This paper describes expansions in terms of Hermite and Laguerre functions similar to the Frazier-Jawerth expansion in Fourier analysis. The wave packets occurring in these expansions are finite linear combinations of Hermite and Laguerre functions. The Shannon sampling formula played an important role in the derivation of the Frazier-Jawerth expansion. In this paper we use the Christoffel-Darboux formula for orthogonal polynomials instead. We obtain estimates on the decay of the Hermite and Laguerre wave packets by investigating two closely related oscillatory integrals.
@article{bwmeta1.element.bwnjournal-article-smv126i3p199bwm,
author = {Jay Epperson},
title = {Hermite and Laguerre wave packet expansions},
journal = {Studia Mathematica},
volume = {122},
year = {1997},
pages = {199-217},
zbl = {0899.42005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv126i3p199bwm}
}
Epperson, Jay. Hermite and Laguerre wave packet expansions. Studia Mathematica, Tome 122 (1997) pp. 199-217. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv126i3p199bwm/
[00000] [1] J. Epperson, Triebel-Lizorkin spaces for Hermite expansions, Studia Math. 114 (1995), 87-103. | Zbl 0828.42017
[00001] [2] M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), 34-170. | Zbl 0716.46031
[00002] [3] C. Markett, Mean Cesàro summability of Laguerre expansions and norm estimates with shifted parameter, Anal. Math. 8 (1982), 19-37. | Zbl 0515.42023
[00003] [4] E. Rainville, Special Functions, Macmillan, New York, 1960.
[00004] [5] G. Szegő, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 23, Providence, R.I., 1975.
[00005] [6] S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Math. Notes 42, Princeton Univ. Press, 1993.