Limits of interpolatory processes
Author:
W. R. Madych
Journal:
Trans. Amer. Math. Soc. 355 (2003), 11091133
MSC (2000):
Primary 41A05, 41A15
Published electronically:
October 25, 2002
MathSciNet review:
1938748
Fulltext PDF Free Access
Abstract 
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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 wellbehaved 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.
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M. Golomb, extensions by splines, J. Approx. Theory, 5, (1972), 238275. MR 49:937
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Yu. Lyubarskii and W. R. Madych, The recovery of irregularly sampled band limited functions via tempered splines, J. Functional Analysis, 155 (1994), 201222. MR 96d:41013
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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), 205229. MR 58:12097b
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 1.
 C. deBoor, Odd degree spline interpolation at a biinfinite knot sequence, in Approximation Theory, R. Schaback and K. Scherer, eds., Lecture Notes in Mathematics, Vol. 556, Springer Verlag, Berlin, 1976, 3053. MR 58:29610
 2.
 C. deBoor, K. Höllig, S. Riemenschneider, Convergence of cardinal series. Proc. Amer. Math. Soc. 98 (1986), 457460. 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), 238275. 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), 201222. 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), 205229. 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
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Additional Information
W. R. Madych
Affiliation:
Department of Mathematics, U9, University of Connecticut, Storrs, Connecticut 062693009
Email:
madych@uconn.edu
DOI:
http://dx.doi.org/10.1090/S0002994702031768
PII:
S 00029947(02)031768
Received by editor(s):
April 11, 2002
Published electronically:
October 25, 2002
Article copyright:
© Copyright 2002
American Mathematical Society
