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.
The preparation of these lectures took place during the tenure of NSF grant DMS-9404316
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References work
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W. C. Connett and A. L. Schwartz. The theory of ultraspherical multipliers. Mem. Amer. Math. Soc, 183:1–92, 1977.
W. C. Connett and A. L. Schwartz. The Littlewood-Paley theory for Jacobi expansions. Trans. Amer. Math. Soc, 251:219–234, 1979.
W. C. Connett and A. L. Schwartz. A Hardy-Littlewood maximal inequality for Jacobi type hypergroups. Proc Amer. Math. Soc, 107:137–143, 1989.
W. C. Connett and A. L. Schwartz. Product formulas, hy-pergroups, and Jacobi polynomials. Bull. Amer. Math. Soc, 22:91–96, 1990.
W. C. Connett and A. L. Schwartz. Subsets of R which support hypergroups with polynomial characters, to appear inJ. Comput. Appl. Math. , 1995.
G. Gasper. Linearization of the product of Jacobi polynomials. II. Canad. J. Math. , 32:582–593, 1970.
G. Gasper. Banach algebras for Jacobi series and positivity of a kernel. Ann. of Math. , 95:261–280, 1972.
I. I. Hirschman, Jr. Harmonic analysis and the ultraspherical polynomials. InSymposium of the Conference on Harmonic Analysis, Cornell, 1956.
I. I. Hirschman Jr. Sur les polynomes ultraspheriques. C. R. Acad. Sci. Paris, 242:2212–2214, 1956.
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.
T. Koornwinder. Jacobi polynomials II. An analytic proof of the product formula. SIAM J. Math. Anal, 5:125–137, 1974.
Other one-variable polynomials
W. Al-Salam, W. R. Allaway, and R. Askey. Sieved ultraspherical polynomials. Trans. Amer. Math. Soc, 284:41–54, 1984.
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.
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.
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.
S. Bochner. Über Sturm-Liouvillische Polynomesysteme. Math. Z. , 29:730–736, 1929.
C. F. Dunkl and D. E. Ramirez. Krawtchouk polynomials and the symmetrization of hypergroups. SIAM J. Math. Anal, 5:351–366, 1974.
J. Favard. Sur les polynomes de Tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935. Sér A-B.
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.
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M. Voit. Central limit theorems for a class of polynomial hypergroups. Adv. in Appl Probab. , 22:68–87, 1990.
Other special functions
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M. Flensted-Jensen and T. Koornwinder. The convolution structure for Jacobi function expansions. Ark. Mat. , 11:245–262, 1973.
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.
M. Flensted-Jensen. Paley-Wiener type theorems for a differential operator connected with symmetric spaces. Ark. Mat. , 10:143–162, 1972.
T. Koornwinder. A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. , 13:145–159, 1975.
C. Markett. Product formulas and convolution structure for Fourier-Bessel series. Constr. Approx. , 5:383–404, 1989.
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Disk polynomials
H. Annabi and K. Trimèche. Convolution généralisée sur le disque unité. C. R. Acad. Sc. Paris, 278:21–24, 1974.
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.
M. Bouhaik and L. Gallardo. A Mehler-Heine formula for disk polynomials. Indag. Math. , 1:9–18, 1991
M. Bouhaik and L. Gallardo. Un théorm¯e limite central dans hypergroupe bidimensionnel. Ann. Inst. H. Poincaré, 28(1):47–61, 1992.
O. Gebuhrer and A. L. Schwartz. Sidon sets and Riesz sets for some measure algebras on the disk, to appearinColloq. Math. , 1996.
H. Heyer and S. Koshi. Harmonic analysis on the disk hy-pergroup. Mathematical Seminar Notes, Tokyo Metropolitan University, 1993.
Y. Kanjin. A convolution measure algebra on the unit disc. Tôhoku Math. J. (2), 28:105–115, 1976.
Y. Kanjin. Banach algebra related to disk polynomials. Tôhoku Math. J. (2), 37:395–404, 1985.
Multivariate polynomials
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.
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.
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.
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.
H. L. Krall and I. M. Sheffer. Orthogonal polynomials in two variables. Ann. Mat. Pura Appl, 76:325–376, 1967.
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.
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Differential equations
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.
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.
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.
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.
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.
Probability on hypergroups
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.
M. Bouhaik and L. Gallardo. Un théorm¯e limite central dans hypergroupe bidimensionnel. Ann. Inst. H. Poincaré, 28(1):47–61, 1992.
L. Gallardo and O. Gebuhrer. Marches aléatoires et hyper-groupes. Exposition. Math. , 5:41–73, 1987.
H. Heyer, editor. Probability Measures on Groups VII, Proceedings Oberwolfach 1983, volume 1064 of Lecture Notes in Math. Springer, Berlin, 1984.
H. Heyer, editor. Probability Measures on Groups VIII, Proceedings Oberwolfach 1985, volume 1210 ofLecture Notes in Math. Springer, Berlin, 1986.
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.
H. Heyer, editor. Probability Measures on Groups X, Proceedings Oberwolf ach 1990. Plenum, New York, 1991.
M. Voit. Central limit theorems for a class of polynomial hypergroups. Adv. in Appl Probab. , 22:68–87, 1990.
Miscellaneous
R. G. M. Brummelhuis. An F. and M. Riesz theorem for bounded symmetric domains. Ann. Inst. Fourier (Grenoble), 37:139–150, 1987.
C. F. Dunkl. Operators and harmonic analysis on the sphere. Trans. Amer. Math. Soc, 125:250–263, 1966.
E. Michael. Topologies on spaces of subsets. Trans. Amer. Math. Soc, 71:152–182, 1951.
W. Al-Salam, W. R. Allaway, and R. Askey. Sieved ultraspherical polynomials. Trans. Amer. Math. Soc, 284:41–54, 1984.
R. Askey and J. Fitch. Integral representations for Jacobi polynomials and some applications. J. Math. Anal. Appi, 26:411–437, 1969.
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.
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.
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.
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.
H. Annabi and K. Trimèche. Convolution généralisée sur le disque unité. C. R. Acad. Sc. Paris, 278:21–24, 1974.
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.
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.
M. Bouhaik and L. Gallardo. A Mehler-Heine formula for disk polynomials. Indag. Math. , 1:9–18, 1991
M. Bouhaik and L. Gallardo. Un théorm¯e limite central dans hypergroupe bidimensionnel. Ann. Inst. H. Poincaré, 28(1):47–61, 1992.
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.
Y. M. Berezanskii and A. A. Kalyuzhnyi. Harmonic Analysis in Hypercomplex Systems. Academia Nauk Ukranii, Institut Matematekii, Kiev Nauko Dumka, Kiev, 1992.
S. Bochner. Über Sturm-Liouvillische Polynomesysteme. Math. Z. , 29:730–736, 1929.
R. G. M. Brummelhuis. An F. and M. Riesz theorem for bounded symmetric domains. Ann. Inst. Fourier (Grenoble), 37:139–150, 1987.
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.
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.
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.
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.
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.
W. C. Connett and A. L. Schwartz. The theory of ultraspherical multipliers. Mem. Amer. Math. Soc, 183:1–92, 1977.
W. C. Connett and A. L. Schwartz. The Littlewood-Paley theory for Jacobi expansions. Trans. Amer. Math. Soc, 251:219–234, 1979.
W. C. Connett and A. L. Schwartz. A Hardy-Littlewood maximal inequality for Jacobi type hypergroups. Proc Amer. Math. Soc, 107:137–143, 1989.
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.
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.
W. C. Connett and A. L. Schwartz. Product formulas, hy-pergroups, and Jacobi polynomials. Bull. Amer. Math. Soc, 22:91–96, 1990.
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.
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.
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.
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.
W. C. Connett and A. L. Schwartz. Subsets of R which support hypergroups with polynomial characters, to appear inJ. Comput. Appl. Math. , 1995.
C. F. Dunkl and D. E. Ramirez. Krawtchouk polynomials and the symmetrization of hypergroups. SIAM J. Math. Anal, 5:351–366, 1974.
C. F. Dunkl. Operators and harmonic analysis on the sphere. Trans. Amer. Math. Soc, 125:250–263, 1966.
C. F. Dunkl. The measure algebra of a locally compact hy-pergroup. Trans. Amer. Math. Soc, 179:331–348, 1973.
R. E. Edwards. Fourier series, volume I, II of New York. Holt, Rinehart and Winston, Inc. , 1967.
A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi. Higher Transcendental Functions, volume I. McGraw-Hill Book Company, New York, 1953.
J. Favard. Sur les polynomes de Tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935. Sér A-B.
M. Flensted-Jensen and T. Koornwinder. The convolution structure for Jacobi function expansions. Ark. Mat. , 11:245–262, 1973.
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.
M. Flensted-Jensen. Paley-Wiener type theorems for a differential operator connected with symmetric spaces. Ark. Mat. , 10:143–162, 1972.
G. Gasper. Linearization of the product of Jacobi polynomials. II. Canad. J. Math. , 32:582–593, 1970.
G. Gasper. Banach algebras for Jacobi series and positivity of a kernel. Ann. of Math. , 95:261–280, 1972.
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.
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.
L. Gallardo and O. Gebuhrer. Marches aléatoires et hyper-groupes. Exposition. Math. , 5:41–73, 1987.
G. Gasper and M. Rahman. Basic Hypergeometric Series, volume 35 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1990.
O. Gebuhrer and A. L. Schwartz. Sidon sets and Riesz sets for some measure algebras on the disk, to appearinColloq. Math. , 1996.
H. Heyer, editor. Probability Measures on Groups VII, Proceedings Oberwolfach 1983, volume 1064 of Lecture Notes in Math. Springer, Berlin, 1984.
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.
H. Heyer, editor. Probability Measures on Groups VIII, Proceedings Oberwolfach 1985, volume 1210 ofLecture Notes in Math. Springer, Berlin, 1986.
H. Heyer, editor. Probability Measures on Groups IX, Proceedings Oberwolf ach 1988, volume 1379 of Lecture Notes in Math. Springer, Berlin, 1989.
H. Heyer, editor. Probability Measures on Groups X, Proceedings Oberwolf ach 1990. Plenum, New York, 1991.
I. I. Hirschman, Jr. Harmonic analysis and the ultraspherical polynomials. InSymposium of the Conference on Harmonic Analysis, Cornell, 1956.
I. I. Hirschman Jr. Sur les polynomes ultraspheriques. C. R. Acad. Sci. Paris, 242:2212–2214, 1956.
H. Heyer and S. Koshi. Harmonic analysis on the disk hy-pergroup. Mathematical Seminar Notes, Tokyo Metropolitan University, 1993.
R. I. Jewett. Spaces with an Abstract convolution of measures. Adv. in Math. , 18:1–101, 1975.
Y. Kanjin. A convolution measure algebra on the unit disc. Tôhoku Math. J. (2), 28:105–115, 1976.
Y. Kanjin. Banach algebra related to disk polynomials. Tôhoku Math. J. (2), 37:395–404, 1985.
J. F. C. Kingman. Random walks with spherical symmetry. Acta Math. , 109:11–53, 1963.
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.
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.
T. Koornwinder. Jacobi polynomials II. An analytic proof of the product formula. SIAM J. Math. Anal, 5:125–137, 1974.
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.
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Schwartz, A.L. (1998). Three lectures on Hypergroups Delhi, December 1995. In: Ross, K.A., Singh, A.I., Anderson, J.M., Sunder, V.S., Litvinov, G.L., Wildberger, N.J. (eds) Harmonic Analysis and Hypergroups. Trends in Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4348-5_8
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