Limits of interpolatory processes

Author:
W. R. Madych

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

MSC (2000):
Primary 41A05, 41A15

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

Published electronically:
October 25, 2002

MathSciNet review:
1938748

Full-text PDF

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.**C. deBoor, Odd degree spline interpolation at a bi-infinite knot sequence, in*Approximation Theory*, R. Schaback and K. Scherer, eds., Lecture Notes in Mathematics, Vol. 556, Springer Verlag, Berlin, 1976, 30-53. MR**58:29610****2.**C. deBoor, K. Höllig, S. Riemenschneider, Convergence of cardinal series. Proc. Amer. Math. Soc. 98 (1986), 457-460. MR**87j:41057****3.**P. J. Davis,*Interpolation and Approximation*, Dover, New York, 1975. MR**52:1089****4.**M. Golomb, -extensions by -splines,*J. Approx. Theory*,**5**, (1972), 238-275. MR**49:937****5.**Yu. Lyubarskii and W. R. Madych, The recovery of irregularly sampled band limited functions via tempered splines,*J. Functional Analysis*,**155**(1994), 201-222. MR**96d:41013****6.**I. J. Schoenberg, Cardinal interpolation and spline function VII: The behavior of cardinal spline interpolation as their degree tends to infinity,*J. Analyse Math.***27**, (1974), 205-229. MR**58:12097b****7.**E. M. Stein and G. Weiss,*Introduction to Fourier Analysis on Euclidean Spaces*, Princeton Univ. Press, Princeton, N. J., 1971. MR**46:4102****8.**A. Zygmund,*Trigonometric Series*, 2nd ed., Cambridge Univ. Press, Cambridge, 1968. MR**38:4882**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
41A05,
41A15

Retrieve articles in all journals with MSC (2000): 41A05, 41A15

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