Geodesic
Rational approximation of functions of several variables with finite Hardy variation
Sbornik. Mathematics, Tome 186 (1995) no. 11, pp. 1599-1620

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The rate of rational approximation of functions of $N$ variables with given modulus of continuity and bounded Hardy variation on the unit N-cube $[0,1]^N$ is considered. In particular, if a function $f(x)$ on $[0,1]^N$ has bounded Hardy variation and $f \in\operatorname{Lip}\alpha$, $0\alpha1$ then it can be seen from the central result of this paper that $$ R_n(f,[0,1]^N)\leqslant C\frac{\ln^2 n}n\,. $$ 
@article{SM_1995_186_11_a2,
     author = {A. P. Bulanov},
     title = {Rational approximation of functions of several variables with finite {Hardy} variation},
     journal = {Sbornik. Mathematics},
     pages = {1599--1620},
     publisher = {mathdoc},
     volume = {186},
     number = {11},
     year = {1995},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1995_186_11_a2/}
}
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A. P. Bulanov. Rational approximation of functions of several variables with finite Hardy variation. Sbornik. Mathematics, Tome 186 (1995) no. 11, pp. 1599-1620. http://geodesic.mathdoc.fr/item/SM_1995_186_11_a2/