On the convergence of double Fourier series of discontinuous functions

In the paper we obtain a formula for the computation of the rectangular partial sums of a double Fourier series of a function of two variables at the point (0, 0), having a discontinuity along any ray at this point. We also give an estimate of the remainder in the abovementioned formula.     

Generalized localization and convergence tests for double trigonometric Fourier series of functions fromL p,p>1

. E , f(x)L _p (T ^N ),P1,f(x)=0 E (E— ^N =[-, ]^N) E , . , .

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In closing the author thanks V. A. Il'in and . A. Alimov for their constant attention paid to the present work.     

On the almost everywhere convergence of double Fourier series of integrable functions

, . . Q _k [0,2],k=1,2, — . F(x, y)L(T), T=[0, 2]², G(x, y)L(T) , G(x,y)=F(x,y) Q=Q ₁ ×Q ₂ - .     

On the convergence of double Fourier series of functions in L P[0, 1]2, where p=(p 1, p 2)

Pointwise convergence of double trigonometric Fourier series of functions in the Lebesgue space L _p[0,2]²was studied by M. I. Dyachenko. In this paper, we also consider the problems of the convergence of double Fourier series in Pringsheim"s sense with respect to the trigonometric as well as the Walsh systems of functions in the Lebesgue space L _p[0,1]², p=(p₁,p₂), endowed with a mixed norm, in the particular case when the coefficients of the series in question are monotone with respect to each of the indices. We shall obtain theorems which generalize those of M. I. Dyachenko to the case when p is a vector. We shall also show that our theorems in the case of trigonometric Fourier series are best possible.     

On convergence of Fourier series of Besicovitch almost periodic functions

The paper deals with convergence of the Fourier series of q-Besicovitch almost periodic functions of the form $$ f(t)\sim \mathop{\sum}\limits_{m=1}^{\infty }{a_m}{{\mathrm{e}}^{{-\mathrm{i}{\uplambda_m}t}}}, $$ where {λm} is a Dirichlet sequence, that is, a strictly increasing sequence of nonnegative numbers tending to infinity. In particular, we show that, for 1 < q < ∞, the Fourier series of f(t) converges in norm to the function f(t) itself with usual order, which is analogous to the convergence in norm of the Fourier series of a function on [0, 2π]. A version of the Carleson–Hunt theorem is also investigated.     

On almost everywhere convergence of Marcinkiewicz sums of double Fourier series

Let f be a function summable on the two-dimensional torus with Fourier series . The Marcinkiewicz means . where is a function defined on [0, 1], are considered. The following theorem is proved. Let > 0 and assume that the function , concave on [0, 1], is such that (0)=1, (1)=0 and its modulus of continuity satisfies the relation (,h)=0 (log^–2–(1+1/k)). Then for almost all x.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 148–156, 1991.     

Convergence of double Fourier series of functions from symmetric spaces

We establish conditions for the convergence of double Fourier series in the trigonometric system of functions belonging to a symmetric space.     

On the convergence of Fourier series in the metric ofL 1

L ¹ .      

On the convergence of double orthogonal series

={_i()} ={_i(y)} (i=1,2,...) — (, ) . (,)&#x0425;(,) {_i(x)_k(y))} (i, k=1, 2, ...). (1.1) , . , , . , , . (i) (1.3), .. « » (1.1), (1.7) . , (ii) « » (1.4) (1.1), (1.8) .     

On the convergence of multiple Fourier series of functions of bounded partial generalized variation

The convergence of multiple Fourier series of functions of bounded partial Λ-variation is investigated. The sufficient and necessary conditions on the sequence Λ = {λ _n } are found for the convergence of multiple Fourier series of functions of bounded partial Λ-variation.     

On convergence and summability of Fourier series

. , , –1<<0. .

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The present work was written on the basis of two earlier works received byAnalysis Mathematica on January 16, 1979, and July 20, 1979.