Geodesic
Approximation of functions of several variables, taking account of the growth of the coefficients of the approximating combinations
Sbornik. Mathematics, Tome 39 (1981) no. 1, pp. 133-143 
V. V. Napalkov. Approximation of functions of several variables, taking account of the growth of the coefficients of the approximating combinations. Sbornik. Mathematics, Tome 39 (1981) no. 1, pp. 133-143. http://geodesic.mathdoc.fr/item/SM_1981_39_1_a6/
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Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that every continuous function defined on the $n$-dimensional rectangular parallelepiped $\{x=(x_1,\dots,x_n)\in\mathbf R^n:0\leqslant x_i\leqslant a_i,\ 1\leqslant i\leqslant n\}$ can be approximated by polynomials of the form $Q(x)=\sum^p_{|\alpha|=0}c_\alpha x^\alpha$, where $c_\alpha=\eta_\alpha M(\alpha)$, with $\sum^p_{|\alpha|=0}|\eta_\alpha|\leqslant1$. Here $M(\alpha)$ is an arbitrary positive function defined on the set of multi-indices, and $\lim_{|\alpha|\to\infty}\sqrt[|\alpha|]{M(\alpha)}=\infty$. Bibliography: 9 titles.

[1] J. D. Stafney, “A permissible restriction on the coefficients in uniform polynomial approximation to $\mathcal C_{[0,1]}$”, Duke Math. J., 34:3 (1967), 393–396 | DOI | MR | Zbl

[2] S. Ya. Khavinson, “Dopustimye velichiny koeffitsientov mnogochlenov pri ravnomernoi approksimatsii nepreryvnykh funktsii”, Matem. zametki, 6:5 (1969), 619–625 | Zbl

[3] S. Ya. Khavinson, “Nekotorye voprosy polnoty sistem”, DAN SSSR, 137:4 (1961), 793–796 | Zbl

[4] O. A. Muradyan, “O roste koeffitsientov approksimiruyuschikh agregatov v teoreme Myuntsa”, Izv. AN Arm. SSR, seriya matem., 8:1 (1973), 70–87 | MR

[5] R. Edvards, Funktsionalnyi analiz, Moskva, 1969

[6] R. P. Boas, Entire functions, Academic Press, New York, 1954 | MR | Zbl

[7] L. Khermander, Vvedenie v teoriyu funktsii neskolkikh kompleksnykh peremennykh, izd-vo “Mir”, Moskva, 1968 | MR

[8] R. Narasimkhan, Analiz na deistvitelnykh i kompleksnykh mnogoobraziyakh, izd-vo “Mir”, Moskva, 1971

[9] V. S. Vladimirov, Metody teorii funktsii mnogikh kompleksnykh peremennykh, izd-vo “Nauka”, Moskva, 1964 | MR