Banach Algebras for Jacobi Series and Positivity of a Kernel

Gasper, George

doi:10.2307/1970800

Cited by 151 publications

(71 citation statements)

References 9 publications

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“…The case of Jacobi polynomials may be considered as a special case of a Sturm-Liouville basis on [0, π/2]. In this situation, both the GKS and the HGP property hold [24,25,26]. Actually, it is a unique situation for orthogonal polynomials, since they are the only ones, up to a linear change of variables, for which the HGP property holds (see [18,17,19]) (under some mild extra condition on the support of the measure which represents the product formula).…”

Section: The Case Of Jacobi Polynomials: Gasper's Theoremmentioning

confidence: 99%

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The Hypergroup Property and Representation of Markov Kernels

Bakry

Huet

2008

Lecture Notes in Mathematics

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Summary. For a given orthonormal basis (fn) on a probability measure space, we want to describe all Markov operators which have the fn as eigenvectors. We introduce for that what we call the hypergroup property. We study this property in three different cases.On finite sets, this property appears as the dual of the GKS property linked with correlation inequalities in statistical mechanics. The representation theory of groups provides generic examples where these two properties are satisfied, although this group structure is not necessary in general.The hypergroup property also holds for Sturm-Liouville bases associated with logconcave symmetric measures on a compact interval, as stated in Achour-Trimèche's theorem. We give some criteria to relax this symmetry condition in view of extensions to a more general context.In the case of Jacobi polynomials with non-symmetric parameters, the hypergroup property is nothing else than Gasper's theorem. The proof we present is based on a natural interpretation of these polynomials as harmonic functions and is related to analysis on spheres. The proof relies on the representation of the polynomials as the moments of a complex variable.

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Section: The Case Of Jacobi Polynomials: Gasper's Theoremmentioning

confidence: 99%

“…These polynomials are traditionally parametrized by α = q−2 2 and β = p−2 2 with α, β > −1, from [39] or [24,25,26].…”

Section: Jacobi Polynomialsmentioning

confidence: 99%

The Hypergroup Property and Representation of Markov Kernels

Bakry

Huet

2008

Lecture Notes in Mathematics

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“…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%

On Algebras With Convolution Structures for Laguerre Polynomials

Kanjin¹

1986

Transactions of the American Mathematical Society

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“…INTRODUCTION. In the last twenty years, a number of concrete convolution measure algebras have been studied in great detail, and new structures continually emerge (see [2,3,4,5,9,15], and the references cited in those articles). Many of these are probability-preserving algebras of measures on an interval; indeed, many are examples of the type II one-dimensional hypergroups discussed in [17].…”

mentioning

confidence: 99%

“…Many of these are probability-preserving algebras of measures on an interval; indeed, many are examples of the type II one-dimensional hypergroups discussed in [17]. When the interval is compact, these examples share many properties of the convolution structure associated with Jacobi polynomials described in [9] and below. In an effort to understand the general behavior of such measure algebras, we introduce the notion of Jacobi type hypergroups, a general class which includes many examples.…”

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confidence: 99%

Analysis of a class of probability preserving measure algebras on compact intervals

Connett¹,

Schwartz²

1990

Trans. Amer. Math. Soc.

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ABSTRACT. The measure algebras of the title are those which are also hypergroups with some regularity conditions, Examples include the convolutions associated with Jacobi polynomial series and Fourier Bessel series, It is shown here that there is a one-to-one correspondence between these hypergroups and a class of Sturm-Liouville problems which have the characters of the hypergroup as eigenfunctions, The interplay between these two characterizations allows a detailed analysis which includes a Hilb-type formula for the characters and asymptotic estimates for the Plancherel measure and the eigenvalues of the associated Sturm-Liouville problem,

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Banach Algebras for Jacobi Series and Positivity of a Kernel

Cited by 151 publications

References 9 publications

The Hypergroup Property and Representation of Markov Kernels

The Hypergroup Property and Representation of Markov Kernels

On Algebras With Convolution Structures for Laguerre Polynomials

Analysis of a class of probability preserving measure algebras on compact intervals

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