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

This article is cited in 14 scientific papers (total in 14 papers)

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

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

Bibliographic databases:

UDC: 519.46
MSC: Primary 44A15; Secondary 33Cxx, 43A85
Received: 08.06.2000

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

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    Citing articles on Google Scholar: Russian citations, English citations
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    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.  crossref  isi
    2. Yu. A. Neretin, “Matrix balls, radial analysis of Berezin kernels, and hypergeometric determinants”, Mosc. Math. J., 1:2 (2001), 157–220  mathnet  mathscinet  zmath  elib
    3. Yu. A. Neretin, “Beta-integrals and finite orthogonal systems of Wilson polynomials”, Sb. Math., 193:7 (2002), 1071–1089  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. 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  mathnet  crossref  crossref  mathscinet  zmath  elib
    5. Neretin, YA, “Plancherel formula for Berezin deformation of L-2 on Riemannian symmetric space”, Journal of Functional Analysis, 189:2 (2002), 336  crossref  mathscinet  zmath  isi  scopus
    6. Yu. A. Neretin, “Rayleigh triangles and non-matrix interpolation of matrix beta integrals”, Sb. Math., 194:4 (2003), 515–540  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. 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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. Yu. A. Neretin, “Perturbations of Jacobi polynomials and piecewise hypergeometric orthogonal systems”, Sb. Math., 197:11 (2006), 1607–1633  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. J. Math. Sci. (N. Y.), 141:4 (2007), 1452–1478  mathnet  crossref  mathscinet  zmath  elib
    10. András Biró, “A relation between triple products of weight 0 and weight 1/2 cusp forms”, Isr. J. Math, 182:1 (2011), 61  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    11. Melby-Thompson Ch.M., Schmidt-Colinet C., “Double Trace Interfaces”, J. High Energy Phys., 2017, no. 11, 110  crossref  mathscinet  zmath  isi  scopus
    12. 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  crossref  mathscinet  zmath  isi  scopus
    13. 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  crossref  mathscinet  zmath  isi  scopus
    14. 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  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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