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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  • 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 Book Co., New York-Toronto, Ont.-London, 1966. MR 0222517
  • 3. P. Erdös, On some convergence properties of the interpolation polynomials, Ann. of Math. (2) 44 (1943), 330–337. MR 0008276
  • 4. Gregory E. Fasshauer and Larry L. Schumaker, Scattered data fitting on the sphere, Mathematical methods for curves and surfaces, II (Lillehammer, 1997), Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, TN, 1998, pp. 117–166. MR 1640548
  • 5. G. Grünwald, On the theory of interpolation, Acta Math. 75 (1943), 219–245. MR 0013466
  • 6. Kurt Jetter, Joachim Stöckler, and Joseph D. Ward, Error estimates for scattered data interpolation on spheres, Math. Comp. 68 (1999), no. 226, 733–747. MR 1642746, 10.1090/S0025-5718-99-01080-7
  • 7. K. Jetter, J. Stöckler, and J. D. Ward, Norming sets and spherical cubature formulas, Advances in computational mathematics (Guangzhou, 1997) Lecture Notes in Pure and Appl. Math., vol. 202, Dekker, New York, 1999, pp. 237–244. MR 1661538
  • 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), no. 2-3, 95–131. MR 1745108, 10.1023/A:1018921319865
  • 12. I. P. Natanson, Constructive function theory. Vol. III. Interpolation and approximation quadratures, Frederick Ungar Publishing Co., New York, 1965. Translated from the Russian by John R. Schulenberger. MR 0196342
  • 13. Siegfried Pawelke, Über die Approximationsordnung bei Kugelfunktionen und algebraischen Polynomen, Tôhoku Math. J. (2) 24 (1972), 473–486 (German). MR 0322412
  • 14. Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0365062
  • 15. J. Szabados, On some convergent interpolatory polynomials, Fourier analysis and approximation theory (Proc. Colloq., Budapest, 1976), Vol. II, Colloq. Math. Soc. János Bolyai, vol. 19, North-Holland, Amsterdam-New York, 1978, pp. 805–815. MR 540356
  • 16. J. Szabados and P. Vértesi, Interpolation of functions, World Scientific Publishing Co., Inc., Teaneck, NJ, 1990. MR 1089431
  • 17. P. Vértesi, Convergent interpolatory processes for arbitrary systems of nodes, Acta Math. Acad. Sci. Hungar. 33 (1979), no. 1-2, 223–234. Special issue dedicated to George Alexits on the occasion of his 80th birthday. MR 515137, 10.1007/BF01903398

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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
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