Approximation of functions of several variables, taking account of the growth of the coefficients of the approximating combinations

V. V. Napalkov

Sbornik. Mathematics, Tome 39 (1981) no. 1, pp. 133-143

Résumé

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.