An application of hypergeometric shift operators to the $𝜒$-spherical Fourier transform

We study the action of hypergeometric shift operators on the Heckman-Opdam hypergeometric functions associated with the

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Published in:Trends in Harmonic Analysis and Its Applications Vol. 650; pp. 143 - 155
Main Authors: Ho, Vivian M., Ólafsson, Gestur
Format: Book Chapter
Language:English
Japanese
Published: Providence, Rhode Island American Mathematical Society 2015
Series:Contemporary Mathematics
Subjects:
symmetric space
<inline-formula content-type="math/mathml"> χ \chi </inline-formula>-spherical Fourier transform
Hypergeometric shift operator
hypergeometric function
Paley-Wiener theorem
ISBN:9781470418793, 1470418797
ISSN:0271-4132, 1098-3627
Online Access:Get full text
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Abstract We study the action of hypergeometric shift operators on the Heckman-Opdam hypergeometric functions associated with the
AbstractList We study the action of hypergeometric shift operators on the Heckman-Opdam hypergeometric functions associated with the
Author Ho, Vivian M.
Ólafsson, Gestur
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  givenname: Vivian M.
  surname: Ho
  fullname: Ho, Vivian M.
  email: vivian@math.lsu.edu
  organization: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
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  givenname: Gestur
  surname: Ólafsson
  fullname: Ólafsson, Gestur
  email: olafsson@math.lsu.edu
  organization: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
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Editor Dann, Susanna
Christensen, Jens G.
Mayeli, Azita
Ólafsson, Gestur
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  organization: Vienna University of Technology, Wien, Austria
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  surname: Ólafsson
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  organization: Louisiana State University, Baton Rouge, LA
EndPage 155
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Keywords symmetric space
<inline-formula content-type="math/mathml"> χ \chi </inline-formula>-spherical Fourier transform
Hypergeometric shift operator
hypergeometric function
Paley-Wiener theorem
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Japanese
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References Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561 (80k:53081)
Nobukazu Shimeno, The Plancherel formula for spherical functions with a one-dimensional KK-type on a simply connected simple Lie group of Hermitian type, J. Funct. Anal. 121 (1994), no. 2, 330–388. MR 1272131 (95b:22036), DOI 10.1006/jfan.1994.1052
E. M. Opdam, Root systems and hypergeometric functions. III, Compositio Math. 67 (1988), no. 1, 21–49. MR 949270 (90k:17021)
Eric M. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), no. 1, 75–121. MR 1353018 (98f:33025), DOI 10.1007/BF02392487
S. G. Gindikin and F. I. Karpelevič, Plancherel measure for symmetric Riemannian spaces of non-positive curvature, Dokl. Akad. Nauk SSSR 145 (1962), 252–255 (Russian). MR 0150239 (27 \#240)
G. J. Heckman and E. M. Opdam, Root systems and hypergeometric functions. I, Compositio Math. 64 (1987), no. 3, 329–352. MR 918416 (89b:58192a)
E. M. Opdam, Some applications of hypergeometric shift operators, Invent. Math. 98 (1989), no. 1, 1–18. MR 1010152 (91h:33024), DOI 10.1007/BF01388841
E. M. Opdam, Root systems and hypergeometric functions. IV, Compositio Math. 67 (1988), no. 2, 191–209. MR 951750 (90c:58079)
Sigurdur Helgason, Groups and geometric analysis, Mathematical Surveys and Monographs, vol. 83, American Mathematical Society, Providence, RI, 2000. Integral geometry, invariant differential operators, and spherical functions; Corrected reprint of the 1984 original. MR 1790156 (2001h:22001), DOI 10.1090/surv/083
G. J. Heckman, Root systems and hypergeometric functions. II, Compositio Math. 64 (1987), no. 3, 353–373. MR 918417 (89b:58192b)
Gerrit Heckman and Henrik Schlichtkrull, Harmonic analysis and special functions on symmetric spaces, Perspectives in Mathematics, vol. 16, Academic Press, Inc., San Diego, CA, 1994. MR 1313912 (96j:22019)
References_xml – reference: Sigurdur Helgason, Groups and geometric analysis, Mathematical Surveys and Monographs, vol. 83, American Mathematical Society, Providence, RI, 2000. Integral geometry, invariant differential operators, and spherical functions; Corrected reprint of the 1984 original. MR 1790156 (2001h:22001), DOI 10.1090/surv/083
– reference: Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561 (80k:53081)
– reference: E. M. Opdam, Root systems and hypergeometric functions. III, Compositio Math. 67 (1988), no. 1, 21–49. MR 949270 (90k:17021)
– reference: E. M. Opdam, Root systems and hypergeometric functions. IV, Compositio Math. 67 (1988), no. 2, 191–209. MR 951750 (90c:58079)
– reference: Gerrit Heckman and Henrik Schlichtkrull, Harmonic analysis and special functions on symmetric spaces, Perspectives in Mathematics, vol. 16, Academic Press, Inc., San Diego, CA, 1994. MR 1313912 (96j:22019)
– reference: S. G. Gindikin and F. I. Karpelevič, Plancherel measure for symmetric Riemannian spaces of non-positive curvature, Dokl. Akad. Nauk SSSR 145 (1962), 252–255 (Russian). MR 0150239 (27 \#240)
– reference: E. M. Opdam, Some applications of hypergeometric shift operators, Invent. Math. 98 (1989), no. 1, 1–18. MR 1010152 (91h:33024), DOI 10.1007/BF01388841
– reference: Eric M. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), no. 1, 75–121. MR 1353018 (98f:33025), DOI 10.1007/BF02392487
– reference: G. J. Heckman and E. M. Opdam, Root systems and hypergeometric functions. I, Compositio Math. 64 (1987), no. 3, 329–352. MR 918416 (89b:58192a)
– reference: G. J. Heckman, Root systems and hypergeometric functions. II, Compositio Math. 64 (1987), no. 3, 353–373. MR 918417 (89b:58192b)
– reference: Nobukazu Shimeno, The Plancherel formula for spherical functions with a one-dimensional KK-type on a simply connected simple Lie group of Hermitian type, J. Funct. Anal. 121 (1994), no. 2, 330–388. MR 1272131 (95b:22036), DOI 10.1006/jfan.1994.1052
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Snippet We study the action of hypergeometric shift operators on the Heckman-Opdam hypergeometric functions associated with the
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Title An application of hypergeometric shift operators to the $𝜒$-spherical Fourier transform
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