Multiple fourier series and integrals

One gives a brief survey of the investigations on the theory of multiple Fourier series and integrals, reviewed in Referativnyi Zhurnal Matematika in the period 1953–1980. Principal attention is given to the following questions: localization principles, uniform convergence and summability, convergence and summability at a point, in the L_p metric, and almost everywhere, absolute convergence, uniqueness theorems, conjugate Fourier series and integrals, equiconvergence and equisummability of Fourier series and integrals, properties of the kernel and of the Lebesgue constant of summation methods of Fourier series, Fourier coefficients and Fourier transforms.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 19, pp. 3–54, 1982.     

On absolute convergence of multiple fourier series

We obtain sufficient conditions for β-absolute convergence (0 < β ≤ 1) of multiple Fourier series of functions of the classes $L^2 ([0,2\pi ]^N ),(\Lambda ^1 ,\Lambda ^2 ,...,\Lambda ^N )BV^{(p)} ([0,2\pi ]^N ),r - BV([0,2\pi ]^N )$ .     

A method of summation of multiple trigonometric series

Institute of Mathematics and Cybernetics, Academy of Sciences of the Lithuanian SSR. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 29, No. 3, pp. 615–620, July–September, 1989.     

On the absolute convergence of multiple fourier series

Let f: R ^N → C 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. This research was started while the first author was a visiting professor at the Department of Mathematics, Texas A&M University, College Station during the fall semester in 2005; and it was also supported by the Hungarian National Foundation for Scientific Research under Grant T 046 192.     

Absolutely convergent multiple fourier series and multiplicative Lipschitz classes of functions

We consider N-multiple trigonometric series whose complex coefficients c _j1,...,j _N , (j ₁,...,j _N ) ∈ ? ^N , form an absolutely convergent series. Then the series $$ \sum\limits_{(j_1 , \ldots ,j_N ) \in \mathbb{Z}^N } {c_{j_1 , \ldots j_N } } e^{i(j_1 x_1 + \ldots + j_N x_N )} = :f(x_1 , \ldots ,x_N ) $$ converges uniformly in Pringsheim’s sense, and consequently, it is the multiple Fourier series of its sum f, which is continuous on the N-dimensional torus $ \mathbb{T} $ ^N , $ \mathbb{T} $ := [?π, π). We give sufficient conditions in terms of the coefficients in order that >f belong to one of the multiplicative Lipschitz classes Lip (α₁,..., α _N ) and lip (α₁,..., α _N ) for some α₁,..., α _N > 0. These multiplicative Lipschitz classes of functions are defined in terms of the multiple difference operator of first order in each variable. The conditions given by us are not only sufficient, but also necessary for a special subclass of coefficients. Our auxiliary results on the equivalence between the order of magnitude of the rectangular partial sums and that of the rectangular remaining sums of related N-multiple numerical series may be useful in other investigations, too.     

Spherical jump of a function and the Bochner-Riesz means of conjugate multiple fourier series and fourier integrals

Mathematical Notes - We introduce the notion of spherical jump of a function of several variables at a given point with respect to a homogeneous harmonic polynomial. Here, if the function is...     

Convergence of multiple fourier series for functions of bounded variation

For functions of bounded variation in the sense of Hardy, we consider the pointwise convergence of the partial sums of Fourier series over a given sequence of bounded sets in the space of harmonics. We obtain sufficient conditions for convergence; necessary and sufficient conditions are obtained for the case in which these sets are convex with respect to each coordinate direction. The Pringsheim convergence of Fourier series in this problem was established by Hardy. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 583–595, April, 1997. Translated by S. A. Telyakovskii and V. N. Temlyakov     

Approximation of fourier integral by Bochner-Riesz means under one-side condition

In this paper we study the approximation of Fourier integral by Bochner-Riesz means under one-side condition and improve the results in[1].     

Multiple Eisenstein series and multiple cotangent functions

We construct the multiple Eisenstein series and we show a relation to the multiple cotangent function. We calculate a limit value of the multiple Eisenstein series.