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 the convergence of multiple Walsh-Fourier series of functions of bounded generalized variation

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

Absolute convergence of multiple Fourier series revisited

In a recent paper [4], Gogoladze and Meskhia generalized the classical results of Bernstein, Szász, Zygmund and others related to absolute convergence of single trigonometric Fourier series. Our aim is to extend these results from single to multiple Fourier series. To this effect, we introduce the notions of multiplicative moduli of continuity and that of smoothness. Multiplicative Lipschitz classes of functions in several variables, and functions of bounded s-variation in the sense of Vitali are also considered.     

Application of Interpolational Methods to the Study of Properties of Functions of Several Variables

In this paper, interpolation theorems for spaces of functions of several variables are used to generalize and refine Hörmander's theorem on the multipliers of the Fourier transform from L _p to L _q and the Hardy--Littlewood--Paley inequality for a class of multiple Fourier series in the multidimensional case.     

Θ-summability of Fourier series

A general summability method of orthogonal series is given with the help of an integrable function Θ. Under some conditions on Θ we show that if the maximal Fejér operator is bounded from a Banach space X to Y, then the maximal Θ-operator is also bounded. As special cases the trigonometric Fourier, Walsh, Walsh--Kaczmarz, Vilenkin and Ciesielski--Fourier series and the Fourier transforms are considered. It is proved that the maximal operator of the Θ-means of these Fourier series is bounded from H _p to L _p (1/2<p≤; ∞) and is of weak type (1,1). In the endpoint case p=1/2 a weak type inequality is derived. As a consequence we obtain that the Θ-means of a function f∈L ₁ converge a.e. to f. Some special cases of the Θ-summation are considered, such as the Weierstrass, Picar, Bessel, Riesz, de la Vallée-Poussin, Rogosinski and Riemann summations. Similar results are verified for several-dimensional Fourier series and Hardy spaces.     

Absolute convergence of double series of Fourier-Haar coefficients for functions of bounded p-variation

We consider functions of two variables of bounded p-variation of the Hardy type on the unit square. For these functions we obtain a sufficient condition for the absolute convergence of series of positive powers of Fourier coefficients with power-type weights with respect to the double Haar system. This condition implies those for the absolute convergence of series of Fourier-Haar coefficients of one-variable functions which have a bounded Wiener p-variation or belong to the class Lip ??. We show that the obtained results are unimprovable. We also formulate N-dimensional analogs of the main result and its corollaries.     

On the kolmogorov theorems on Fourier series and conjugate functions

We consider Fourier series of summable functions from spaces ??wider?? than L ₁. We describe classes ??(L) which contain conjugate functions, where their conjugate Fourier series converge. The obtained results are more general than A. N. Kolmogorov theorems on the convergence of Fourier series in metrics weaker than that of L ₁.     

Convergence of fourier series with respect to multiplicative systems and the p-fluctuation continuity modulus

We obtain some sufficient conditions for convergence of series of the Fourier coefficients with respect to multiplicative systems for functions of bounded p-fluctuation. In some cases we establish the unimprovability of these conditions.     

Strong convergence theorems for Walsh–Fejér means

As main result we prove that Fejér means of Walsh–Fourier series are uniformly bounded operators from H _p to H _p (0<p≦1/2).     

A Dini type test on the pointwise convergence of double Fourier integrals

We give sufficient conditions for the convergence of the double Fourier integral of a complex-valued function f ∈ L ¹(?²) with bounded support at a given point (x ₀,y ₀) ∈ ?². It turns out that this convergence essentially depends on the convergence of the single Fourier integrals of the marginal functions f(x,y ₀), x ∈ ?, and f(x ₀,y), y ∈ ?, at the points x:= x ₀ and y:= y ₀, respectively. Our theorem applies to functions in the multiplicative Zygmund classes of functions in two variables.     

Convergence and localization of multiple Fourier series for classes of bounded Λ-variation

Previously obtained results for convergence and localization of multiple trigonometric Fourier series for functions from classes of bounded Λ-variation and embedding of these classes into each other are strengthened in the paper. The case when sequences Λ and M have a limit of the ratio Σ _n=1 ^N 1/λ _n /Σ _n=1 ^N 1/µ _n is considered. A more strict condition, the existence of a limit for the ratio λ _n /µ _n was considered before.     

Stability of the solution of the Cauchy problem for the Laplace equation on a weak compactum

The paper investigates the stability of the Cauchy problem for the Laplace equation under the a priori assumption that the solution is bounded. A special metrization of the weak topology in the space L₂ and the standard Fourier series technique are applied to obtain stability bounds for the solution of the Cauchy problem on the class of absolutely bounded functions.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 44–50, 1985.     

A quantified version of the Dirichlet-Jordan test inL 1-norm

We introduce the notion of bounded variation in the sense ofL ¹-norm for periodic functions and prove a version of the classical Dirichlet-Jordan test for the convergence of Fourier series inL ¹-norm. We also give an estimate of the rate of convergence.     

Pringsheim type tests on the pointwise convergence of integrals conjugate to double fourier integrals

We prove sufficient conditions for the convergence of the integrals conjugate to the double Fourier integral of a complex-valued function f ∈ L ¹ (?²) with bounded support at a given point (x ₀, g ₀) ∈ ?². It turns out that this convergence essentially depends on the convergence of the integral conjugate to the single Fourier integral of the marginal functions f(x, y ₀), x ∈ ?, and f(x ₀, y), y ∈ ?, at x:= x ₀ and y:= y ₀, respectively. Our theorems apply to functions in the multiplicative Lipschitz and Zygmund classes introduced in this paper.     

Sufficient conditions for the Lebesgue integrability of Fourier transforms

We give sufficient conditions for the Lebesgue integrability of the Fourier transform of a function f ∈ L ^p (?) for some 1 < p ≤ 2. These sufficient conditions are in terms of the L ^p integral modulus of continuity of f; in particular, they apply for functions in the integral Lipschitz class Lip(α, p) and for functions of bounded s-variation for some 0 < s < p. Our theorems are nonperiodic versions of the classical theorems of Bernstein, Szász, Zygmund and Salem, and recent theorems of Gogoladze and Meskhia on the absolute convergence of Fourier series.     

Pointwise Behavior of Double Fourier Series of Functions of Bounded Variation

We extend some recent results of S. A. Telyakovskii on the uniform boundedness of the partial sums of Fourier series of functions of bounded variation to periodic functions of two variables, which are of bounded variation in the sense of Hardy. As corollaries, we obtain the classical Parseval formula, the convergence theorem of the series involving the sine Fourier coefficients, and a lower estimate of the best approximation by trigonometric polynomials in the metric of L in a sharpened version.     

Marcinkiewicz-summability of multi-dimensional Fourier transforms and Fourier series

A generalization of Marcinkiewicz-summability of multi-dimensional Fourier transforms and Fourier series is investigated with the help of a continuous function θ. Under some weak conditions on θ we show that the maximal operator of the Marcinkiewicz-θ-means of a tempered distribution is bounded from Hp(Xd) to Lp(Xd) for all d/(d+α)<p?∞ and, consequently, is of weak type (1,1), where 0<α?1 is depending only on θ and X=R or X=T. As a consequence we obtain a generalization of a summability result due to Marcinkiewicz and Zhizhiashvili for d-dimensional Fourier transforms and Fourier series, more exactly, the Marcinkiewicz-θ-means of a function f∈L₁(Xd) converge a.e. to f. Moreover, we prove that the Marcinkiewicz-θ-means are uniformly bounded on the spaces Hp(Xd) and so they converge in norm (d/(d+α)<p<∞). Similar results are shown for conjugate functions. Some special cases of the Marcinkiewicz-θ-summation are considered, such as the Fejér, Cesàro, Weierstrass, Picar, Bessel, de La Vallée-Poussin, Rogosinski and Riesz summations.