In 1967 R.P. Boas Jr. found necessary and sufficient conditions of belonging of a function to a Lipschitz class. Later Boas’s findings were generalized by many authors (M. and S. Izumi (1969), L.-Y. Chan (1991) and others). Recently, L. Leindler (2000) and J. Nemeth (2001) have published two papers, in which they have generalized all the previous results. The authors have considered the case, when the order of modulus of smoothness equals one (L. Leindler) or two (J. Nemeth). In this paper, we prove theorems of Boas-type for the modulus of smoothness of any order. Furthermore, we solve the inverse problem. Also, we discuss some conditions on a majorant which are equivalent to the well-known conditions of Bari-Stechkin.

https://www.ems-ph.org/journals/show_pdf.php?issn=0232-2064&vol=23&iss=4&rank=6

On Generalized Lipschitz Classes and Fourier Series

We investigate the order of magnitude of the modulus of continuity of a function f with absolutely convergent Fourier series. We give sucient conditions in terms of the Fourier coecients in order that f belong to one of the generalized Lipschitz classes Lip(;L ) and Lip(; 1=L), where 0 1 and L = L(x) is a positive, nondecreasing, slowly varying function such that L(x) ! 1 as x ! 1. For example, a 2 -periodic function f is said to belong to the class Lip(;L ) if jf(x +h) f(x)j Ch L(1=h) for all x2 T; h > 0;

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Absolutely convergent fourier series and generalized lipschitz classes of functions

We establish Wiener type theorems and Paley type theorems for Laguerre polynomial expansions and disk polynomial expansions with nonnegative coefficients.

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Laguerre and Disk Polynomial Expansions with Nonnegative Coefficients

In this paper we consider spaces that are “close” to L(T): L itself; the space of functions f with positive Fourier coefficients that have |f |p integrable near 0; the space of functions whose Fourier coefficients are in p ; the space of functions whose Fourier coefficients {cn} satisfy ∑|cn|pnp−2 < ∞; and the mixed norm spaces p ,2, 1 < p < 2. We shall describe several relationships between these spaces. Let T be the interval [−π,π]. For every 1 ≤ p < ∞, we say that a measurable function f is in L = L(T) if ‖f ‖p p = 1 2π ∫

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Wiener's positive Fourier coefficients theorem in variants of Lp spaces

Local and global properties of functions and their Fourier transforms

Extending a result of N. Wiener, it is shown that functions on the circle with positive Fourier coefficients that are pth power integrable near 0, 1 < P < 2, have Fourier coefficients in lp . The following result was proved (but never published) by Norbert Wiener in the early 1950's. (See [1, pp. 242, 250] and [3].) WIENER'S THEOREM. // YLcneint is the Fourier series of a function f € L1(—7T, 7t) with cn > 0 for all n, and f restricted to a neighborhood (—6,6) of the origin belong to L2(—6,6), then f belongs to L2(—tt,tt). A question which immediately arises in connection with this result is the following: does the theorem remain true if one replaces L2(—6,6) and L2(—tt,tt) in its statement respectively by Lp(—6,6) and Lp(—-k,tt), with 1 < p < oo? In 1969 Stephen Wainger showed, by ingenious counterexamples, that the answer is negative for 1 < p < 2 [4]. If p is an even integer or oo it is very easy to see that the answer is "yes." For every other exponent between 2 and oo it is "no," as was shown in 1975 by Harold S. Shapiro [3]. These negative results have been extended to compact abelian groups [2]. However, the conclusion of Wiener's theorem can be stated equivalently as "then J2 cn < °°-" This suggested the following theorem. THEOREM. If¿~2cneint is the Fourier series of a function f S Lx(—tt, tt) with cn > 0 for all n, and f restricted to a neighborhood (—6,6) of the origin belongs to Lp(—6,6) with 1 < p < 2, then ^cn < °°> where p' = p/(p — 1). PROOF. (See [3, p. 12].) Let h(t) be the 27r-periodic function which for |t| < -k is defined by M-KIA M Si, I 0, 6 < \t\ < it. Received by the editors March 2, 1987. 1980 Mathematics Subject Classification. Primary 42A32, 42A16; Secondary 43A15. 1 The research presented here was supported in part by a grant from the University Research Council of DePaul University. ©1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page

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Michael Rains

Papers

Fourier series with positive coefficients

Shorter Notes: Fourier Series with Positive Coefficients