Convergence of multi-dimensional integral operators and applications

Abstract

In this paper we investigate a general multi-dimensional integral operator \(V_{T}\). Under the condition that the kernel function of \(V_{T}\) is in a suitable Herz space, we get several convergence theorems about norm and almost everywhere convergence and convergence at Lebesgue points. The multi-dimensional convergence is investigated over cones and cone-like sets. As special cases we consider three multi-dimensional integral operators, the \(\theta \)-summation of Fourier transforms and Fourier series and the discrete wavelet transforms. The convergence results are formulated for functions from the Wiener amalgam spaces and variable Lebesgue spaces, too.