The approximation of functions of many variables by their F\'ej\'er sums

V. A. Yudin

Matematičeskie zametki, Tome 13 (1973) no. 6, pp. 817-828.

Résumé

We construct elliptic Féjér polynomials $K_n(x)$ of $m$ variables. We prove some of their properties: a) the Féjér polynomials are positive on the $m$-dimensional torus $T^m$, $K_n(x)\ge0$, b) $\min\limits_{x\in T^m}K_n(x)=O(n^{-1})$, as $n\to\infty$, c) we calculate their norms in the spaces $L[T^m]$ and $C[T^m]$. We estimate the deviation of the Féjér sum $\sigma_n(x,f)$ from the function $f(x)$. For the space $C[T^m]$: $$ \sup_{f\in K\operatorname{Lip}\{\alpha,C[T^m]\}}\|f(x)-\sigma_n(x,f)\|_{C[T^m]}= \begin{cases} c_{\alpha,m}n^{-\alpha}+O(n^{-1}),0\alpha1,\\c_{1,m}n^{-1}\ln n+O(n^{-1}),\alpha=1, \end{cases} $$ where $c_{\alpha,m}$ , $c_{1,m}$ are constants.