Remarks on Fourier series

We prove the following propositions. An even integrable function whose Fourier coefficients form a convex sequence is absolutely continuous if and only if its Fourier series converges absolutely. If the function f(t)is convex on [0, ],f(t)=f(—t), then for odd n while for even n, b₀=0.Translated from Matematicheskie Zametki, Vol. 3, No. 5, pp. 597–604, May, 1968.     

On Bohr's theorem for multiple Fourier series

The problem of Pringsheim uniform convergence of multiple Fourier series in the trigonometric system is considered. A multidimensional analog of Bohr's theorem on the uniform convergence of the Fourier series of a continuous function after a homeomorphic chance of variable is proved.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 913–924, December, 1998.     

Strong spherical means of multiple Fourier series

The present paper is a survey of some results related to the strong means of spherical partial sums, their integral analogs and the Bochner-Riesz sums of Fourier series of functions defined almost everywhere on the m-dimensional torus.     

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.     

The Littlewood-Paley theory for multiple Fourier series

We study the Littlewood-Paley theory for multiple Fourier series with arbitrary period lattice. It is shown that the constants in the Littlewood-Paley inequality can be chosen to be independent of the mutual arrangement of the period lattice and the set of dyadic parallelepipeds. Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 155–169.     

Paley inequality for Bellman transform of multiple Fourier series

A Hardy-Littlewood-Paley inequality is obtained for Bellman type means of Fourier coefficients of functions from anisotropic Lorentz spaces.     

Summability of multiple trigonometric Fourier series by linear methods

The paper deals with the problem of estimation of deviations of functions of several variables from linear means of their multiple trigonometric Fourier series. An approach of reducing this problem to the corresponding problem for functions of single variable is developed.     

Strong summability of multiple Fourier series and Sidon-type inequalities

We study different versions of strong summation of N-dimensional Fourier series over polyhedrons and related estimates for integral norms of linear means of the Dirichlet kernels (Sidon-type inequalities). Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1630–1635, December, 1998.     

On strong Carleman means of multiple trigonometric Fourier series

We prove the strong Carleman summability of the Fourier series of continuous functions on the m-dimensional torus, with partial sums constructed over polyhedra of a certain class.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 275–279, February, 1992.