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ERROR ESTIMATES FOR DOMINICI’S HERMITE FUNCTION ASYMPTOTIC FORMULA AND SOME APPLICATIONS

Published online by Cambridge University Press:  04 December 2009

R. KERMAN
Affiliation:
Department of Mathematics, Brock University, St. Catharines, Ontario, Canada L2S 3A1 (email: [email protected])
M. L. HUANG*
Affiliation:
Department of Mathematics, Brock University, St. Catharines, Ontario, Canada L2S 3A1 (email: [email protected])
M. BRANNAN
Affiliation:
Department of Mathematics, Brock University, St. Catharines, Ontario, Canada L2S 3A1 (email: [email protected])
*
For correspondence; e-mail: [email protected]
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Abstract

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The aim of this paper is to find a concrete bound for the error involved when approximating the nth Hermite function (in the oscillating range) by an asymptotic formula due to D. Dominici. This bound is then used to study the accuracy of certain approximations to Hermite expansions and to Fourier transforms. A way of estimating an unknown probability density is proposed.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2009

References

[1]Arfken, G., Mathematical methods for physicists, 2nd edn (Academic Press, New York, 1970).Google Scholar
[2]Dominici, D., “Asymptotic analysis of the Hermite polynomials from their differential-difference equation”, J. Difference Equ. Appl. 13(12) (2007) 11151128.CrossRefGoogle Scholar
[3]Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A., Wavelets, approximation, and statistical applications (Springer, New York, 1998).CrossRefGoogle Scholar
[4]Schwartz, S. C., “Estimation of probability density by an orthogonal series”, Ann. Math. Statist. 38 (1967) 12611265.CrossRefGoogle Scholar
[5]Slater, L. J., Confluent hypergeometric functions (Cambridge University Press, Cambridge, 1960).Google Scholar
[6]Walter, G. G., “Properties of Hermite series estimation of probability density”, Ann. Statist. 5(6) (1977) 12581264.CrossRefGoogle Scholar
[7]Whittaker, E. T. and Watson, G. N., A course of modern analysis, 4th edn (Cambridge University Press, Cambridge, 1927).Google Scholar
[8]Wiener, N., The Fourier integral and certain of its applications (Dover, New York, 1933).Google Scholar