Index hypergeometric transform and imitation of analysis of Berezin kernels on hyperbolic spaces

Yurii Neretin

Veröffentlichungen: Beitrag in FachzeitschriftArtikelPeer Reviewed

Abstract

The index hypergeometric transform (also called the Olevskii? transform or the Jacobi transform) generalizes the spherical transform in L2 on rank 1 symmetric spaces (that is, real, complex, and quaternionic Lobachevskii? spaces). The aim of this paper is to obtain properties of the index hypergeometric transform imitating the analysis of Berezin kernels on rank 1 symmetric spaces. The problem of the explicit construction of a unitary operator identifying L2 and a Berezin space is also discussed. This problem reduces to an integral expression (the A-function), which apparently cannot be expressed in a finite form in terms of standard special functions. (Only for certain special values of the parameter can this expression be reduced to the so-called Volterra type special functions.) Properties of this expression are investigated. For some series of symmetric spaces of large rank the above operator of unitary equivalence can be expressed in terms of the determinant of a matrix of A-functions.
OriginalspracheEnglisch
Seiten (von - bis)403-432
Seitenumfang30
FachzeitschriftSbornik Mathematics
Jahrgang192
Ausgabenummer3-4
PublikationsstatusVeröffentlicht - 2001

ÖFOS 2012

  • 1010 Mathematik

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