Positivity and the Convolution Structure for Jacobi Series

Gasper, George

doi:10.2307/1970755

Cited by 138 publications

(48 citation statements)

References 6 publications

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“…In fact, there is a dearth of concrete examples which can be examined to gain insight; the only really well understood examples are the Jacobi polynomials and functions and the associated dual structures [11,12,13,16]. The knowledge we have in those cases is based upon detailed information about the special function systems.…”

Section: Jhmentioning

confidence: 99%

Convolution and hypergroup structures associated with a class of Sturm-Liouville systems

Connett¹,

Markett²,

Schwartz³

1992

Trans. Amer. Math. Soc.

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When D(£,, 8, ) is nonnegative (which can be guaranteed by a simple restriction on the differential operator of the Sturm-Liouville problem), it is possible to define a convolution with respect to which M[0, n] becomes a Banach algebra with the functions Uk(Ç)/uo(Ç) as its characters. In fact this measure algebra is a Jacobi type hypergroup. It is possible to completely describe the maximal ideal space and idempotents of this measure algebra.

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Section: Jhmentioning

confidence: 99%

Convolution and hypergroup structures associated with a class of Sturm-Liouville systems

Connett¹,

Markett²,

Schwartz³

1992

Trans. Amer. Math. Soc.

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“…Gasper [17] proved that for a ≥ b ≥ −1/2 and 0 < h < π the operator T (a,b) h is positive. Therefore, we have…”

Section: Let Also R (Ab)mentioning

confidence: 99%

On approximation of functions by algebraic polynomials in Hölder spaces

Kolomoitsev

Lomako

Prestin

2016

Mathematische Nachrichten

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Abstract. We study approximation of functions by algebraic polynomials in the Hölder spaces corresponding to the generalized Jacobi translation and the Ditzian-Totik moduli of smoothness. By using modifications of the classical moduli of smoothness, we give improvements of the direct and inverse theorems of approximation and prove the criteria of the precise order of decrease of the best approximation in these spaces. Moreover, we obtain strong converse inequalities for some methods of approximation of functions. As an example, we consider approximation by the Durrmeyer-Bernstein polynomial operators.

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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%

“…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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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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Positivity and the Convolution Structure for Jacobi Series

Cited by 138 publications

References 6 publications

Convolution and hypergroup structures associated with a class of Sturm-Liouville systems

Convolution and hypergroup structures associated with a class of Sturm-Liouville systems

On approximation of functions by algebraic polynomials in Hölder spaces

The Hypergroup Property and Representation of Markov Kernels

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