol 39 (1) ◽  
pp. 63-74
Author(s):  
Zivorad Tomovski
Keyword(s):  
Fourier Series ◽  
Trigonometric Series ◽  
Sufficient Conditions ◽  
Sigma 1 ◽  
Complex Valued ◽  
Derivatives Of

In this paper we study $ L^1 $-convergence of the $ r $-th derivatives of Fourier series with complex-valued coefficients. Namely new necessary-sufficient conditions for $L^1$-convergence of the $ r $-th derivatives of Fourier series are given. These results generalize corresponding theorems proved by several authors (see [7], [10], [13], [19]). Applying the Wang-Telyakovskii class $ ({\bf B}{\bf V})_r^\sigma $, $ \>\sigma>0 $, $ \>r=0,1,2,\ldots\, $ we generalize also the theorem proved by Garrett, Rees and Stanojevi\'{c} in [5]. Finally, for $ \sigma=1 $ some corollaries of this theorem are given.

scholarly journals On extension of bi-derivations to the bidual of Banach algebras

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2261-2267
Author(s):  
Attar Erfanian ◽  
S. Barootkoob ◽  
Vishki Ebrahimi
Keyword(s):  
Banach Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
C Algebras ◽  
Necessary And Sufficient

We present some necessary and sufficient conditions such that the (Arens) extensions of a bi-derivation on Banach algebras are again bi-derivations. We then examine our results for some Banach algebras. In particular, we show that the (Arens) extensions of a bi-derivation on C*-algebras are biderivations. Some results on extensions of an inner bi-derivation are also included.

Right and Left Weak Approximation Properties in Banach Spaces

2009 ◽  
Vol 52 (1) ◽  
pp. 28-38 ◽  
Author(s):  
Changsun Choi ◽  
Ju Myung Kim ◽  
Keun Young Lee
Keyword(s):  
Banach Spaces ◽  
Approximation Property ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weak Approximation ◽  
Weak Version ◽  
Approximation Properties ◽  
Necessary And Sufficient

AbstractNew necessary and sufficient conditions are established for Banach spaces to have the approximation property; these conditions are easier to check than the known ones. A shorter proof of a result of Grothendieck is presented, and some properties of a weak version of the approximation property are addressed.

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