Convergence of Double Fourier Series after a Change of Variable

Abstract

In this paper, we prove that for any compact set \(\Omega \subset C({\mathbb{T}}^2 )\) there exists a homeomorphism τ of the closed interval \({\mathbb{T}} = [ - \pi ,\pi ]\) such that for an arbitrary function f ∈ Ω the Fourier series of the function F(x,y) = f(τ(x),τ(y)) converges uniformly on \(C({\mathbb{T}}^2)\) simultaneously over rectangles, over spheres, and over triangles.