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
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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.
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- 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
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F. R. Gantmacher, Teoriya matrits, English translation by K. A. Hirsch, The theory of matrices, Vol. I, Chelsea, N. Y., 1959
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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)
Hwa-Shan Ho, Convergent approximations of problems of impulsively loaded structures, J. Appl. Mech. 38, 852–860 (1971)
Hwa-Shan Ho, Stepwise convergence in linear systems employing non-orthogonal finite sums (manuscript)
L. V. Kantorovich and V. I. Krylov, Priblizhennye metody vysshego analiza, English translation by C. D. Benster, Approximate methods of higher analysis, Interscience, Noordhoff, Groningen, Netherlands, 1958
J. W. Bettman, Mathematical methods in physics and engineering, McGraw-Hill, N. Y., 1962
R. Courant and D. Hilbert, Methods of mathematical physics, Vol. I, Interscience, N. Y., 1953
F. R. Gantmacher, Teoriya matrits, English translation by K. A. Hirsch, The theory of matrices, Vol. I, Chelsea, N. Y., 1959
K. Hoffman and R. Kunze, Linear algebra, Prentice-Hall, Englewood, N. J., 1964
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)
Hwa-Shan Ho, Convergent approximations of problems of impulsively loaded structures, J. Appl. Mech. 38, 852–860 (1971)
Hwa-Shan Ho, Stepwise convergence in linear systems employing non-orthogonal finite sums (manuscript)
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Article copyright:
© Copyright 1973
American Mathematical Society
