Abstract
We continue with the theme studied in [2] and examine certain questions pertaining to best methods to recover classes of periodic functions from function evaluations. The main feature that distinguishes this work from [2] is that we are concerned with functions of more than one variable, and this leads us to difficulties not encountered in [2]. We treat periodic function classes similar to those studied in [2], which, because of their greater generality, include those bivariate spaces appearing in [7]. Nevertheless, we follow [7] and [3,4,1] and show how blending interpolation operators yield better error estimates, per function evaluation, than the available tensor product methods. Surprisingly, however, we will also demonstrate that tensor products of equally spaced sampling schemes are optimal in a wide variety of cases. Outstanding unresolved problems remain in determining optimal sampling schemes even for standard spaces of periodic functions. This will be mentioned at the end of the paper.
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References
G. Baszenski and F. J. Delvos, Boolean methods in Fourier approximation, in Topics in Multivariate Approximation, C. K. Chui, L. L. Schumaker, and F. Utreras (eds.), Academic Press, N.Y., 1989.
W. Dahmen, C. A. Micchelli, and P. W. Smith, Asymptotically optimal sampling schemes for periodic functions, Math. Proc. Camb. Phil. Soc., 99 (1986), 171–177.
F. J. Delvos, d-Variate Boolean interpolation, J. Approx Theory, 34 (1982), 99–114.
F. J. Delvos and H. Posdorf, Nth order blending, in Constructive Theory of Functions of Several Variables, W. Schempp and K. Zeller (eds.), Springer Verlag, Heidelberg, 1976.
Dinh Dung, Number of integral points in a certain set and the approximation of functions of several variables, Math. Notes, 36 (1984), 736–744.
C. A. Micchelli and T. J. Rivlin, A survey of optimal recovery, in Optimal Estimation in Approximation Theory, C. A. Micchelli and T. J. Rivlin (eds.), Plenum Press, 1976.
G. Wahba, Interpolating surfaces: High order convergence rates and their associated designs, with application to X-ray image reconstruction, Report #523, Univ. of Wisconsin Stat., Madison, 1978.
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© 1989 Birkhäuser Verlag Basel
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Chen, Hl., Chui, C.K., Micchelli, C.A. (1989). Asymptotically Optimal Sampling Schemes for Periodic Functions II: The Multivariate Case. In: Multivariate Approximation Theory IV. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 90. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-7298-0_8
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DOI: https://doi.org/10.1007/978-3-0348-7298-0_8
Publisher Name: Birkhäuser Basel
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