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Transactions of the American Mathematical Society

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Some transplantation theorems for the generalized Mehler transform and related asymptotic expansions


Author: Susan Schindler
Journal: Trans. Amer. Math. Soc. 155 (1971), 257-291
MSC: Primary 42.26; Secondary 44.00
DOI: https://doi.org/10.1090/S0002-9947-1971-0279528-X
MathSciNet review: 0279528
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Abstract: Let $ P_{ - 1/2 + ix}^m(z)$ be the associated Legendre function of order $ m$ and degree $ - 1/2 + ix$. We give, here, two integral transforms $ {G^m}$ and $ {H^m}$, arising naturally from the generalized Mehler transform, which is induced by $ P_{ - 1/2 + ix}^m(\cosh y)$, such thatb $ {H^m}{G^m}$ = Identity (formally). We show that if $ 1 < p < \infty , - 1/p < \alpha < 1 - 1/p,m \leqq 1/2$ or $ m = 1,2, \ldots ,$ then $ \vert\vert{G^m}f\vert{\vert _{p,\alpha }} \leqq A_{p,\alpha }^m\vert\vert\hat f\vert{\vert _{p,\alpha }}$ and $ \vert\vert{H^m}f\vert{\vert _{p,\alpha }} \leqq A_{p,\alpha }^m\vert\vert\hat f\vert{\vert _{p,\alpha }}$, where $ ^ \wedge $ denotes the Fourier cosine transform. We also prove that $ {G^m}f,{H^m}f$ exist as limits in $ {L^{p,\alpha }}$ of partial integrals, and we prove inequalities equivalent to the above pair: $ \vert\vert{G^m}\hat f\vert{\vert _{p,\alpha }} \leqq A_{p,\alpha }^m\vert\vert f\vert{\vert _{p,\alpha }}$ and $ \vert\vert{H^m}\hat f\vert{\vert _{p,\alpha }} \leqq A_{p,\alpha }^m\vert\vert f\vert{\vert _{p,\alpha }}$. These we dualize to $ \vert\vert{({H^m}f)^ \wedge }\vert{\vert _{p,\alpha }} \leqq A_{p,\alpha }^m\vert\vert f\vert{\vert _{p,\alpha }}$, and $ \vert\vert{({G^m}f)^ \wedge }\vert{\vert _{p,\alpha }} \leqq A_{p,\alpha }^m\vert\vert f\vert{\vert _{p,\alpha }}$.

$ {G^m}$ and $ {H^m}$ are given by $ {G^m}(f;y) = \int_0^\infty {f(x){K^m}(x,y)dx} $ and $ {H^m}(f;x) = \int_0^\infty {f(y){K^m}(x,y)dy\;} (0 \leqq y < \infty )$, where

$\displaystyle {K^m}(x,y) = \vert\Gamma (1/2 - m - ix)/\Gamma ( - ix)/{(\sinh y)^{1/2}}P_{ - 1/2 + ix}^m(\cosh y).$

The principal method of proving the inequalities involves getting asymptotic expansions for $ {K^m}(x,y)$; these are in terms of sines and cosines for large $ y$, and in terms of Bessel functions for $ y$ small. Then we can use Fourier and Hankel multiplier theorems.

The main consequences of our results are the typical ones for transplantation theorems: mean convergence and multiplier theorems. They can easily be restated in terms of the more usual Mehler transform pair

$\displaystyle g(y) = \int_0^\infty {f(x){P_{ - 1/2 + ix}}(y)dx} $

and $ f(x) = {\pi ^{ - 1}}x\sinh \pi x \cdot \Gamma (1/2 - m + ix)\Gamma (1/2 - m - ix)\int_0^\infty {g(y){P_{ - 1/2 + ix}}(y)dy.} $

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0279528-X
Keywords: Legendre function, hypergeometric function, Bessel function, transplantation theorem, mean convergence, Fourier multiplier, Hankel transform
Article copyright: © Copyright 1971 American Mathematical Society

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