Geodesic
Sequences of rectangular Fourier sums of continuous functions with given majorants of the~mixed moduli of smoothness
Sbornik. Mathematics, Tome 187 (1996) no. 7, pp. 981-1004

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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. 
@article{SM_1996_187_7_a1,
     author = {O. V. Davydov},
     title = {Sequences of rectangular {Fourier} sums of continuous functions with given majorants of the~mixed moduli of smoothness},
     journal = {Sbornik. Mathematics},
     pages = {981--1004},
     publisher = {mathdoc},
     volume = {187},
     number = {7},
     year = {1996},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1996_187_7_a1/}
}
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O. V. Davydov. Sequences of rectangular Fourier sums of continuous functions with given majorants of the~mixed moduli of smoothness. Sbornik. Mathematics, Tome 187 (1996) no. 7, pp. 981-1004. http://geodesic.mathdoc.fr/item/SM_1996_187_7_a1/