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

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