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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.

References [Enhancements On Off] (What's this?)

  • 1. N. I. Achieser, Vorlesungen über Approximationstheorie, Akademie-Verlag, Berlin, 1953. MR 15:867c
  • 2. N. I. Achieser, Elements of the Theory of Elliptic Functions, Nauka, Moscow, 1970 (Russian).MR 44:5517
  • 3. Bateman Manuscript Project, Higher Transcendental Functions, Vol. II, McGraw-Hill, New York, 1953.MR 15:419i
  • 4. S. D. Fisher, Function Theory on Planar Domains: A Second Course in Complex Analysis, Wiley-Interscience, New York, 1983.MR 85d:30001
  • 5. S. D. Fisher and C. A. Micchelli, The $n$-width of sets of analytic functions, Duke Math. J. 47 (1980), 789-801.MR 82b:30035
  • 6. S. D. Fisher and C. A. Micchelli, Optimal sampling of holomorphic functions, Amer. J. Math. 106 (1984), 593-609.MR 85j:30008
  • 7. S. D. Fisher and C. A. Micchelli, Optimal sampling of holomorphic functions. II, Math. Ann. 273 (1985), 131-147.MR 87b:30009
  • 8. C. A. Micchelli and T. J. Rivlin, A survey of optimal recovery, Optimal Estimation in Approximation Theory (C. A. Micchelli and T. J. Rivlin, eds.), Plenum Press, New York, 1977, pp. 1-54.MR 58:29718
  • 9. C. A. Micchelli and T. J. Rivlin, Lectures on optimal recovery, Lecture Notes in Mathematics, vol. 1129, Springer-Verlag, Berlin, 1985, pp. 21-93. MR 87a:41036
  • 10. K. Yu. Osipenko, Best approximation of analytic functions from information about their values at a finite number of points, Mat. Zametki 19 (1976), 29-40; English transl. in Math. Notes 19 (1976), 17-23. MR 56:12271
  • 11. K. Yu. Osipenko, On $n$-widths, optimal quadrature formulas, and optimal recovery of functions analytic in a strip, Izv. Ross. Akad. Nauk, Ser. Mat. 58 (1994), 55-79; English transl. in Russian Acad. Sci. Izv. Math. 45 (1995), 55-78.MR 96e:30091
  • 12. K. Yu. Osipenko, On $n$-widths of holomorphic functions of several variables, J. Approx. Theory 82 (1995), 135-155. MR 96d:32005
  • 13. K. Yu. Osipenko, Exact $n$-widths of Hardy-Sobolev classes, Constr. Approx. 13 (1997), 17-27. CMP 97:05
  • 14. M. P. Ovchincev, Best methods for approximating regular bounded functions in an annulus by their values in given points, Izv. Vuzov. Mat. 5 (1989), 32-39 (Russian).
  • 15. A. Pinkus, $n$-Widths in Approximation Theory, Springer-Verlag, Berlin, 1985. MR 86k:41001
  • 16. J. F. Traub and H. Wozniakowski, A General Theory of Optimal Algorithms, Academic Press, New York, 1980. MR 84m:68041
  • 17. K. Wilderotter, Optimal recovery of bounded analytic functions in an annulus, Optimal Recovery (B. Bojanov and H. Wozniakowski, eds.), Nova Science Publishers, New York, 1992, pp. 311-323. MR 94d:30072
  • 18. K. Wilderotter, On $n$-widths of bounded periodic holomorphic functions, Ukrainian Math. J. 47 (1995), 1170-1175. MR 96k:41037
  • 19. K. Wilderotter, Optimal sampling of periodic analytic functions, J. Approx. Theory 82 (1995), 304-316. MR 96h:30077

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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

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