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Book Review

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Book Information:

Authors: Walter R. Bloom and Herbert Heyer
Title: The harmonic analysis of probability measures on hypergroups
Additional book information: de Gruyter Stud. Math., vol. 20, de Gruyter, Berlin and Hawthorne, NY, 1995, vi + 601 pp., ISBN 3-11-012105-0, $125.00$

William C. Connett, Marc-Olivier Gebuhrer, and Alan L. Schwartz (eds.), Applications of hypergroups and related measure algebras, Contemporary Mathematics, vol. 183, American Mathematical Society, Providence, RI, 1995. MR 1334767, DOI 10.1090/conm/183

William C. Connett, Clemens Markett, and Alan L. Schwartz, Convolution and hypergroup structures associated with a class of Sturm-Liouville systems, Trans. Amer. Math. Soc. 332 (1992), no. 1, 365–390. MR 1053112, DOI 10.1090/S0002-9947-1992-1053112-6

William C. Connett, Clemens Markett, and Alan L. Schwartz, Product formulas and convolutions for angular and radial spheroidal wave functions, Trans. Amer. Math. Soc. 338 (1993), no. 2, 695–710. MR 1104199, DOI 10.1090/S0002-9947-1993-1104199-4

W. C. Connett and A. L. Schwartz, The theory of ultraspherical multipliers, Mem. Amer. Math. Soc. 9 (1977), no. 183, iv+92. MR 435708, DOI 10.1090/memo/0183

William C. Connett and Alan L. Schwartz, Product formulas, hypergroups, and the Jacobi polynomials, Bull. Amer. Math. Soc. (N.S.) 22 (1990), no. 1, 91–96. MR 1003862, DOI 10.1090/S0273-0979-1990-15850-1

T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058

Charles F. Dunkl, The measure algebra of a locally compact hypergroup, Trans. Amer. Math. Soc. 179 (1973), 331–348. MR 320635, DOI 10.1090/S0002-9947-1973-0320635-2

George Gasper, Linearization of the product of Jacobi polynomials. II, Canadian J. Math. 22 (1970), 582–593. MR 264136, DOI 10.4153/CJM-1970-065-4

George Gasper, Banach algebras for Jacobi series and positivity of a kernel, Ann. of Math. (2) 95 (1972), 261–280. MR 310536, DOI 10.2307/1970800

[GS95]

O. Gebuhrer and A. L. Schwartz, Sidon sets and Riesz sets for some measure algebras on the disk, to appear, 1995.

Herbert Heyer, Probability measures on locally compact groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 94, Springer-Verlag, Berlin-New York, 1977. MR 0501241

H. Heyer (ed.), Probability measures on groups. VII, Lecture Notes in Mathematics, vol. 1064, Springer-Verlag, Berlin, 1984. MR 772396, DOI 10.1007/BFb0073628

Herbert Heyer, Probability theory on hypergroups: a survey, Probability measures on groups, VII (Oberwolfach, 1983) Lecture Notes in Math., vol. 1064, Springer, Berlin, 1984, pp. 481–550. MR 772428, DOI 10.1007/BFb0073660

H. Heyer (ed.), Probability measures on groups. VIII, Lecture Notes in Mathematics, vol. 1210, Springer-Verlag, Berlin, 1986. MR 878988, DOI 10.1007/BFb0077166

H. Heyer (ed.), Probability measures on groups. IX, Lecture Notes in Mathematics, vol. 1379, Springer-Verlag, Berlin, 1989. MR 1020516, DOI 10.1007/BFb0087839

Herbert Heyer (ed.), Probability measures on groups. X, Plenum Press, New York, 1991. MR 1178970, DOI 10.1007/978-1-4899-2364-6

[Hey95]

------, (ed.), Probability measures on groups and related structures XI, Proceedings Oberwolfach 1994, World Scientific, Singapore, 1995.

[Hir56a]

I. I. Hirschman, Jr., Harmonic analysis and the ultraspherical polynomials, Symposium of the Conference on Harmonic Analysis (Cornell), 1956.

Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3

Robert I. Jewett, Spaces with an abstract convolution of measures, Advances in Math. 18 (1975), no. 1, 1–101. MR 394034, DOI 10.1016/0001-8708(75)90002-X

Yûichi Kanjin, A convolution measure algebra on the unit disc, Tohoku Math. J. (2) 28 (1976), no. 1, 105–115. MR 397319, DOI 10.2748/tmj/1178240884

J. F. C. Kingman, Random walks with spherical symmetry, Acta Math. 109 (1963), 11–53. MR 149567, DOI 10.1007/BF02391808

[KS95]

T. H. Koornwinder and A. L. Schwartz, Product formulas and associated hypergroups for orthogonal polynomials on the simplex and on a parabolic biangle, preprint, 1995.

G. L. Litvinov, Hypergroups and hypergroup algebras, Current problems in mathematics. Newest results, Vol. 26, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985, pp. 57–106, 260 (Russian). MR 849785

Clemens Markett, Product formulas and convolution structure for Fourier-Bessel series, Constr. Approx. 5 (1989), no. 4, 383–404. MR 1014305, DOI 10.1007/BF01889617

R. Spector, Mesures invariantes sur les hypergroupes, Trans. Amer. Math. Soc. 239 (1978), 147–165 (French, with English summary). MR 463806, DOI 10.1090/S0002-9947-1978-0463806-1

Ryszard Szwarc, Orthogonal polynomials and a discrete boundary value problem. I, II, SIAM J. Math. Anal. 23 (1992), no. 4, 959–964, 965–969. MR 1166568, DOI 10.1137/0523052

Review Information:

Reviewer: Alan L. Schwartz
Affiliation: University of Missouri-St. Louis
Email: schwartz@arch.umsl.edu

Journal: Bull. Amer. Math. Soc. 33 (1996), 355-362
DOI: https://doi.org/10.1090/S0273-0979-96-00658-1
Additional Notes: This review was prepared during the tenure of National Science Foundation grant DMS-9404316.
Review copyright: © Copyright 1996 American Mathematical Society