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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Approximation with interpolatory constraints


Authors: H. N. Mhaskar, F. J. Narcowich, N. Sivakumar and J. D. Ward
Journal: Proc. Amer. Math. Soc. 130 (2002), 1355-1364
MSC (2000): Primary 41A17, 42C10, 41A05; Secondary 65D32, 65D05
Published electronically: December 27, 2001
MathSciNet review: 1879957
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Abstract: Given a triangular array of points on $[-1,1]$ satisfying certain minimal separation conditions, a classical theorem of Szabados asserts the existence of polynomial operators that provide interpolation at these points as well as a near-optimal degree of approximation for arbitrary continuous functions on the interval. This paper provides a simple, functional-analytic proof of this fact. This abstract technique also leads to similar results in general situations where an analogue of the classical Jackson-type theorem holds. In particular, it allows one to obtain simultaneous interpolation and a near-optimal degree of approximation by neural networks on a cube, radial-basis functions on a torus, and Gaussian networks on Euclidean space. These ideas are illustrated by a discussion of simultaneous approximation and interpolation by polynomials and also by zonal-function networks on the unit sphere in Euclidean space.


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Additional Information

H. N. Mhaskar
Affiliation: Department of Mathematics, California State University, Los Angeles, California 90032

F. J. Narcowich
Affiliation: Center for Approximation Theory, Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: fnarc@math.tamu.edu

N. Sivakumar
Affiliation: Center for Approximation Theory, Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: n.sivakumar@math.tamu.edu

J. D. Ward
Affiliation: Center for Approximation Theory, Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: jward@math.tamu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06240-2
PII: S 0002-9939(01)06240-2
Received by editor(s): July 12, 2000
Published electronically: December 27, 2001
Additional Notes: The research of the first, third, and fourth authors was supported by grants DMS-9971846, DMS-9706583, and DMS-9971276, respectively, from the National Science Foundation. The work of the second and fourth authors was supported by grant F49620-98-1-0204 from AFOSR
Dedicated: Dedicated to Professor Ambikeshwarji Sharma, on the occasion of his 80th birthday
Communicated by: David R. Larson
Article copyright: © Copyright 2001 American Mathematical Society