Optimal approximation in Hilbert spaces with reproducing kernel functions
HTML articles powered by AMS MathViewer
- by F. M. Larkin PDF
- Math. Comp. 24 (1970), 911-921 Request permission
Abstract:
Characterisations of optimal linear estimation rules are given in terms of the reproducing kernel function of a suitable Hilbert space. The results are illustrated by means of three different, useful function spaces, showing, among other things, how Gaussian quadrature rules, and the Whittaker Cardinal Function, relate to optimal linear estimation rules in particular spaces.References
- N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404. MR 51437, DOI 10.1090/S0002-9947-1950-0051437-7
- Michael Golomb and Hans F. Weinberger, Optimal approximation and error bounds, On numerical approximation. Proceedings of a Symposium, Madison, April 21-23, 1958, Publication of the Mathematics Research Center, U.S. Army, the University of Wisconsin, no. 1, University of Wisconsin Press, Madison, Wis., 1959, pp. 117–190. Edited by R. E. Langer. MR 0121970
- Louis de Branges, Hilbert spaces of entire functions, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1968. MR 0229011
- Vladimir Ivanovich Krylov, Approximate calculation of integrals, The Macmillan Company, New York-London, 1962, 1962. Translated by Arthur H. Stroud. MR 0144464
- Herbert Meschkowski, Hilbertsche Räume mit Kernfunktion, Die Grundlehren der mathematischen Wissenschaften, Band 113, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1962 (German). MR 0140912
- Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR 1451142, DOI 10.1090/coll/019
- Arthur Sard, Best approximate integration formulas; best approximation formulas, Amer. J. Math. 71 (1949), 80–91. MR 29283, DOI 10.2307/2372095
- M. D. Stern, Optimal quadrature formulae, Comput. J. 9 (1967), 396–403. MR 213020, DOI 10.1093/comjnl/9.4.396 E. T. Whittaker, "On the functions which are represented by the expansions of the interpolation theory," Proc. Edinburg Math. Soc., v. 35, 1915, pp. 181–194. N. Richter & P. Rabinowitz, Weizmann Institute of Science, Rehovot, Israel. (Private communication.)
- R. Fletcher and C. M. Reeves, Function minimization by conjugate gradients, Comput. J. 7 (1964), 149–154. MR 187375, DOI 10.1093/comjnl/7.2.149
Additional Information
- © Copyright 1970 American Mathematical Society
- Journal: Math. Comp. 24 (1970), 911-921
- MSC: Primary 65.20; Secondary 41.00
- DOI: https://doi.org/10.1090/S0025-5718-1970-0285086-9
- MathSciNet review: 0285086