A theorem of Bohr for multiple trigonometric series (Q752389)

The following theorem due to Bohr [see \textit{B. S. Kashin} and the author, Orthogonal series (Russian, 1984; Zbl 0632.42017; English transl., 1989; Zbl 0668.42011)] is well known: For an arbitrary \(f\in C(-\pi,\pi)\), \(f(- \pi)=f(\pi)\), there exists a homeomorphism \(\tau\) (x) of the interval \([- \pi,\pi]\) (i.e., a one-to-one continuous mapping of the interval \([- \pi,\pi]\) onto itself) such that the Fourier series of the superposition \(f\circ \tau (x)\) converges uniformly on \([-\pi,\pi]\). Here we prove an analogue of this theorem in the multidimensional case using a multidimensional analogue of a Faber-Schauder system.