Sequences of rectangular Fourier sums of continuous functions with given majorants of the mixed moduli of smoothness

Approximation by the rectangular Fourier sums $S_N(f)$ is studied for classes of functions of several variables defined in terms of the orders of decrease of the mixed moduli of smoothness. The problem of the existence of a single (independent of $N$) function $f$ on which the order of the approximation in the corresponding class is realized is solved.