Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-01-06240-2
Published electronically: December 27, 2001
MathSciNet review: 1879957
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. S. Bernstein, Sur une modification de la formule d'interpolation de Lagrange, Comm. Soc. Math. Kharkov, 5 (1932), 49-57.
  • 2. E. W. Cheney, ``Introduction to approximation theory'', McGraw-Hill Inc., New York, 1966. MR 36:5568
  • 3. P. Erdos, On some convergence properties of the interpolation polynomials, Ann. Math., 44 (1943), 330-337. MR 4:273e
  • 4. G. E. Fasshauer and L. L. Schumaker, Scattered data fitting on the sphere, in ``Mathematical Methods for Curves and Surfaces II'', M. Dæhlen, T. Lyche, and L. L. Schumaker (eds.), Vanderbilt University Press, Nashville & London, 1998, 117-166. MR 99d:41054
  • 5. G. Grünwald, On the theory of interpolation, Acta Math., 75 (1943), 219-245. MR 7:157e
  • 6. K. Jetter, J. Stöckler, and J. D. Ward, Error estimates for scattered data interpolation, Math. Comp., 68 (1999), 733-747. MR 99i:41032
  • 7. K. Jetter, J. Stöckler, and J. D. Ward, Norming sets and spherical cubature formulas, in ``Computational Mathematics'', Z. Chen, Y. Li, C. Micchelli, and Y. Xu (eds.), Marcel Dekker, New York, 1998, 237-245. MR 99i:65022
  • 8. H. N. Mhaskar, F. J. Narcowich, and J. D. Ward, Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature, Math. Comp., 70 (2001), 1113-1130. CMP 2001:11
  • 9. H. N. Mhaskar, F. J. Narcowich, and J. D. Ward, Approximation properties of zonal function networks using scattered data on the sphere, Adv. Comput. Math., 11 (1999), 121-137. CMP 2000:06
  • 10. H. N. Mhaskar, F. J. Narcowich, and J. D. Ward, Representing and analyzing scattered data on spheres, in ``Multivariate Approximation and Applications'', N. Dyn, D. Leviaton, D. Levin, and A. Pinkus (eds.), Cambridge Univ. Press, Cambridge, U. K., 2001.
  • 11. H. N. Mhaskar and J. Prestin, On the detection of singularities of a periodic function, Adv. Comput. Math., 12 (2000), 95-131. MR 2001a:42003
  • 12. I. P. Natanson, ``Constructive function theory, Vol. 3'', Frederick Ungar Publishing Co., New York, 1965. MR 33:4529c
  • 13. S. Pawelke, Über die Approximationsordnung bei Kugelfunktionen und algebraischen Polynomen, Tôhoku Math. J., 24 (1972), 473-486. MR 48:774
  • 14. W. Rudin, ``Functional analysis'', McGraw-Hill Inc., New York, 1973. MR 51:1315
  • 15. J. Szabados, On some convergent interpolatory polynomials, in ``Fourier Analysis and Approximation Theory'', Colloq. Math. Soc. János Bolyai, Vol. 19, North-Holland Publishing Co., Amsterdam, 1976, 805-815. MR 81g:41009
  • 16. J. Szabados and P. Vertesi, ``Interpolation of functions'', World Scientific Publishing Co., Singapore, 1990. MR 92j:41009
  • 17. P. Vertesi, Convergent interpolatory processes for arbitrary systems of nodes, Acta Math. Acad. Sci. Hungar., 33 (1979), 223-234. MR 80g:41007

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 41A17, 42C10, 41A05, 65D32, 65D05

Retrieve articles in all journals with MSC (2000): 41A17, 42C10, 41A05, 65D32, 65D05


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: https://doi.org/10.1090/S0002-9939-01-06240-2
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

American Mathematical Society