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Three lectures on Hypergroups Delhi, December 1995

  • Alan L. Schwartz
Conference paper
  • 287 Downloads
Part of the Trends in Mathematics book series (TM)

Abstract

LECTURE 1. Cosines, Legendre polynomials, and Bessel functions are examples of eigenfunctions of Sturm-Liouville problems which are also characters of hypergroups. There are many additional examples from the classical special functions. Indeed, it is possible to give conditions on a Sturm-Liouville problem so that its eigenfunctions must be characters of a hyper group. The converse will also be discussed. If a hypergroup consists of measures on a real (compact or not) interval, then with adequate regularity conditions, it must be the case that the characters of the hypergroup are eigenfunctions of a Sturm-Liouville problem.

LECTURE 2. A family of orthogonal polynomials on an interval I may be the characters of a hypergroup of measures supported on I (which would be called a continuous polynomial hypergroup), and the family may also supply the characters of a hypergroup of measures on the discrete set {0, 1, 2, } (which would be called a discrete polynomial hypergroup). The entire category of continuous polynomial hypergroups can be explicitly described, but the full category of discrete polynomial hypergroups has not yet been characterized, though there are some fairly general theorems.

LECTURE 3. The same issues in the second talk raise analagous questions for multivariate orthogonal polynomials. A family of multivariate orthogonal polynomials is a much more subtle object than a family of one-variable orthogonal polynomials. Some progress has been made in the classification problem, and these results will be discussed as well as recently discovered examples.

Keywords

Orthogonal Polynomial Haar Measure Legendre Polynomial Product Formula Measure Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References work

  1. [BH95]
    W. R. Bloom and H. Heyer. Harmonie analysis of probability measures on hypergroups, volume 20 of de Gruyter Studies in Mathematics, de Gruyter, Berlin, New York, 1995.CrossRefGoogle Scholar
  2. [Edw67]
    R. E. Edwards. Fourier series, volume I, II of New York. Holt, Rinehart and Winston, Inc. , 1967.Google Scholar
  3. [EMOT53]
    A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi. Higher Transcendental Functions, volume I. McGraw-Hill Book Company, New York, 1953.Google Scholar
  4. [GR90]
    G. Gasper and M. Rahman. Basic Hypergeometric Series, volume 35 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1990.Google Scholar
  5. [Hey84a]
    H. Heyer, editor. Probability Measures on Groups VII, Proceedings Oberwolfach 1983, volume 1064 of Lecture Notes in Math. Springer, Berlin, 1984.Google Scholar
  6. [Hey86]
    H. Heyer, editor. Probability Measures on Groups VIII, Proceedings Oberwolfach 1985, volume 1210 ofLecture Notes in Math. Springer, Berlin, 1986.Google Scholar
  7. [Hey89]
    H. Heyer, editor. Probability Measures on Groups IX, Proceedings Oberwolf ach 1988, volume 1379 of Lecture Notes in Math. Springer, Berlin, 1989.Google Scholar
  8. [Hey91]
    H. Heyer, editor. Probability Measures on Groups X, Proceedings Oberwolf ach 1990. Plenum, New York, 1991.Google Scholar
  9. [Jew75]
    R. I. Jewett. Spaces with an Abstract convolution of measures. Adv. in Math. , 18:1–101, 1975.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [Rud62]
    W. Rudin. Fourier Analysis on Groups. Interscience Publishers, 1962.zbMATHGoogle Scholar
  11. [Rud74]
    W. Rudin. Real and complex analysis. McGraw-Hill Book Company, New York, second edition, 1974.zbMATHGoogle Scholar
  12. [Sze67]
    G. Szegö. Orthogonal Polynomials, volume 23 ofColloquium Publications. Amer. Math. Soc, Providence, RI, second edition, 1967.Google Scholar
  13. [Wat66]
    G. Watson. A Treatise on the Theory of Bessel Functions. Cambridge University Press, 1966.zbMATHGoogle Scholar

Hypergroups (and related systems in general)

  1. [BH95]
    W. R. Bloom and H. Heyer. Harmonie analysis of probability measures on hypergroups, volume 20 of de Gruyter Studies in Mathematics, de Gruyter, Berlin, New York, 1995.CrossRefGoogle Scholar
  2. [BK92]
    Y. M. Berezanskii and A. A. Kalyuzhnyi. Harmonic Analysis in Hypercomplex Systems. Academia Nauk Ukranii, Institut Matematekii, Kiev Nauko Dumka, Kiev, 1992.Google Scholar
  3. [CGS95]
    W. C. Connett, O. Gebuhrer, and A. L. Schwartz, editors. Applications of hypergroups and related measure algebras. Providence, R. I. , 1995. American Mathematical Society. Contemporary Mathematics,183.zbMATHGoogle Scholar
  4. [CS92]
    W. C. Connett and A. L. Schwartz. Fourier analysis off groups. In A. Nagel and L. Stout, editors,The Madison symposium on complex analysis, pages 169–176, Providence, R. I. , 1992. American Mathematical Society. Contemporary Mathematics,137. CrossRefGoogle Scholar
  5. [DR74]
    C. F. Dunkl and D. E. Ramirez. Krawtchouk polynomials and the symmetrization of hypergroups. SIAM J. Math. Anal, 5:351–366, 1974.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [Dun73]
    C. F. Dunkl. The measure algebra of a locally compact hy-pergroup. Trans. Amer. Math. Soc, 179:331–348, 1973.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [Geb89]
    O. Gebuhrer. Analyse harmonique sur les espaces de Gel’fand-Levitan et applications ala theorie des semi-groupes de convolution. PhD thesis, Universite Louis Pasteur, Strasbourg, France, 1989.Google Scholar
  8. [Geb95]
    M. -O. Gebuhrer. Bounded measure algebras: A fixed point approach. In O. Gebuhrer W. C. Connett and A. L. Schwartz, editors,Applications of hypergroups and related measure algebras, pages 171–190, Providence, R. I. , 1995. American Mathematical Society. Contemporary Mathematics,183.CrossRefGoogle Scholar
  9. [Jew75]
    R. I. Jewett. Spaces with an Abstract convolution of measures. Adv. in Math. , 18:1–101, 1975.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [Lev64]
    B. M. Levitan. Generalized Translation Operators. Israel Program for Scientific Translations, Jerusalem, 1964.zbMATHGoogle Scholar
  11. [Lit87]
    G. L. Litvinov. Hypergroups and hypergroup algebras. J. Soviet Math. , 38:1734–1761, 1987.zbMATHCrossRefGoogle Scholar
  12. [Ros77]
    K. A. Ross. Hypergroups and centers of measure algebras. Ist Naz. Alta Mat. (Symposia Math. ), 22:189–203, 1977.Google Scholar
  13. [Ros78]
    K. A. Ross. Centers of hypergroups. Trans. Amer. Math. Soc, 243:251–269, 1978.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [Ros95]
    K. A. Ross. Signed hypergroups — a survey. In O. Gebuhrer W. C. Connett and A. L. Schwartz, editors,Applications of hypergroups and related measure algebras, pages 319–329, Providence, R. I. , 1995. American Mathematical Society. Contemporary Mathematics,183.CrossRefGoogle Scholar
  15. [Sch. 74]
    A. L. Schwartz. Generalized convolutions and positive definite functions associated with general orthogonal series. Pacific J. Math. , 55:565–582, 1974.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [Sch77]
    A. L. Schwartz, 11-convolution algebras: representation and factorization. Z. Wahrsch. Verw. Gebiete, 41:161–176, 1977.CrossRefGoogle Scholar
  17. [Sch88]
    A. L. Schwartz. Classification of one-dimensional hypergroups. Proc. Amer. Math. Soc, 103:1073–1081, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [Spe78]
    R. Spector. Mesures invariantes sur les hypergroupes. Trans. Amer. Math. Soc, 239:147–165, 1978.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [Zeu89]
    H. Zeuner. One-dimensional hypergroups. Adv. in Math. , pages 1–18, 1989.Google Scholar
  20. [Zeu91]
    H. Zeuner. Duality of commutative hypergroups. In H. Heyer, editor,Probability Measures on Groups X, Proceedings Oberwolfach 1990, New York, 1991. Plenum.Google Scholar

Ultraspherical and Jacobi polynomials

  1. [AF69]
    R. Askey and J. Fitch. Integral representations for Jacobi polynomials and some applications. J. Math. Anal. Appi, 26:411–437, 1969.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [Ask74]
    R. Askey. Jacobi polynomials, I. New proofs of Koorn-winder’s Laplace type integral representation and Bateman’s bilinear sum. SIAM J. Math. Anal, 5:119–124, 1974.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [CMS91]
    W. C. Connett, C. Markett, and A. L. Schwartz. Jacobi polynomials and related hyper group structures. In H. Heyer, editor,Probability Measures on Groups X, Proceedings Ober-wolfach 1990, pages 45–81, New York, 1991. Plenum.Google Scholar
  4. [CS77]
    W. C. Connett and A. L. Schwartz. The theory of ultraspherical multipliers. Mem. Amer. Math. Soc, 183:1–92, 1977.MathSciNetGoogle Scholar
  5. [CS79]
    W. C. Connett and A. L. Schwartz. The Littlewood-Paley theory for Jacobi expansions. Trans. Amer. Math. Soc, 251:219–234, 1979.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [CS89]
    W. C. Connett and A. L. Schwartz. A Hardy-Littlewood maximal inequality for Jacobi type hypergroups. Proc Amer. Math. Soc, 107:137–143, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [CS90c]
    W. C. Connett and A. L. Schwartz. Product formulas, hy-pergroups, and Jacobi polynomials. Bull. Amer. Math. Soc, 22:91–96, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [CS95b]
    W. C. Connett and A. L. Schwartz. Subsets of R which support hypergroups with polynomial characters, to appear inJ. Comput. Appl. Math. , 1995.Google Scholar
  9. [Gas70]
    G. Gasper. Linearization of the product of Jacobi polynomials. II. Canad. J. Math. , 32:582–593, 1970.MathSciNetCrossRefGoogle Scholar
  10. [Gas72]
    G. Gasper. Banach algebras for Jacobi series and positivity of a kernel. Ann. of Math. , 95:261–280, 1972.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [Hir56a]
    I. I. Hirschman, Jr. Harmonic analysis and the ultraspherical polynomials. InSymposium of the Conference on Harmonic Analysis, Cornell, 1956.Google Scholar
  12. [Hir56b]
    I. I. Hirschman Jr. Sur les polynomes ultraspheriques. C. R. Acad. Sci. Paris, 242:2212–2214, 1956.MathSciNetzbMATHGoogle Scholar
  13. [Koo72]
    T. H. Koornwinder. The addition formula for Jacobi polynomials, II, the Laplace type integral representation and the product formula. Technical Report TW 133/72, Mathematisch Centrum, Amsterdam, 1972.Google Scholar
  14. [Koo74a]
    T. Koornwinder. Jacobi polynomials II. An analytic proof of the product formula. SIAM J. Math. Anal, 5:125–137, 1974.MathSciNetzbMATHCrossRefGoogle Scholar

Other one-variable polynomials

  1. [AAA84]
    W. Al-Salam, W. R. Allaway, and R. Askey. Sieved ultraspherical polynomials. Trans. Amer. Math. Soc, 284:41–54, 1984.MathSciNetCrossRefGoogle Scholar
  2. [AI83]
    R. Askey and M. E. H. Ismail. A generalization of ultraspherical polynomials. In P. Erdos, editor,Studies in Pure Mathematics, pages 55–78, Boston, 1983. Birkhauser.Google Scholar
  3. [AKR86]
    R. Askey, T. H. Koornwinder, and M. Rahman. An integral of products of ultraspherical functions and aq-extension. J. London Math. Soc, 33:133–148, 1986.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [Ask70]
    R. Askey. Linearization of the product of orthogonal polynomials. In R. Gunning, editor,Problems in Analysis, pages 223–228, Priceton, NJ, 1970. Princeton University Press.Google Scholar
  5. [Boc29]
    S. Bochner. Über Sturm-Liouvillische Polynomesysteme. Math. Z. , 29:730–736, 1929.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [DR74]
    C. F. Dunkl and D. E. Ramirez. Krawtchouk polynomials and the symmetrization of hypergroups. SIAM J. Math. Anal, 5:351–366, 1974.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [Fav35]
    J. Favard. Sur les polynomes de Tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935. Sér A-B.Google Scholar
  8. [Koo78]
    T. H. Koornwinder. Positivity proofs for linearization and connection coefficients of orthogonal polynomials satisfying an addition formula. J. London Math. Soc, (2) 18:101–114, 1978.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [Lai80]
    T. P. Laine. The product formula and convolution structure for the generalized Chebyshev polynomials. SIAM J. Math. Anal, 11:133–146, 1980.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [Las83]
    R. Lasser. Bochner theorems for hypergroups and their applications to orthogonal polynomial expansions. J. Approx. Theory, 37:311–325, 1983.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [L091]
    R. Lasser and J. Obermaier. On Fejér means with respect to orthogonal polynomials: a hypergroup-theoretic approach. Progress in Approximation Theory, pages 551–565, 1991.Google Scholar
  12. [Mar94]
    C. Markett. Linearization of the product of symmetrical orthogonal polynomials. Constr. Approx. , 10:317–338, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [Rah86]
    M. Rahman. A product formula for the continuousq-Jacobi polynomials. J. Math. Anal. Appl, 118:309–322, 1986.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [Sho36]
    J. Shohat. The relation of the classical orthogonal polynomials to the polynomials of Appell. Amer. J. Math. , 58:453–464, 1936.MathSciNetCrossRefGoogle Scholar
  15. [Soa91]
    P. Soardi. Bernstein polynomials and random walks on hypergroups. preprint, 1991.Google Scholar
  16. [Szw92a]
    R. Szwarc. Orthogonal polynomials and a discrete boundary value problem I. SI AM J. Math. Anal, 23:959–964, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [Szw92b]
    R. Szwarc. Orthogonal polynomials and a discrete boundary value problem II. SI AM J. Math. Anal, 23:965–969, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [Voi90]
    M. Voit. Central limit theorems for a class of polynomial hypergroups. Adv. in Appl Probab. , 22:68–87, 1990.MathSciNetzbMATHCrossRefGoogle Scholar

Other special functions

  1. [CMS93]
    W. C. Connett, C. Markett, and A. L. Schwartz. Product formulas and convolutions for angular and radial spheroidal wave functions. Trans. Amer. Math. Soc, 338:695–710, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [FK73]
    M. Flensted-Jensen and T. Koornwinder. The convolution structure for Jacobi function expansions. Ark. Mat. , 11:245–262, 1973.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [FK79]
    M. Flensted-Jensen and T. H. Koornwinder. Positive definite spherical functions on a non-compact rank one symmetric space. In P. Eymard, J. Faraut, G. Schiffman, and R. Taka-hashi, editors,Analyse harmonique sur les groupes de Lie, II, pages 249–282. Springer, 1979. Lecture Notes in Math. , 739.CrossRefGoogle Scholar
  4. [Fle72]
    M. Flensted-Jensen. Paley-Wiener type theorems for a differential operator connected with symmetric spaces. Ark. Mat. , 10:143–162, 1972.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [Koo75a]
    T. Koornwinder. A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. , 13:145–159, 1975.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [Mar89]
    C. Markett. Product formulas and convolution structure for Fourier-Bessel series. Constr. Approx. , 5:383–404, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [Per86]
    M. Perlstadt. Polynomial analogs of prolate spheroidal wave functions and uncertainty. SI AM J. Math. Anal, 17:242–248, 1986.MathSciNetCrossRefGoogle Scholar
  8. [Sch71]
    A. L. Schwartz. The structure of the algebra of Hankel and Hankel-Stieltjes transforms. Canad. J. Math. , 23:236–246, 1971.MathSciNetzbMATHCrossRefGoogle Scholar

Disk polynomials

  1. [AT74]
    H. Annabi and K. Trimèche. Convolution généralisée sur le disque unité. C. R. Acad. Sc. Paris, 278:21–24, 1974.zbMATHGoogle Scholar
  2. [BG90]
    M. Bouhaik and L. Gallardo. Une loi des grandes nombres et un théorème limite central pour les chaines de Markov sur N2associés aux polynômes discaux. C. R. Acad. Sci. Paris, 310:739–744, 1990.MathSciNetzbMATHGoogle Scholar
  3. [BG91]
    M. Bouhaik and L. Gallardo. A Mehler-Heine formula for disk polynomials. Indag. Math. , 1:9–18, 1991MathSciNetGoogle Scholar
  4. [BG92]
    M. Bouhaik and L. Gallardo. Un théorm¯e limite central dans hypergroupe bidimensionnel. Ann. Inst. H. Poincaré, 28(1):47–61, 1992.MathSciNetzbMATHGoogle Scholar
  5. [GS96]
    O. Gebuhrer and A. L. Schwartz. Sidon sets and Riesz sets for some measure algebras on the disk, to appearinColloq. Math. , 1996.Google Scholar
  6. [HK93]
    H. Heyer and S. Koshi. Harmonic analysis on the disk hy-pergroup. Mathematical Seminar Notes, Tokyo Metropolitan University, 1993.Google Scholar
  7. [Kan76]
    Y. Kanjin. A convolution measure algebra on the unit disc. Tôhoku Math. J. (2), 28:105–115, 1976.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [Kan85]
    Y. Kanjin. Banach algebra related to disk polynomials. Tôhoku Math. J. (2), 37:395–404, 1985.MathSciNetzbMATHCrossRefGoogle Scholar

Multivariate polynomials

  1. [CS95a]
    W. C. Connett and A. L. Schwartz. Continuous 2-variable polynomial hypergroups. In O. Gebuhrer W. C. Connett and A. L. Schwartz, editors,Applications of hypergroups and related measure algebras, pages 89–109, Providence, R. I. , 1995. American Mathematical Society. Contemporary Mathematics,183. CrossRefGoogle Scholar
  2. [KMT91]
    E. G. Kalnins, W. Miller, Jr. , and M. V. Tratnik. Families of orthogonal and biorthogonal polynomials on the iV-sphere. SIAM J. Math. Anal, 22:272–294, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [Koo74b]
    T. H. Koornwinder. Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. I, II, III, IV. Indag. Math. , 36:48–58, 59–66, 357–369, 370–381, 1974.MathSciNetGoogle Scholar
  4. [Koo75b]
    T. H. Koornwinder. Two-variable analogues of the classical orthogonal polynomials. In Richard A. Askey, editor,Theory and Applications of Special Functions, pages 435–495, New York, 1975. Academic Press, Inc.Google Scholar
  5. [KS67]
    H. L. Krall and I. M. Sheffer. Orthogonal polynomials in two variables. Ann. Mat. Pura Appl, 76:325–376, 1967.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [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.Google Scholar
  7. [Mac90]
    I. G. Macdonald. Orthogonal polynomials associated with root systems. In P. Nevai, editor,Orthogonal polynomials: theory and practice: (proceedings of the NATO Advanced Study Institute on “Orthogonal Polynomials and Their Applications,” the Ohio State University, Columbus, Ohio, U. S. A. , May 22-June 3, 1989), pages 311–318. Kluwer Academic Publishers, 1990.Google Scholar
  8. [Tra91]
    M. V. Tratnik. Some multivariable orthogonal polynomials of the Askey tableau-continuous families. J. Math. Phys. , 32:2065–2073, 1991.MathSciNetzbMATHCrossRefGoogle Scholar

Differential equations

  1. [AT79]
    A. Achour and K. Trimèche. Opérateurs de translation généralisée associés á un opérateur différentiel singulier sur un intervalle borné. C. R. Acad. Sci. Paris, 288:399–402, 1979.zbMATHGoogle Scholar
  2. [Ché72]
    H. Chébli. Sur la positivité des opérateurs de “translation généralisée” associés à un opérateur de Sturm-Liouville sur]0,∞[. C. R. Acad. Sci. Paris, 275:601–604, 1972.zbMATHGoogle Scholar
  3. [CMS92]
    W. C. Connett, C. Markett, and A. L. Schwartz. Convolution and hypergroup structures associated with a class of Sturm-Liouville systems. Trans. Amer. Math. Soc, 332:365–390, 1992.MathSciNetzbMATHGoogle Scholar
  4. [CS90a]
    W. C. Connett and A. L. Schwartz. Analysis of a class of probability preserving measure algebras on a compact interval. Trans. Amer. Math. Soc, 320:371–393, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [CS90b]
    W. C. Connett and A. L. Schwartz. Positive product formulas and hypergroups associated with singular Sturm-Liouville problems on a compact interval. Colloq. Math. , LX/LXL:525–535, 1990.MathSciNetGoogle Scholar

Probability on hypergroups

  1. [BG90]
    M. Bouhaik and L. Gallardo. Une loi des grandes nombres et un théorème limite central pour les chaines de Markov sur N2associés aux polynômes discaux. C. R. Acad. Sci. Paris, 310:739–744, 1990.MathSciNetzbMATHGoogle Scholar
  2. [BG92]
    M. Bouhaik and L. Gallardo. Un théorm¯e limite central dans hypergroupe bidimensionnel. Ann. Inst. H. Poincaré, 28(1):47–61, 1992.MathSciNetzbMATHGoogle Scholar
  3. [GG87]
    L. Gallardo and O. Gebuhrer. Marches aléatoires et hyper-groupes. Exposition. Math. , 5:41–73, 1987.MathSciNetzbMATHGoogle Scholar
  4. [Hey84a]
    H. Heyer, editor. Probability Measures on Groups VII, Proceedings Oberwolfach 1983, volume 1064 of Lecture Notes in Math. Springer, Berlin, 1984.Google Scholar
  5. [Hey86]
    H. Heyer, editor. Probability Measures on Groups VIII, Proceedings Oberwolfach 1985, volume 1210 ofLecture Notes in Math. Springer, Berlin, 1986.Google Scholar
  6. [Hey84b]
    H. Heyer. Probability theory on hypergroups: a survey. In H. Heyer, editor,Probability Measures on Groups VII, Proceedings Oberwolfach 1983, pages 481–550, Berlin, 1984. Springer. Lecture Notes in Math. , vol. 1064.Google Scholar
  7. [Hey91]
    H. Heyer, editor. Probability Measures on Groups X, Proceedings Oberwolf ach 1990. Plenum, New York, 1991.Google Scholar
  8. [Voi90]
    M. Voit. Central limit theorems for a class of polynomial hypergroups. Adv. in Appl Probab. , 22:68–87, 1990.MathSciNetzbMATHCrossRefGoogle Scholar

Miscellaneous

  1. [Bru87]
    R. G. M. Brummelhuis. An F. and M. Riesz theorem for bounded symmetric domains. Ann. Inst. Fourier (Grenoble), 37:139–150, 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [Dun66]
    C. F. Dunkl. Operators and harmonic analysis on the sphere. Trans. Amer. Math. Soc, 125:250–263, 1966.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [Mic51]
    E. Michael. Topologies on spaces of subsets. Trans. Amer. Math. Soc, 71:152–182, 1951.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [AAA84]
    W. Al-Salam, W. R. Allaway, and R. Askey. Sieved ultraspherical polynomials. Trans. Amer. Math. Soc, 284:41–54, 1984.MathSciNetCrossRefGoogle Scholar
  5. [AF69]
    R. Askey and J. Fitch. Integral representations for Jacobi polynomials and some applications. J. Math. Anal. Appi, 26:411–437, 1969.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [AI83]
    R. Askey and M. E. H. Ismail. A generalization of ultraspherical polynomials. In P. Erdos, editor, Studies in Pure Mathematics, pages 55–78, Boston, 1983. Birkhauser.Google Scholar
  7. [AKR86]
    R. Askey, T. H. Koornwinder, and M. Rahman. An integral of products of ultraspherical functions and aq-extension. J. London Math. Soc, 33:133–148, 1986.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [Ask70]
    R. Askey. Linearization of the product of orthogonal polynomials. In R. Gunning, editor,Problems in Analysis, pages 223–228, Priceton, NJ, 1970. Princeton University Press.Google Scholar
  9. [Ask74]
    R. Askey. Jacobi polynomials, I. New proofs of Koorn-winder’s Laplace type integral representation and Bateman’s bilinear sum. SIAM J. Math. Anal, 5:119–124, 1974.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [AT74]
    H. Annabi and K. Trimèche. Convolution généralisée sur le disque unité. C. R. Acad. Sc. Paris, 278:21–24, 1974.zbMATHGoogle Scholar
  11. [AT79]
    A. Achour and K. Trimèche. Opérateurs de translation généralisée associés á un opérateur différentiel singulier sur un intervalle borné. C. R. Acad. Sci. Paris, 288:399–402, 1979.zbMATHGoogle Scholar
  12. [BG90]
    M. Bouhaik and L. Gallardo. Une loi des grandes nombres et un théorème limite central pour les chaines de Markov sur N2associés aux polynômes discaux. C. R. Acad. Sci. Paris, 310:739–744, 1990.MathSciNetzbMATHGoogle Scholar
  13. [BG91]
    M. Bouhaik and L. Gallardo. A Mehler-Heine formula for disk polynomials. Indag. Math. , 1:9–18, 1991MathSciNetGoogle Scholar
  14. [BG92]
    M. Bouhaik and L. Gallardo. Un théorm¯e limite central dans hypergroupe bidimensionnel. Ann. Inst. H. Poincaré, 28(1):47–61, 1992.MathSciNetzbMATHGoogle Scholar
  15. [BH95]
    W. R. Bloom and H. Heyer. Harmonie analysis of probability measures on hypergroups, volume 20 of de Gruyter Studies in Mathematics, de Gruyter, Berlin, New York, 1995.CrossRefGoogle Scholar
  16. [BK92]
    Y. M. Berezanskii and A. A. Kalyuzhnyi. Harmonic Analysis in Hypercomplex Systems. Academia Nauk Ukranii, Institut Matematekii, Kiev Nauko Dumka, Kiev, 1992.Google Scholar
  17. [Boc29]
    S. Bochner. Über Sturm-Liouvillische Polynomesysteme. Math. Z. , 29:730–736, 1929.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [Bru87]
    R. G. M. Brummelhuis. An F. and M. Riesz theorem for bounded symmetric domains. Ann. Inst. Fourier (Grenoble), 37:139–150, 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [CGS95]
    W. C. Connett, O. Gebuhrer, and A. L. Schwartz, editors. Applications of hypergroups and related measure algebras. Providence, R. I. , 1995. American Mathematical Society. Contemporary Mathematics,183.zbMATHGoogle Scholar
  20. [Ché72]
    H. Chébli. Sur la positivité des opérateurs de “translation généralisée” associés à un opérateur de Sturm-Liouville sur]0,∞[. C. R. Acad. Sci. Paris, 275:601–604, 1972.zbMATHGoogle Scholar
  21. [CMS91]
    W. C. Connett, C. Markett, and A. L. Schwartz. Jacobi polynomials and related hyper group structures. In H. Heyer, editor,Probability Measures on Groups X, Proceedings Ober-wolfach 1990, pages 45–81, New York, 1991. Plenum.Google Scholar
  22. [CMS92]
    W. C. Connett, C. Markett, and A. L. Schwartz. Convolution and hypergroup structures associated with a class of Sturm-Liouville systems. Trans. Amer. Math. Soc, 332:365–390, 1992.MathSciNetzbMATHGoogle Scholar
  23. [CMS93]
    W. C. Connett, C. Markett, and A. L. Schwartz. Product formulas and convolutions for angular and radial spheroidal wave functions. Trans. Amer. Math. Soc, 338:695–710, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [CS77]
    W. C. Connett and A. L. Schwartz. The theory of ultraspherical multipliers. Mem. Amer. Math. Soc, 183:1–92, 1977.MathSciNetGoogle Scholar
  25. [CS79]
    W. C. Connett and A. L. Schwartz. The Littlewood-Paley theory for Jacobi expansions. Trans. Amer. Math. Soc, 251:219–234, 1979.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [CS89]
    W. C. Connett and A. L. Schwartz. A Hardy-Littlewood maximal inequality for Jacobi type hypergroups. Proc Amer. Math. Soc, 107:137–143, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [CS90a]
    W. C. Connett and A. L. Schwartz. Analysis of a class of probability preserving measure algebras on a compact interval. Trans. Amer. Math. Soc, 320:371–393, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [CS90b]
    W. C. Connett and A. L. Schwartz. Positive product formulas and hypergroups associated with singular Sturm-Liouville problems on a compact interval. Colloq. Math. , LX/LXL:525–535, 1990.MathSciNetGoogle Scholar
  29. [CS90c]
    W. C. Connett and A. L. Schwartz. Product formulas, hy-pergroups, and Jacobi polynomials. Bull. Amer. Math. Soc, 22:91–96, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [CS92]
    W. C. Connett and A. L. Schwartz. Fourier analysis off groups. In A. Nagel and L. Stout, editors,The Madison symposium on complex analysis, pages 169–176, Providence, R. I. , 1992. American Mathematical Society. Contemporary Mathematics,137. CrossRefGoogle Scholar
  31. groups. In A. Nagel and L. Stout, editors,The Madison symposium on complex analysis, pages 169–176, Providence, R. I. , 1992. American Mathematical Society. Contemporary Mathematics,137.Google Scholar
  32. [CS95a]
    W. C. Connett and A. L. Schwartz. Continuous 2-variable polynomial hypergroups. In O. Gebuhrer W. C. Connett and A. L. Schwartz, editors,Applications of hypergroups and related measure algebras, pages 89–109, Providence, R. I. , 1995. American Mathematical Society. Contemporary Mathematics,183.Google Scholar
  33. [CS95a]
    W. C. Connett and A. L. Schwartz. Continuous 2-variable polynomial hypergroups. In O. Gebuhrer W. C. Connett and A. L. Schwartz, editors,Applications of hypergroups and related measure algebras, pages 89–109, Providence, R. I. , 1995. American Mathematical Society. Contemporary Mathematics,183.Google Scholar
  34. [CS95b]
    W. C. Connett and A. L. Schwartz. Subsets of R which support hypergroups with polynomial characters, to appear inJ. Comput. Appl. Math. , 1995.Google Scholar
  35. [DR74]
    C. F. Dunkl and D. E. Ramirez. Krawtchouk polynomials and the symmetrization of hypergroups. SIAM J. Math. Anal, 5:351–366, 1974.MathSciNetzbMATHCrossRefGoogle Scholar
  36. [Dun66]
    C. F. Dunkl. Operators and harmonic analysis on the sphere. Trans. Amer. Math. Soc, 125:250–263, 1966.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [Dun73]
    C. F. Dunkl. The measure algebra of a locally compact hy-pergroup. Trans. Amer. Math. Soc, 179:331–348, 1973.MathSciNetzbMATHCrossRefGoogle Scholar
  38. [Edw67]
    R. E. Edwards. Fourier series, volume I, II of New York. Holt, Rinehart and Winston, Inc. , 1967.Google Scholar
  39. [EMOT53]
    A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi. Higher Transcendental Functions, volume I. McGraw-Hill Book Company, New York, 1953.Google Scholar
  40. [Fav35]
    J. Favard. Sur les polynomes de Tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935. Sér A-B.Google Scholar
  41. [FK73]
    M. Flensted-Jensen and T. Koornwinder. The convolution structure for Jacobi function expansions. Ark. Mat. , 11:245–262, 1973.MathSciNetzbMATHCrossRefGoogle Scholar
  42. [FK79]
    M. Flensted-Jensen and T. H. Koornwinder. Positive definite spherical functions on a non-compact rank one symmetric space. In P. Eymard, J. Faraut, G. Schiffman, and R. Taka-hashi, editors,Analyse harmonique sur les groupes de Lie, II, pages 249–282. Springer, 1979. Lecture Notes in Math. , 739.CrossRefGoogle Scholar
  43. [Fle72]
    M. Flensted-Jensen. Paley-Wiener type theorems for a differential operator connected with symmetric spaces. Ark. Mat. , 10:143–162, 1972.MathSciNetzbMATHCrossRefGoogle Scholar
  44. [Gas70]
    G. Gasper. Linearization of the product of Jacobi polynomials. II. Canad. J. Math. , 32:582–593, 1970.MathSciNetCrossRefGoogle Scholar
  45. [Gas72]
    G. Gasper. Banach algebras for Jacobi series and positivity of a kernel. Ann. of Math. , 95:261–280, 1972.MathSciNetzbMATHCrossRefGoogle Scholar
  46. [Geb89]
    O. Gebuhrer. Analyse harmonique sur les espaces de Gel’fand-Levitan et applications ala theorie des semi-groupes de convolution. PhD thesis, Universite Louis Pasteur, Strasbourg, France, 1989.Google Scholar
  47. [Geb95]
    M. -O. Gebuhrer. Bounded measure algebras: A fixed point approach. In O. Gebuhrer W. C. Connett and A. L. Schwartz, editors,Applications of hypergroups and related measure algebras, pages 171–190, Providence, R. I. , 1995. American Mathematical Society. Contemporary Mathematics,183.CrossRefGoogle Scholar
  48. [GG87]
    L. Gallardo and O. Gebuhrer. Marches aléatoires et hyper-groupes. Exposition. Math. , 5:41–73, 1987.MathSciNetzbMATHGoogle Scholar
  49. [GR90]
    G. Gasper and M. Rahman. Basic Hypergeometric Series, volume 35 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1990.Google Scholar
  50. [GS96]
    O. Gebuhrer and A. L. Schwartz. Sidon sets and Riesz sets for some measure algebras on the disk, to appearinColloq. Math. , 1996.Google Scholar
  51. [Hey84a]
    H. Heyer, editor. Probability Measures on Groups VII, Proceedings Oberwolfach 1983, volume 1064 of Lecture Notes in Math. Springer, Berlin, 1984.Google Scholar
  52. [Hey84b]
    H. Heyer. Probability theory on hypergroups: a survey. In H. Heyer, editor,Probability Measures on Groups VII, Proceedings Oberwolfach 1983, pages 481–550, Berlin, 1984. Springer. Lecture Notes in Math. , vol. 1064.Google Scholar
  53. [Hey86]
    H. Heyer, editor. Probability Measures on Groups VIII, Proceedings Oberwolfach 1985, volume 1210 ofLecture Notes in Math. Springer, Berlin, 1986.Google Scholar
  54. [Hey89]
    H. Heyer, editor. Probability Measures on Groups IX, Proceedings Oberwolf ach 1988, volume 1379 of Lecture Notes in Math. Springer, Berlin, 1989.Google Scholar
  55. [Hey91]
    H. Heyer, editor. Probability Measures on Groups X, Proceedings Oberwolf ach 1990. Plenum, New York, 1991.Google Scholar
  56. [Hir56a]
    I. I. Hirschman, Jr. Harmonic analysis and the ultraspherical polynomials. InSymposium of the Conference on Harmonic Analysis, Cornell, 1956.Google Scholar
  57. [Hir56b]
    I. I. Hirschman Jr. Sur les polynomes ultraspheriques. C. R. Acad. Sci. Paris, 242:2212–2214, 1956.MathSciNetzbMATHGoogle Scholar
  58. [HK93]
    H. Heyer and S. Koshi. Harmonic analysis on the disk hy-pergroup. Mathematical Seminar Notes, Tokyo Metropolitan University, 1993.Google Scholar
  59. [Jew75]
    R. I. Jewett. Spaces with an Abstract convolution of measures. Adv. in Math. , 18:1–101, 1975.MathSciNetzbMATHCrossRefGoogle Scholar
  60. [Kan76]
    Y. Kanjin. A convolution measure algebra on the unit disc. Tôhoku Math. J. (2), 28:105–115, 1976.MathSciNetzbMATHCrossRefGoogle Scholar
  61. [Kan85]
    Y. Kanjin. Banach algebra related to disk polynomials. Tôhoku Math. J. (2), 37:395–404, 1985.MathSciNetzbMATHCrossRefGoogle Scholar
  62. [Kin63]
    J. F. C. Kingman. Random walks with spherical symmetry. Acta Math. , 109:11–53, 1963.MathSciNetzbMATHCrossRefGoogle Scholar
  63. [KMT91]
    E. G. Kalnins, W. Miller, Jr. , and M. V. Tratnik. Families of orthogonal and biorthogonal polynomials on the iV-sphere. SIAM J. Math. Anal, 22:272–294, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  64. [Koo72]
    T. H. Koornwinder. The addition formula for Jacobi polynomials, II, the Laplace type integral representation and the product formula. Technical Report TW 133/72, Mathematisch Centrum, Amsterdam, 1972.Google Scholar
  65. [Koo74a]
    T. Koornwinder. Jacobi polynomials II. An analytic proof of the product formula. SIAM J. Math. Anal, 5:125–137, 1974.MathSciNetzbMATHCrossRefGoogle Scholar
  66. [Koo74b]
    T. H. Koornwinder. Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. I, II, III, IV. Indag. Math. , 36:48–58, 59–66, 357–369, 370–381, 1974.MathSciNetGoogle Scholar
  67. [Koo75a]
    T. Koornwinder. A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. , 13:145–159, 1975.MathSciNetzbMATHCrossRefGoogle Scholar
  68. [Koo75b]
    T. H. Koornwinder. Two-variable analogues of the classical orthogonal polynomials. In Richard A. Askey, editor,Theory and Applications of Special Functions, pages 435–495, New York, 1975. Academic Press, Inc.Google Scholar
  69. [Koo78]
    T. H. Koornwinder. Positivity proofs for linearization and connection coefficients of orthogonal polynomials satisfying an addition formula. J. London Math. Soc, (2) 18:101–114, 1978.MathSciNetzbMATHCrossRefGoogle Scholar
  70. [KS67]
    H. L. Krall and I. M. Sheffer. Orthogonal polynomials in two variables. Ann. Mat. Pura Appl, 76:325–376, 1967.MathSciNetzbMATHCrossRefGoogle Scholar
  71. [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.Google Scholar
  72. [Lai80]
    T. P. Laine. The product formula and convolution structure for the generalized Chebyshev polynomials. SIAM J. Math. Anal, 11:133–146, 1980.MathSciNetzbMATHCrossRefGoogle Scholar
  73. [Las83]
    R. Lasser. Bochner theorems for hypergroups and their applications to orthogonal polynomial expansions. J. Approx. Theory, 37:311–325, 1983.MathSciNetzbMATHCrossRefGoogle Scholar
  74. [Lev64]
    B. M. Levitan. Generalized Translation Operators. Israel Program for Scientific Translations, Jerusalem, 1964.zbMATHGoogle Scholar
  75. [Lit87]
    G. L. Litvinov. Hypergroups and hypergroup algebras. J. Soviet Math. , 38:1734–1761, 1987.zbMATHCrossRefGoogle Scholar
  76. [L091]
    R. Lasser and J. Obermaier. On Fejér means with respect to orthogonal polynomials: a hypergroup-theoretic approach. Progress in Approximation Theory, pages 551–565, 1991.Google Scholar
  77. [Mac90]
    I. G. Macdonald. Orthogonal polynomials associated with root systems. In P. Nevai, editor,Orthogonal polynomials: theory and practice: (proceedings of the NATO Advanced Study Institute on “Orthogonal Polynomials and Their Applications,” the Ohio State University, Columbus, Ohio, U. S. A. , May 22-June 3, 1989), pages 311–318. Kluwer Academic Publishers, 1990.Google Scholar
  78. [Mar89]
    C. Markett. Product formulas and convolution structure for Fourier-Bessel series. Constr. Approx. , 5:383–404, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  79. [Mar94]
    C. Markett. Linearization of the product of symmetrical orthogonal polynomials. Constr. Approx. , 10:317–338, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  80. [Mic51]
    E. Michael. Topologies on spaces of subsets. Trans. Amer. Math. Soc, 71:152–182, 1951.MathSciNetzbMATHCrossRefGoogle Scholar
  81. [Per86]
    M. Perlstadt. Polynomial analogs of prolate spheroidal wave functions and uncertainty. SI AM J. Math. Anal, 17:242–248, 1986.MathSciNetCrossRefGoogle Scholar
  82. [Rah86]
    M. Rahman. A product formula for the continuousq-Jacobi polynomials. J. Math. Anal. Appl, 118:309–322, 1986.MathSciNetzbMATHCrossRefGoogle Scholar
  83. [Ros77]
    K. A. Ross. Hypergroups and centers of measure algebras. Ist Naz. Alta Mat. (Symposia Math. ), 22:189–203, 1977.Google Scholar
  84. [Ros78]
    K. A. Ross. Centers of hypergroups. Trans. Amer. Math. Soc, 243:251–269, 1978.MathSciNetzbMATHCrossRefGoogle Scholar
  85. [Ros95]
    K. A. Ross. Signed hypergroups — a survey. In O. Gebuhrer W. C. Connett and A. L. Schwartz, editors,Applications of hypergroups and related measure algebras, pages 319–329, Providence, R. I. , 1995. American Mathematical Society. Contemporary Mathematics,183.CrossRefGoogle Scholar
  86. [Rud62]
    W. Rudin. Fourier Analysis on Groups. Interscience Publishers, 1962.zbMATHGoogle Scholar
  87. [Rud74]
    W. Rudin. Real and complex analysis. McGraw-Hill Book Company, New York, second edition, 1974.zbMATHGoogle Scholar
  88. [Sch71]
    A. L. Schwartz. The structure of the algebra of Hankel and Hankel-Stieltjes transforms. Canad. J. Math. , 23:236–246, 1971.MathSciNetzbMATHCrossRefGoogle Scholar
  89. [Sch. 74]
    A. L. Schwartz. Generalized convolutions and positive definite functions associated with general orthogonal series. Pacific J. Math. , 55:565–582, 1974.MathSciNetzbMATHCrossRefGoogle Scholar
  90. [Sch77]
    A. L. Schwartz, 11-convolution algebras: representation and factorization. Z. Wahrsch. Verw. Gebiete, 41:161–176, 1977.CrossRefGoogle Scholar
  91. [Sch88]
    A. L. Schwartz. Classification of one-dimensional hypergroups. Proc. Amer. Math. Soc, 103:1073–1081, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  92. [Sho36]
    J. Shohat. The relation of the classical orthogonal polynomials to the polynomials of Appell. Amer. J. Math. , 58:453–464, 1936.MathSciNetCrossRefGoogle Scholar
  93. [Soa91]
    P. Soardi. Bernstein polynomials and random walks on hypergroups. preprint, 1991.Google Scholar
  94. [Spe78]
    R. Spector. Mesures invariantes sur les hypergroupes. Trans. Amer. Math. Soc, 239:147–165, 1978.MathSciNetzbMATHCrossRefGoogle Scholar
  95. [Sze67]
    G. Szegö. Orthogonal Polynomials, volume 23 ofColloquium Publications. Amer. Math. Soc, Providence, RI, second edition, 1967.Google Scholar
  96. [Szw92a]
    R. Szwarc. Orthogonal polynomials and a discrete boundary value problem I. SI AM J. Math. Anal, 23:959–964, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  97. [Szw92b]
    R. Szwarc. Orthogonal polynomials and a discrete boundary value problem II. SI AM J. Math. Anal, 23:965–969, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  98. [Tra91]
    M. V. Tratnik. Some multivariable orthogonal polynomials of the Askey tableau-continuous families. J. Math. Phys. , 32:2065–2073, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  99. [Voi90]
    M. Voit. Central limit theorems for a class of polynomial hypergroups. Adv. in Appl Probab. , 22:68–87, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  100. [Wat66]
    G. Watson. A Treatise on the Theory of Bessel Functions. Cambridge University Press, 1966.zbMATHGoogle Scholar
  101. [Zeu89]
    H. Zeuner. One-dimensional hypergroups. Adv. in Math. , pages 1–18, 1989.Google Scholar
  102. [Zeu91]
    H. Zeuner. Duality of commutative hypergroups. In H. Heyer, editor,Probability Measures on Groups X, Proceedings Oberwolfach 1990, New York, 1991. Plenum.Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Alan L. Schwartz
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of Missouri-St. LouisSt. LouisUSA

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