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Stone's theorem and completeness of orthogonal systems

Published online by Cambridge University Press:  09 April 2009

B. D. Craven
B. D. Craven
Affiliation:
University of Melbourne
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It is well known (e.g. Stone [1]) that the Stone-Weierstrass approximation theorem can be used to prove the completeness of various systems of orthogonal polynomials, e.g. Chebyshev polynomials. In this paper, Stone's theorem is used to prove a more general completeness theorem, which includes as special cases Plancherel's theorem, the corresponding theorem for Hankel transforms, the completeness of various polynomial systems, and certain expansions in Jacobian elliptic functions. The essential feature common to all these systems is a certain algebraic structure — if S is an appropriate vector space spanned by orthogonal functions, then the algebra A generated by S is contained in the closure of S in a suitable norm.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 12 , Issue 2 , May 1971 , pp. 211 - 223
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Stone, M. H., A generalized Weierstrass approximation theorem (Studies in Mathematics, Vol. 1, Math. Assoc. of America, 1962).Google Scholar
[2]Naimark, M. A., Normed Rings (Noordhoff, Groningen, 1964).Google Scholar
[3]Milne-Thomson, L. M., Jacobian Elliptic Function Tables (Dover, New York, 1950).Google Scholar
[4]Magnus, W. and Oberhettinger, F., Formulas and theorems for the functions of mathematical physics (Chelsea, New York, 1954).Google Scholar