Convergence of Double Fourier Series after a Change of Variable

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