Almost everywhere convergence of Fourier integrals revisited

A condition of proved worth guarantees almost everywhere convergence of Fourier integrals of functions from an essentially wider class than known earlier.     

Almost everywhere summability of multiple Walsh-Fourier series

We prove that for the N-dimensional Walsh-Fourier series the maximal operator of the Marcinkiewicz means is of weak type (1,1). Moreover, the Marcinkiewicz means σ_nf of the function f∈L¹ converge a.e. to f as n→∞.     

Almost everywhere convergence of inverse Fourier transforms

We show that if , then the inverse Fourier transform of converges almost everywhere. Here the partial integrals in the Fourier inversion formula come from dilates of a closed bounded neighbourhood of the origin which is star shaped with respect to 0. Our proof is based on a simple application of the Rademacher-Menshov Theorem. In the special case of spherical partial integrals, the theorem was proved by Carbery and Soria. We obtain some partial results when and . We also consider sequential convergence for general elements of .

     

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.     

The generalized localization for multiple Fourier integrals

In this paper we investigate almost-everywhere convergence properties of the Bochner-Riesz means of N-fold Fourier integrals under summation over domains bounded by the level surfaces of the elliptic polynomials. It is proved that if the order of the Bochner-Riesz means s?(N−1)(1/p−1/2), then the Bochner-Riesz means of a function f∈Lp(RN), 1?p?2 converge to zero almost-everywhere on RN?supp(f).     

On almost everywhere summability of conjugate Bochner-Riesz means

In this paper, we discuss the conjugate Bochner-Riesz means and get 4 theorems.     

The Lebesgue summability of trigonometric integrals

Motivated by the notion of Lebesgue summability of trigonometric series, we define the Lebesgue summability of trigonometric integrals in terms of the symmetric differentiability of the sum of the formally integrated trigonometric integral in question. We extend two theorems of Zygmund from trigonometric series to integrals, and one of them even in a more general form.