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Summability of Hermite polynomial expansions of generalized functions
Published online by Cambridge University Press: 24 October 2008
Extract
1. Introduction: The well-known result that every function f(x) ∈ L2(–∞, ∞) can be expanded in L2(– ∞,∞) by where and was recently followed by an expansion theorem for generalized functions. In papers by Wildlund(8), Giertz(4) and Zemanian(9) it was shown that a tempered distribution f∈S′ can be expanded by Hermite functions that is
for all ψ∈S (the space of rapidly decreasing test functions).
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 68 , Issue 1 , July 1970 , pp. 129 - 139
- Copyright
- Copyright © Cambridge Philosophical Society 1970
References
REFERENCES
(1)Bilodeau, G. G.On the summability of series of Hennite polynomials. J. Math. Anal. Appl. 8 (1964), 406–422.CrossRefGoogle Scholar
(2)Hirschmann, I. I. and Widder, D. V.The Convolution Transform. (Princeton University Press, 1955).Google Scholar
(3)Gelfand, I. M. and Shilov, G. E.Generalized Functions. Vol. 2. (English translation, Academic Press, 1968.)Google Scholar
(4)Giebtz, M.On the expansion of certain generalized functions of series of orthogonal func-tions. Proc. London Math. Soc. (3), 14 (1964), 45–52.Google Scholar
(5)Korevaar, J.Pansion and the theory of Fourier transforms Trans. Amer. Math. Soc. 31 (1959), 53–101.CrossRefGoogle Scholar
(7)Szego, G.Orthogonal polynomials. Amer. Math. Soc. Colloq. Publ. Vol. XXIII (New York, 1959).Google Scholar
(8)Widlund, O.On the expansion of generalized functions in series of Hermite functions. Kungl. Tekn. Högsk. Handl. Stockholm 173 (1961).Google Scholar
(9)Zemanian, A.Orthonormal Series Expansions of certain Distributions and Transform calculus. J. Math. Anal. Appl. 14 (1966), 263–275.CrossRefGoogle Scholar
(10)Zemanian, A.A generalized Weierstrass transformation. SIAM J. Appl. Math. 15 (1967), 1088–1105.CrossRefGoogle Scholar
(11)Zemanian, A.Generalized integral transformation with Applications (Interscience, 1968).Google Scholar