Limits of interpolatory processes

Author:
W. R. Madych

Journal:
Trans. Amer. Math. Soc. **355** (2003), 1109-1133

MSC (2000):
Primary 41A05, 41A15

Published electronically:
October 25, 2002

MathSciNet review:
1938748

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Abstract | References | Similar Articles | Additional Information

Abstract: Given distinct real numbers and a positive approximation of the identity , which converges weakly to the Dirac delta measure as goes to zero, we investigate the polynomials which solve the interpolation problem

with prescribed data . More specifically, we are interested in the behavior of when the data is of the form for some prescribed function . One of our results asserts that if is sufficiently nice and has sufficiently well-behaved moments, then converges to a limit which can be completely characterized. As an application we identify the limits of certain fundamental interpolatory splines whose knot set is , where is an arbitrary finite subset of the integer lattice , as their degree goes to infinity.

**1.**Carl de Boor,*Odd-degree spline interpolation at a biinfinite knot sequence*, Approximation theory (Proc. Internat. Colloq., Inst. Angew. Math., Univ. Bonn, Bonn, 1976) Springer, Berlin, 1976, pp. 30–53. MR**0613677****2.**Carl de Boor, Klaus Höllig, and Sherman Riemenschneider,*Convergence of cardinal series*, Proc. Amer. Math. Soc.**98**(1986), no. 3, 457–460. MR**857940**, 10.1090/S0002-9939-1986-0857940-1**3.**Philip J. Davis,*Interpolation and approximation*, Dover Publications, Inc., New York, 1975. Republication, with minor corrections, of the 1963 original, with a new preface and bibliography. MR**0380189****4.**Michael Golomb,*ℋ^{𝓂,𝓅}-extensions by ℋ^{𝓂,𝓅}-splines*, J. Approximation Theory**5**(1972), 238–275. Collection of articles dedicated to J. L. Walsh on his 75th birthday, III (Proc. Internat. Conf. Approximation Theory, Related Topics and their Applications, Univ. Maryland, College Park, Md., 1970). MR**0336161****5.**Yu. Lyubarskiĭ and W. R. Madych,*The recovery of irregularly sampled band limited functions via tempered splines*, J. Funct. Anal.**125**(1994), no. 1, 201–222. MR**1297019**, 10.1006/jfan.1994.1122**6.**I. J. Schoenberg,*Cardinal interpolation and spline functions. VII. The behavior of cardinal spline interpolants as their degree tends to infinity*, J. Analyse Math.**27**(1974), 205–229. MR**0493058****7.**Elias M. Stein and Guido Weiss,*Introduction to Fourier analysis on Euclidean spaces*, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR**0304972****8.**A. Zygmund,*Trigonometric series: Vols. I, II*, Second edition, reprinted with corrections and some additions, Cambridge University Press, London-New York, 1968. MR**0236587**

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Additional Information

**W. R. Madych**

Affiliation:
Department of Mathematics, U-9, University of Connecticut, Storrs, Connecticut 06269-3009

Email:
madych@uconn.edu

DOI:
https://doi.org/10.1090/S0002-9947-02-03176-8

Received by editor(s):
April 11, 2002

Published electronically:
October 25, 2002

Article copyright:
© Copyright 2002
American Mathematical Society