Uniform Convergence of Hyperbolic Partial Sums of Multiple Fourier Series

It follows from results of A. Yudin, V. Yudin, E. Belinskii, and I. Liflyand that if $m\ge2$ and a $2\pi$-periodic (in each variable) function $f(\mathbf x)\in C(T^m)$ belongs to the Nikol'skii class $h_\infty^{(m-1)/2}(T^m)$, then its multiple Fourier series is uniformly convergent over hyperbolic crosses. In this paper, we establish the finality of this result. More precisely, there exists a function in the class $h_\infty^{(m-1)/2}(T^m)$ whose Fourier series is divergent over hyperbolic crosses at some point.