## Approximation with interpolatory constraints

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- by H. N. Mhaskar, F. J. Narcowich, N. Sivakumar and J. D. Ward PDF
- Proc. Amer. Math. Soc.
**130**(2002), 1355-1364 Request permission

## 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.## References

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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
- MR Author ID: 129435
- 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
- MR Author ID: 180590
- Email: jward@math.tamu.edu
- 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
- Communicated by: David R. Larson
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**130**(2002), 1355-1364 - MSC (2000): Primary 41A17, 42C10, 41A05; Secondary 65D32, 65D05
- DOI: https://doi.org/10.1090/S0002-9939-01-06240-2
- MathSciNet review: 1879957

Dedicated: Dedicated to Professor Ambikeshwarji Sharma, on the occasion of his 80th birthday