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.