Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Mean convergence of approximation to a function by general finite sums

Author: Hwa Shan Ho
Journal: Quart. Appl. Math. 31 (1973), 177-184
MSC: Primary 41A30
DOI: https://doi.org/10.1090/qam/410191
MathSciNet review: 410191
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The approximation of a function by a general finite sum (linear combination of non-orthogonal functions) is considered here. It is shown that the mean error of such an approximation, defined in the sense of any weighted inner product in the Hilbert space, is positive semi-definitely decreasing as the number of terms in the expansion increases. Conditions under which the mean error is stationary are thoroughly discussed. Some interesting properties of such approximations are revealed by related theorems. The theorems are proven for complex variables, and are valid of course for real variables.

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

  • [1] L. V. Kantorovich and V. I. Krylov, Approximate methods of higher analysis, Translated from the 3rd Russian edition by C. D. Benster, Interscience Publishers, Inc., New York; P. Noordhoff Ltd., Groningen, 1958. MR 0106537
  • [2] John W. Dettman, Mathematical methods in physics and engineering, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., Inc., New York-San Francisco-Toronto-London, 1962. MR 0141246
  • [3] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
  • [4] F. R. Gantmacher, Teoriya matrits, English translation by K. A. Hirsch, The theory of matrices, Vol. I, Chelsea, N. Y., 1959
  • [5] Kenneth Hoffman and Ray Kunze, Linear algebra, Second edition, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0276251
  • [6] J. B. Martin and P. S. Symonds, Mode approximations for impulsively loaded rigid-plastic structures, J. Engineering Mechanics Division, Amer. Soc. Civil Engineers 92, 43-66 (1966)
  • [7] Hwa-Shan Ho, Convergent approximations of problems of impulsively loaded structures, J. Appl. Mech. 38, 852-860 (1971)
  • [8] Hwa-Shan Ho, Stepwise convergence in linear systems employing non-orthogonal finite sums (manuscript)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 41A30

Retrieve articles in all journals with MSC: 41A30

Additional Information

DOI: https://doi.org/10.1090/qam/410191
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society