Banach algebra related to the Jacobi polynomials

Igari, Satoru; Uno, Yoshikazu

doi:10.2748/tmj/1178242910

Cited by 14 publications

(9 citation statements)

References 2 publications

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“…See his survey paper [1] for further references. Igari and Uno [10] have found the maximal ideal space (as hoped for, [-1,1]) of the algebra of absolutely convergent Jacobi series.…”

Section: Proposition 38 Fae Span Of H Is Sup-norm Dense In C(h)mentioning

confidence: 99%

The measure algebra of a locally compact hypergroup

Dunkl¹

1973

Trans. Amer. Math. Soc.

157

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ABSTRACT. A hypergroup is a locally compact space on which the space of finite regular Borel measures has a convolution structure preserving the probability measures. This paper deals only with commutative hypergroups. § 1 contains definitions, a discussion of invariant measures, and a characterization of idempotent probability measures. §2 deals with the characters of a hypergroup. §3 is about hypergroups, which have generalized translation operators (in the sense of Levitan), and subhypergroups of such. In this case the set of characters provides much information. Finally §4 discusses examples, such as the space of conjugacy classes of a compact group, certain compact homogeneous spaces, ultraspherical series, and finite hypergroups.A hypergroup is a locally compact space on which the space of finite regular Borel measures has a convolution structure preserving the probability measures. Such a structure can arise in several ways in harmonic analysis. Two major examples are furnished by the space of conjugacy classes of a compact nonabelian group, and by the two-sided cosets of certain nonnormal closed subgroups of a compact group. Another example is given by series of Jacobi polynomials.The class of hypergroups includes the class of locally compact topological semigroups. In this paper we will show that many well-known group theorems extend to the commutative hypergroup case. In §1 we discuss some basic structure and determine the idempotent probability measures. In §2 we present some elementary theory of characters of a hypergroup. In §3 we look at a restricted class of hypergroups, namely those on which there is a generalized translation in the sense of Levitan ([11], or see [12, p. 427]). (The notation of the present paper would seem to have two advantages over Levitan's: ours is compatible with current notation for compact groups, and in Levitan's notation, it is almost impossible to express correctly commutativity and associativity.) The theory for these hypergroups looks much like locally compact abelian group theory, yet covers a much wider range of examples. Finally in §4 we discuss some examples and further questions.1. Basic properties. We recall some standard notation (in the following, A is a locally compact Hausdorff space):

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“…See his survey paper [1] for further references. Igari and Uno [10] have found the maximal ideal space (as hoped for, [-1,1]) of the algebra of absolutely convergent Jacobi series.…”

Section: Proposition 38 Fae Span Of H Is Sup-norm Dense In C(h)mentioning

confidence: 99%

The measure algebra of a locally compact hypergroup

Dunkl¹

1973

Trans. Amer. Math. Soc.

157

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“…(iv) λ 2n + λ 2n+1 ≥ 1, λ 2n−1 + λ 2n ≤ 1 for n ≥ 1, and r 2n−1 ≤ r 2n+1 for n ≥ 1. P r o o f. First we will show that λ 2 n is a chain sequence whose minimal parameter sequence α n satisfies (8) α n+2 ≥ α n .…”

mentioning

confidence: 99%

“…The product of two of these polynomials can be written as a sum of these polynomials: The constants c(n, m, k) of this expansion are called the linearization coefficients. The nonnegativity of these coefficients leads to a Banach algebra structure associated with the polynomials which is analogous to the algebra of absolutely convergent Fourier series on a torus (see Askey [1], Askey-Wainger [2] and Igari-Uno [8] for examples).…”

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confidence: 99%

Nonnegative linearization of orthogonal polynomials

Szwarc¹

1996

Colloq. Math.

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“…By means of our results in §3, we may obtain that the structure of A(a'T^ is simpler than that of A(T) and is similar to that of the algebra of absolutely convergent Jacobi polynomial series or the algebra of Hankel transforms. See Igari and Uno [7] and Schwartz [11].…”

Section: Introductionmentioning

confidence: 99%

“…Although the Laguerre translation is not positive, our results show that the structures of Ma and La axe similar to those of the algebras for ultraspherical polynomials or Jacobi polynomials with positive convolution structures. See Dunkl [2], Gasper [4], Igari [6], and also [11]. for every x > 0.…”

Section: Introductionmentioning

confidence: 99%