Orthogonal Polynomials and Point and Weak Amenability of \(\ell ^1\) -Algebras of Polynomial Hypergroups

详细信息    查看全文

• 作者：Stefan Kahler
• 关键词：Orthogonal polynomials ; Hypergroups ; Weak amenability ; Point derivations ; Jacobi polynomials ; Pollaczek and associated ultraspherical polynomials ; 33C45 ; 33C47 ; 42C05 ; 43A20 ; 43A62 ; 46H20
• 刊名：Constructive Approximation
• 出版年：2015
• 出版时间：August 2015
• 年：2015
• 卷：42
• 期：1
• 页码：31-63
• 全文大小：711 KB
• 参考文献：1.Andrews, G.E., Askey, R., Roy, R.: Special Functions, Encyclopedia of Mathematics and Its Applications, vol. 71. Cambridge University Press, Cambridge (1999)
  2.Askey, R.: Orthogonal expansions with positive coefficients. II. SIAM J. Math. Anal. 2, 340-46 (1971)MATH MathSciNet View Article
  3.Askey, R., Ismail, M.E.H.: Recurrence relations, continued fractions, and orthogonal polynomials. Mem. Am. Math. Soc. 49(300), iv+108 (1984)MathSciNet
  4.Azimifard, A.: On the amenability of compact and discrete hypergroup algebras. arXiv:-908.-590v2 [math.FA] (September 2009)
  5.Bade, W.G., Curtis Jr, P.C., Dales, H.G.: Amenability and weak amenability for Beurling and Lipschitz algebras. Proc. Lond. Math. Soc. 55(2), 359-77 (1987)MATH MathSciNet View Article
  6.Bary, N.K.: A treatise on trigonometric series. In: Margaret, F., Mullins, A. (eds.) Pergamon Press Book, vol. I and vol. II. The Macmillan Co., New York (1964)
  7.Bavinck, H.: On absolute convergence of Jacobi series. J. Approx. Theory 4, 387-00 (1971)MATH MathSciNet View Article
  8.Benedetto, J.J., Czaja, W.: Integration and modern analysis, Birkh?user Advanced Texts: Basler Lehrbücher. [Birkh?user Advanced Texts: Basel Textbooks], Birkh?user Boston Inc., Boston (2009)
  9.Dacorogna, B.: Introduction to the Calculus of Variations. Imperial College Press, London (2004). Translated from the 1992 French originalMATH View Article
  10.Dales, H.G.: Banach algebras and automatic continuity, London Mathematical Society Monographs, vol. 24. Oxford University Press, Oxford (2000)
  11.Gasper, G.: Linearization of the product of Jacobi polynomials. Canad. J. Math. 22, 582-93 (1970)MATH MathSciNet View Article
  12.Ismail, M.E.H.: Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2009 (With two chapters by Walter Van Assche, With a foreword by Richard A. Askey, Reprint of the 2005 original)
  13.Johnson, B.E.: Cohomology in Banach Algebras. American Mathematical Society, Providence (1972)
  14.Johnson, B.E.: Derivations from \(L^1(G)\) into \(L^1(G)\) and \(L^\infty (G)\) , Harmonic analysis (Luxembourg, 1987) Lecture Notes in Math., vol. 1359, pp. 191-98. Springer, Berlin (1988)
  15.Johnson, B.E.: Weak amenability of group algebras. Bull. Lond. Math. Soc. 23(3), 281-84 (1991)MATH View Article
  16.Lasser, R.: Orthogonal polynomials and hypergroups. Rend. Mat. 3(2), 185-09 (1983)MATH MathSciNet
  17.Lasser, R.: Orthogonal polynomials and hypergroups. II. The symmetric case. Trans. Am. Math. Soc. 341(2), 749-70 (1994)MATH MathSciNet
  18.Lasser, R.: Discrete commutative hypergroups, Inzell Lectures on Orthogonal Polynomials, Adv. Theory Spec. Funct. Orthogonal Polynomials, vol. 2, pp. 55-02. Nova Science Publishers, Hauppauge (2005)
  19.Lasser, R.: Amenability and weak amenability of \(l^1\) -algebras of polynomial hypergroups. Studia Math. 182(2), 183-96 (2007)MATH MathSciNet View Article
  20.Lasser, R.: Point derivations on the \(L^1\) -algebra of polynomial hypergroups. Colloq. Math. 116(1), 15-0 (2009)MATH MathSciNet View Article
  21.Lasser, R.: Various amenability properties of the \(L^1\) -algebra of polynomial hypergroups and applications. J. Comput. Appl. Math. 233(3), 786-92 (2009)MATH MathSciNet View Article
  22.Lasser, R., Obermaier, J.: A new characterization of ultraspherical polynomials. Proc. Am. Math. Soc. 136(7), 2493-498 (2008)MATH MathSciNet View Article
  23.Lasser, R., Perreiter, E.: Homomorphisms of \(l^1\) -algebras on signed polynomial hypergroups. Banach J. Math. Anal. 4(2), 1-0 (2010)MATH MathSciNet View Article
  24.Máté, A., Nevai, P., Totik, V.: Strong and weak convergence of orthogonal polynomials. Am. J. Math. 109(2), 239-81 (1987)MATH View Article
  25.Nevai, P.: Orthogonal polynomials. Mem. Am. Math. Soc. 18(213), v+185 (1979)MathSciNet
  26.Nevai, P.: Géza, Freud, orthogonal polynomials and Christoffel functions: A case study. J. Approx. Theory 48(1), 167 (1986)MathSciNet View Article
  27.Nevai, P., Erdélyi, T., Magnus, A.P.: Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials. SIAM J. Math. Anal. 25(2), 602-14 (1994)MATH MathSciNet View Article
  28.Perreiter, E.: \({L}^1\) -algebras on commutative hypergroups: structure and properties arising from harmonic analysis, Dissertation, Technische Universit?t München (2011)
  29.Rahman, M.: A nonnegative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33(4), 915-28 (1981)MATH MathSciNet View Article
  30.Runckel, H.-J.: On the zeros of the hypergeometric function. Math. Ann. 191, 53-8 (1971)MATH MathSciNet View Article
  31.Szeg?, G.: Orthogonal polynomials, 4th edn. American Mathematical Society, Providence (1975)
  32.Szwarc, R.: Connection coefficients of orthogonal polynomials. Canad. Math. Bull. 35(4), 548-56 (1992)MA
• 作者单位：Stefan Kahler (1)
  
  1. Department of Mathematics, Chair for Mathematical Modelling, Chair for Mathematical Modeling of Biological Systems, Technische Universit?t München, Boltzmannstr. 3, 85747, Garching b. München, Germany
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Analysis
  
• 出版者：Springer New York
• ISSN：1432-0940

文摘

In contrast to the group case, amenability and many of its various generalizations are rather strong conditions on the \(\ell ^1\)-algebra of a polynomial hypergroup. In this paper, we study weak amenability and investigate the nonexistence of nonzero bounded point derivations w.r.t. symmetric characters; originally coming from cohomology theory, here these notions correspond to rather concrete problems concerning derivatives of orthogonal polynomials. We give some general results and show that there are polynomial hypergroups with weakly amenable, but nonamenable \(\ell ^1\)-algebra—the latter answers a question which has been open for some years. Moreover, we characterize both point and weak amenability for the classes of ultraspherical, Jacobi, symmetric Pollaczek, random walk, associated ultraspherical and cosh-polynomials by identifying the corresponding parameter regions. Our methods are from the theory of orthogonal polynomials and special functions, and from harmonic and functional analysis; in particular, we shall use approximations, expansions, and asymptotics.