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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Optimal information for approximating
periodic analytic functions

Authors: K. Yu. Osipenko and K. Wilderotter
Journal: Math. Comp. 66 (1997), 1579-1592
MSC (1991): Primary 65E05, 41A46; Secondary 30E10
MathSciNet review: 1433267
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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):=% \operatornamewithlimits {inf\vphantom p}_{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.

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Additional Information

K. Yu. Osipenko
Affiliation: Department of Mathematics, Moscow State University of Aviation Technology, Petrovka 27, Moscow 103767, Russia

K. Wilderotter
Affiliation: Mathematisches Seminar der Landwirtschaftlichen Fakultät, Rheinische Friedrich–Wilhelms–Universität Bonn, Nußallee 15, 53115 Bonn, Germany

Keywords: Optimal recovery, optimal information, periodic Blaschke products
Received by editor(s): March 25, 1996
Additional Notes: The first author was supported in part by RFBR Grant #96-01-00325.
Article copyright: © Copyright 1997 American Mathematical Society