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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Some transplantation theorems for the generalized Mehler transform and related asymptotic expansions
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by Susan Schindler PDF
Trans. Amer. Math. Soc. 155 (1971), 257-291 Request permission

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 $||{G^m}f|{|_{p,\alpha }} \leqq A_{p,\alpha }^m||\hat f|{|_{p,\alpha }}$ and $||{H^m}f|{|_{p,\alpha }} \leqq A_{p,\alpha }^m||\hat f|{|_{p,\alpha }}$, 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: $||{G^m}\hat f|{|_{p,\alpha }} \leqq A_{p,\alpha }^m||f|{|_{p,\alpha }}$ and $||{H^m}\hat f|{|_{p,\alpha }} \leqq A_{p,\alpha }^m||f|{|_{p,\alpha }}$. These we dualize to $||{({H^m}f)^ \wedge }|{|_{p,\alpha }} \leqq A_{p,\alpha }^m||f|{|_{p,\alpha }}$, and $||{({G^m}f)^ \wedge }|{|_{p,\alpha }} \leqq A_{p,\alpha }^m||f|{|_{p,\alpha }}$. ${G^m}$ 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 \[ {K^m}(x,y) = |\Gamma (1/2 - m - ix)/\Gamma ( - ix)/{(\sinh y)^{1/2}}P_{ - 1/2 + ix}^m(\cosh y).\] 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 \[ g(y) = \int _0^\infty {f(x){P_{ - 1/2 + ix}}(y)dx} \] and $f(x) = {\pi ^{ - 1}}x\sinh \pi x \cdot \Gamma (1/2 - m + ix)\Gamma (1/2 - m - ix)\int _0^\infty {g(y){P_{ - 1/2 + ix}}(y)dy.}$
References
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • 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