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 Free Access

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.

**[1]**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 MR**0106537****[2]**J. W. Bettman,*Mathematical methods in physics and engineering*, McGraw-Hill, N. Y., 1962 MR**0141246****[3]**R. Courant and D. Hilbert,*Methods of mathematical physics*, Vol. I, Interscience, 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]**K. Hoffman and R. Kunze,*Linear algebra*, Prentice-Hall, Englewood, N. J., 1964 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)

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