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
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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.
 1.
Carl
de Boor, Odddegree 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
(58 #29610)
 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 (87j:41057), http://dx.doi.org/10.1090/S00029939198608579401
 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
(52 #1089)
 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
(49 #937)
 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
(96d:41013), http://dx.doi.org/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 (58 #12097b)
 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
(46 #4102)
 8.
A.
Zygmund, Trigonometric series: Vols. I, II, Second edition,
reprinted with corrections and some additions, Cambridge University Press,
LondonNew York, 1968. MR 0236587
(38 #4882)
 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
