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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the singularities of the continuous Jacobi transform when $ \alpha+\beta=0$


Author: Ahmed I. Zayed
Journal: Proc. Amer. Math. Soc. 101 (1987), 67-75
MSC: Primary 44A20; Secondary 33A70
DOI: https://doi.org/10.1090/S0002-9939-1987-0897072-0
MathSciNet review: 897072
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Abstract: Let $ \alpha ,\beta > - 1$ and $ \mathcal{P}_\lambda ^{(\alpha ,\beta )}(x) = {(1 - x)^\alpha }{(1 + x)^\beta }P_\lambda ^{(\alpha ,\beta )}(x)$, where $ P_\lambda ^{(\alpha ,\beta )}(x)$ is the Jacobi function of the first kind, $ \lambda \geq - (\alpha + \beta + 1)/2$, and $ - 1 < x \leq 1$. Let

$\displaystyle {F^{(\alpha ,\beta )}}(\lambda ) = \frac{1} {{{2^{\alpha + \beta ... ... \beta + 1}}}}\int_{ - 1}^1 {f(x)\mathcal{P}_\lambda ^{(\alpha ,\beta )}(x)dx} $

whenever the integral exists. It is known that for $ \alpha + \beta = 0$, we have (*)

$\displaystyle f(x) = \mathop {\lim }\limits_{n \to \infty } 4\int_0^n {{F^{(\al... .../2)}}{{\Gamma (\lambda + \alpha + 1/2)\Gamma (\lambda + \beta + 1/2)}}d\lambda $

almost everywhere in $ [-1,1]$.

In this paper, we devise a technique to continue $ f(x)$ analytically to the complex $ z$-plane and locate the singularities of $ f(z)$ by relating them to the singularities of

$\displaystyle g(t) = \int_0^\infty {{e^{ - \lambda t}}{F^{(\alpha ,\beta )}}(\lambda )} \frac{{d\lambda }}{{\Gamma (\lambda + \alpha + 1)}}.$

However, this will be done in the more general case where the limit in (*) exists in the sense of Schwartz distributions and defines a generalized function $ f(x)$. In this case, we pass from $ f(x)$ to its analytic representation

$\displaystyle \hat f(z) = \frac{1} {{2\pi i}}\left\langle {f(x),\frac{1} {{x - z}}} \right\rangle ,\quad z \notin \operatorname{supp} f,$

and then relate the singularities of $ \hat f(z)$ to those of $ g(t)$.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1987-0897072-0
Keywords: Continuous Jacobi transform, analytic continuation, singular points
Article copyright: © Copyright 1987 American Mathematical Society