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Zbl 0584.43010
Koornwinder, Tom H.
Jacobi functions and analysis on noncompact semisimple Lie groups. (English)
[A] Special functions: Group theoretical aspects and applications, Math. Appl., D. Reidel Publ. Co. 18, 1-85 (1984).

[For the entire collection see Zbl 0543.00007.] \par The Jacobi functions $\Phi\sb{\lambda}\sp{(\lambda,\beta)}(t)$, $\alpha$ $\ne -1,-2,...$, are the continuous analogue of the Jacobi polynomials. They occur for special parameters ($\alpha$,$\beta)$ as spherical functions on rank one symmetric spaces. They also occur as a special way of parametrizing the Gauss hypergeometric function ${}\sb 2F\sb 1(a,b;c;z)$, taking $a=(\alpha +\beta +1-i\lambda)$, $b=(\alpha +\beta +1+i\lambda)$, $c=\alpha +1$, $z=-\sinh\sp 2 t.$ \par The paper under review is a well written and most valuable survey of the theory developed around Jacobi functions and the interplay between these special functions and Lie group theory. The emphasis is on harmonic analysis. There is a large and very usefull bibliography of the subject. \par The content is in short described by: The Jacobi transform and its inverse. A short introduction to group theoretic tools needed. Examples of the Jacobi transform from group theory. The Abel transform. The Paley- Wiener and Plancherel theorems. Convolution structures and addition formulas. Further results and references.
[M.Flensted-Jensen]

MSC 2000:: *43A85 Analysis on homogeneous spaces
22E46 Semi-simple Lie groups and their representations
33C80 Connections of theory of special functions with groups and algebras
43-02 Research monographs (abstract harmonic analysis)
33C45 Orthogonal polynomials and functions of hypergeometric type

Keywords: Paley-Wiener theorem; Plancherel theorem; Jacobi functions; Jacobi polynomials; spherical functions; rank one symmetric spaces; survey; bibliography; Jacobi transform; Abel transform; addition formulas

Citations: Zbl 0543.00007

Cited in: Zbl 1087.42001 Zbl 0796.60010 Zbl 0584.43009

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