Edwin Hewitt

Abstract Harmonic Analysis

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The Gibbs- Wilbraham Phenomenon: An Episode in Fourier Analysis

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Singular measures with absolutely continuous convolution squares

plays an essential r61e in computing the amount of this overshoot. While teaching a course in the theory of functions of a real variable, E. HEWITT found the value 1.71... listed for the integral (1) in HARDY & ROGOSINSKI [271, page 36. This anomaly, as well as others encountered in the literature, led us to a study of the GIBBS phenomenon and its history. In the course of this study we uncovered a maze of forgotten results, interesting and difficult generalizations, faulty constants, and some details about the GIBBS phenomenon that have escaped the attention of many writers on the subject. Despite the familiarity of our theme, we therefore entertain a hope that readers of the Archive will find some interest in a discussion of this corner of FOURIER analysis. The paper is divided into three Parts. In Part I, we examine GIBBSs phenomenon in some detail. In Part II, we take up its curious history and describe briefly some of its congeners. In Part III, we offer some conclusions. The computations given in this paper were carried out on two computers: a Hewlett-Packard 9810 and a Univac 1110. The graphs (barring the simplest) were drawn by a Hewlett-Packard 9862A plotter. All finite decimal expansions are truncated decimal expansions. It is a pleasure to record our indebtedness to GERALD B. FOLLAND, THOMAS L. HANKINS, EINAR HILLE, and STEPHEN P. KEELER, who have made valuable suggestions to us.

SOME THEOREMS ON LACUNARY FOURIER SERIES, WITH EXTENSIONS TO COMPACT GROUPS

Introduction . A famous construction of Wiener and Wintner ((13)), later refined by Salem ((11)) and extended by Schaeffer ((12)) and Ivasev-Musatov ((8)), produces a non-negative, singular, continuous measure μ on [ − π,π[ such that for every ∈ > 0. It is plain that the convolution μ * μ is absolutely continuous and in fact has Lebesgue–Radon–Nikodým derivative f such that For general locally compact Abelian groups, no exact analogue of (1 · 1) seems possible, as the character group may admit no natural order. However, it makes good sense to ask if μ* μ is absolutely continuous and has p th power integrable derivative. We will construct continuous singular measures μ on all non-discrete locally compact Abelian groups G such that μ * μ is a absolutely continuous and for which the Lebesgue–Radon–Nikodým derivative of μ * μ is in, for all real p > 1.

Singular measures with absolutely continuous convolution squares.(Corrigendum)

1. Introduction. 1.1. The theory of lacunary Fourier series, developed by Banach, Kolmogorov, Sidon, Zygmund, and others, deals largely with extraordinary properties that such series possess. The Fourier series for a bounded function converges absolutely if the function has a lacunary Fourier series; an integrable function is square integrable if its Fourier series is lacunary. Somewhat different theorems show that very general functions on lacunary sets can be represented by Fourier or Fourier-Stieltjes transforms. Many of these theorems are given in [7, Chapter VI and Chapter IX].

Linear functions on almost periodic functions

Mr Denis Lichtman has kindly pointed out to us that an added hypothesis is needed in Theorem (4·1). The revised theorem follows.

SOME MULTIPLICATIVE LINEAR FUNCTIONALS ON M(G)

Our aim is, first, to present two realizations of the space 1* of all bounded complex linear functionals on 21, and, second, to use these realizations for the study of positive definite functions which are not necessarily continuous. In this fashion, we obtain a generalization of Bochners representation theorem for continuous positive definite functions as well as various facts concerning positive definite functions and their structure. 0.2 Throughout the present paper, the symbol R denotes the real numbers, considered either as an additive group or as a field; K the field of complex numbers; T the multiplicative group of complex numbers of absolute value 1; Tm the complete Cartesian product of m groups each identical with T, m being any cardinal number greater than 1. The characteristic function of a subset B of a set X is denoted by XB. If G is any locally compact Abelian group, we denote the group of all continuous characters of G by the symbol G]. G] is given the usual compact-open topology. If X is any topological space, we denote the set of all complex-valued continuous functions on X which are bounded in absolute value by the symbol C(X). The space of all trigonometric polynomials 0.1.1 is denoted by 93; the space of all almost periodic continuous functions on R by 2f. For a normed complex linear space V, we denote the space of all bounded complex linear functionals on V by the symbol V*.

Fourier Multipliers for Certain Spaces of Functions with Compact Supports

acter on point measures. In the present paper, we replace the independent sets P by Cantor sets S containing many dependent elements. The behavior of multiplicative linear functionals for measures with support contained in the sets S is far more restricted than is the case for independent Cantor sets. Nevertheless some very curious multiplicative linear functionals can be constructed. Details are given in ? 4; the general theorem on which our constructions depend appears in ? 3. The present paper was suggested by the interesting paper [4] of Yu. A.

The Rôle of Compactness in Analysis

This is perhaps the final word on how large overall the Fourier coefficients of continuous periodic functions can be. (See the remarks in Paley [,11] on this point.) Helgason [--53, Theorem 2, generalized the Orlicz-Paley-Sidon theorem to all compact groups, Abelian and non-Abelian. An independent treatment of the compact Abelian case appears in Edwards [-3]. Hewitt and Ritter have given a slightly more general theorem for system of functions in [,6], w 4.

On a theorem of P. J. Cohen and H. Davenport

(1960). The Role of Compactness in Analysis. The American Mathematical Monthly: Vol. 67, No. 6, pp. 499-516.