Best mean-square approximation of functions of $m$ variables

V. Yu. Popov

Matematičeskie zametki, Tome 14 (1973) no. 6, pp. 913-924

Résumé

Let $E_\sigma(f;l_q)_{L_2(R_m)}$ be the best mean-square approximation of a function $f(x)\in L_2(R_m)$ ($m=1,2,\dots$) by integral functions of the exponential spherical type (in the sense of the $l_q$, $0<q\le\infty$) with $\sigma>0$, $\omega(f;\pi/\sigma;l_p)_{L_2(R_m)}$ is the spherical (in the sense of the metric $l_p$, $0<p\le\infty$) continuity module of the function $f(x)\in L_2(R_m)$. For the quantity $C_\sigma(m;q;p)=\sup\limits_{f\in L_2}\{E_\sigma(f;l_q):\omega(f;\pi/\sigma;l_p)\}$ two-sided estimates are obtained which are uniform in the parameters $m$, $q$, and $p$. Similar results are also obtained in the case of $q=p=2$ for classes of functions $W_2^\rho(R_m)$ ($\rho=1,2,\dots$).