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Abstract  

We study when sums of trigonometric series belong to given function classes. For this purpose we describe the Nikol’skii class of functions and, in particular, the generalized Lipschitz class. Results for series with positive and general monotone coefficients are presented.

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Abstract.

We consider complex-valued functions fL 1(ℝ+), where ℝ+:=[0,∞), and prove sufficient conditions under which the sine Fourier transform and the cosine Fourier transform belong to one of the Lipschitz classes Lip (α) and lip (α) for some 0<α≦1, or to one of the Zygmund classes Zyg (α) and zyg (α) for some 0<α≦2. These sufficient conditions are best possible in the sense that they are also necessary if f(x)≧0 almost everywhere.

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Abstract

We consider the double Walsh orthonormal system

ea
on the unit square , where {w m(x)} is the ordinary Walsh system on the unit interval in the Paley enumeration. Our aim is to give sufficient conditions for the absolute convergence of the double Walsh–Fourier series of a function for some 1<p≦2. More generally, we give best possible sufficient conditions for the finiteness of the double series
eb
where {a mn} is a given double sequence of nonnegative real numbers satisfying a mild assumption and 0<r<2. These sufficient conditions are formulated in terms of (either global or local) dyadic moduli of continuity of f.

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Abstract  

Let f: R NC be a periodic function with period 2π in each variable. We prove suffcient conditions for the absolute convergence of the multiple Fourier series of f in terms of moduli of continuity, of bounded variation in the sense of Vitali or Hardy and Krause, and of the mixed partial derivative in case f is an absolutely continuous function. Our results extend the classical theorems of Bernstein and Zygmund from single to multiple Fourier series.

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Abstract  

We describe the functions from Nikol’skii class in terms of behavior of their Fourier coefficients. Results for series with general monotone coefficients are presented. The problem of strong approximation of Fourier series is also studied.

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