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The convergence of least squares approximations for dual orthogonal series

Published online by Cambridge University Press:  18 May 2009

Robert P. Feinerman  and
Robert B. Kelman
Robert P. Feinerman
Affiliation:
Herbert H. Lehman College, Bronx, New York 10468
Robert B. Kelman
Affiliation:
Computer Science Section, Colorado State University, Ft. Collins, Colorado 80521
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The convergence of least squares approximations for dual orthogonal series in Hilbert space is established, thus providing a theorem applicable to practically all dual orthogonal series (such as dual trigonometric series, dual Bessel series, etc.) that have appeared in the literature. Our results establish for such dual series the existence of a sequence of functions satisfying in the L2norm the dual series relation, with an error tending to zero and, in particular, rigorously justify the calculations in [2] which showed least squares to be a practical approximation procedure for dual trigonometric equations. In fact, the desire to provide a rigorous convergence theorem for [2] motivated this study.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 15 , Issue 1 , March 1974 , pp. 82 - 84
Copyright
Copyright © Glasgow Mathematical Journal Trust 1974

References

REFERENCES

1.Achiezer, N. I. and Glasmann, I. M., Theorie der linearen Operatoren in Hilbert-Raum (Berlin, 1954).Google Scholar
2.Kelman, R. B. and Koper, C. A. Jr, Least squares approximations for dual trigonometric series, Glasgow Math. J. 14 (1973), 111–119.CrossRefGoogle Scholar
3.Mikhlin, S. G. and Smolitskiy, K. L., Approximate methods for solution of differential and integral equations (New York, 1967).Google Scholar