RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Sb.: Year: Volume: Issue: Page: Find

 Mat. Sb., 2001, Volume 192, Number 3, Pages 83–114 (Mi msb552)

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

Yu. A. Neretin

Moscow State Institute of Electronics and Mathematics

Abstract: The index hypergeometric transform (also called the Olevskii transform or the Jacobi transform) generalizes the spherical transform in $L^2$ 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 $L^2$ and a Berezin space is also discussed. This problem reduces to an integral expression (the $\Lambda$-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 $\Lambda$-functions.

DOI: https://doi.org/10.4213/sm552

Full text: PDF file (409 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2001, 192:3, 403–432

Bibliographic databases:

UDC: 519.46
MSC: Primary 44A15; Secondary 33Cxx, 43A85

Citation: Yu. A. Neretin, “Index hypergeometric transform and imitation of analysis of Berezin kernels on hyperbolic spaces”, Mat. Sb., 192:3 (2001), 83–114; Sb. Math., 192:3 (2001), 403–432

Citation in format AMSBIB
\Bibitem{Ner01} \by Yu.~A.~Neretin \paper Index hypergeometric transform and imitation of analysis of Berezin kernels on hyperbolic spaces \jour Mat. Sb. \yr 2001 \vol 192 \issue 3 \pages 83--114 \mathnet{http://mi.mathnet.ru/msb552} \crossref{https://doi.org/10.4213/sm552} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1836308} \zmath{https://zbmath.org/?q=an:1054.33004} \elib{https://elibrary.ru/item.asp?id=13371115} \transl \jour Sb. Math. \yr 2001 \vol 192 \issue 3 \pages 403--432 \crossref{https://doi.org/10.1070/sm2001v192n03ABEH000552} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000169973700005} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-0035534764} 

• http://mi.mathnet.ru/eng/msb552
• https://doi.org/10.4213/sm552
• http://mi.mathnet.ru/eng/msb/v192/i3/p83

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. El Hamma M., Daher R., “Equivalence of K-Functionals and Modulus of Smoothness Constructed By Generalized Jacobi Transform”, Integral Transform. Spec. Funct.
2. Platonov S.S., “Fourier-Jacobi Harmonic Analysis and Some Problems of Approximation of Functions on the Half-Axis in l-2 Metric: Nikol'Skii-Besov Type Function Spaces”, Integral Transform. Spec. Funct.
3. Yu. A. Neretin, “Matrix balls, radial analysis of Berezin kernels, and hypergeometric determinants”, Mosc. Math. J., 1:2 (2001), 157–220
4. Yu. A. Neretin, “Beta-integrals and finite orthogonal systems of Wilson polynomials”, Sb. Math., 193:7 (2002), 1071–1089
5. Yu. A. Neretin, “The action of an overalgebra on the Plancherel decomposition and shift operators in the imaginary direction”, Izv. Math., 66:5 (2002), 1035–1046
6. Neretin, YA, “Plancherel formula for Berezin deformation of L-2 on Riemannian symmetric space”, Journal of Functional Analysis, 189:2 (2002), 336
7. Yu. A. Neretin, “Rayleigh triangles and non-matrix interpolation of matrix beta integrals”, Sb. Math., 194:4 (2003), 515–540
8. Yu. A. Neretin, “Some Continuous Analogs of the Expansion in Jacobi Polynomials and Vector-Valued Orthogonal Bases”, Funct. Anal. Appl., 39:2 (2005), 106–119
9. Yu. A. Neretin, “Perturbations of Jacobi polynomials and piecewise hypergeometric orthogonal systems”, Sb. Math., 197:11 (2006), 1607–1633
10. J. Math. Sci. (N. Y.), 141:4 (2007), 1452–1478
11. András Biró, “A relation between triple products of weight 0 and weight 1/2 cusp forms”, Isr. J. Math, 182:1 (2011), 61
12. Melby-Thompson Ch.M., Schmidt-Colinet C., “Double Trace Interfaces”, J. High Energy Phys., 2017, no. 11, 110
13. Sousa R., Yakubovich S., “The Spectral Expansion Approach to Index Transforms and Connections With the Theory of Diffusion Processes”, Commun. Pure Appl. Anal, 17:6 (2018), 2351–2378
14. Platonov S.S., “Fourier-Jacobi Harmonic Analysis and Some Problems of Approximation of Functions on the Half-Axis in l-2 Metric: Jackson'S Type Direct Theorems”, Integral Transform. Spec. Funct., 30:4 (2019), 264–281
15. Neretin Yu.A., “The Fourier Transform on the Group G(l)2(R) and the Action of the Overalgebra Gl(4)”, J. Fourier Anal. Appl., 25:2 (2019), 488–505
•  Number of views: This page: 433 Full text: 142 References: 64 First page: 4