Optimal information for approximating periodic analytic functions

Abstract:

Let $S_{\beta }:=\{z\in \mathbb {C}:| \operatorname {Im}z|<\beta \}$ be a strip in the complex plane. For fixed integer $r\ge 0$ let $H^r_{\infty ,\beta }$ denote the class of $2\pi$-periodic functions $f$, which are analytic in $S_{\beta }$ and satisfy $|f^{(r)}(z)|\le 1$ in $S_{\beta }$. Denote by $H^{r,\mathbb R}_{\infty ,\beta }$ the subset of functions from $H^r_{\infty ,\beta }$ that are real-valued on the real axis. Given a function $f\in H^r_{\infty ,\beta }$, we try to recover $f(\zeta )$ at a fixed point $\zeta \in \mathbb {R}$ by an algorithm $A$ on the basis of the information \begin{equation*}If=(a_{0}(f),a_{1}(f),\dots ,a_{n-1}(f),b_{1}(f),\dots ,b_{n-1}(f)), \end{equation*} where $a_{j}(f)$, $b_{j}(f)$ are the Fourier coefficients of $f$. We find the intrinsic error of recovery \begin{equation*} E(H^r_{\infty ,\beta } ,I):= \operatorname * {inf}_{A\colon \mathbb {C}^{2n-1}\to \mathbb {C}} \sup _{f\in H^r_{\infty ,\beta } }|f(\zeta )- A(If)|. \end{equation*} Furthermore the $(2n-1)$-dimensional optimal information error, optimal sampling error and $n$-widths of $H^{r,\mathbb R}_{\infty ,\beta }$ in $C$, the space of continuous functions on $[0,2\pi ]$, are determined. The optimal sampling error turns out to be strictly greater than the optimal information error. Finally the same problems are investigated for the class $H_{p,\beta }$, consisting of all $2\pi$-periodic functions, which are analytic in $S_{\beta }$ with $p$-integrable boundary values. In the case $p=2$ sampling fails to yield optimal information as well in odd as in even dimensions.