Contributions to the Hypergeometric Function Theory of Heckman and Opdam: Sharp Estimates, Schwartz Space, Heat Kernel

Schapira, Bruno

doi:10.1007/s00039-008-0658-7

Cited by 99 publications

(103 citation statements)

References 17 publications

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“…We denote by C(a) W the subspace of W-invariant functions, which we identify with their restriction to a + . We have seen in [33] that…”

Section: Preliminariesmentioning

confidence: 96%

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The Heckman–Opdam Markov processes

Schapira

2006

Probab. Theory Relat. Fields

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We introduce and study the natural counterpart of the Dunkl Markov processes in a negatively curved setting. We give a semimartingale decomposition of the radial part, and some properties of the jumps. We prove also a law of large numbers, a central limit theorem, and the convergence of the normalized process to the Dunkl process. Eventually we describe the asymptotic behavior of the infinite loop as it was done by Anker, Bougerol and Jeulin in the symmetric spaces setting in (Iberoamericana 18:41-97, 2002).

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“…We denote by C(a) W the subspace of W-invariant functions, which we identify with their restriction to a + . We have seen in [33] that…”

Section: Preliminariesmentioning

confidence: 96%

“…, ξ n } is any orthonormal basis of a (L is independent of the chosen basis). The Laplacian L is given explicitly by (see [33])…”

Section: Preliminariesmentioning

confidence: 99%

The Heckman–Opdam Markov processes

Schapira

2006

Probab. Theory Relat. Fields

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“…The proof is inspired by [31]. If we take complex conjugates in (3.2), we see that Ψ A,ε (λ, ·) and Ψ A,ε (λ, ·) satisfy the same system (3.2).…”

Section: Intertwining Operatorsmentioning

confidence: 99%

Intertwining Operators Associated to a Family of Differential-Reflection Operators

2016

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We introduce a family of differential-reflection operators ΛA,ε acting on smooth functions defined on R. Here, A is a Sturm-Liouville function with additional hypotheses and −1 ≤ ε ≤ 1. For special pairs (A, ε), we recover Dunkl's, Heckman's and Cherednik's operators (in one dimension). The spectral problem for the operators ΛA,ε is studied. In particular, we obtain suitable growth estimates for the eigenfunctions of ΛA,ε. As the operators ΛA,ε are a mixture of d/dx and reflection operators, we prove the existence of an intertwining operator VA,ε between ΛA,ε and the usual derivative. The positivity of VA,ε is also established.Mathematics Subject Classification. 34K99, 34B25, 33E30.

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“…According to (13), there is a constant k > 0 such that |c(|λ|)| −2 ≥ k|λ| 2α+1 for all λ ∈ R with |λ| ≥ 1. Therefore…”

Section: Hardy and Cowling-price Theoremsmentioning

confidence: 99%

“…Such operators have been used by Heckmann and Opdam to develop a theory generalizing harmonic analysis on symmetric spaces (see [9], [12]). For recent important results in this direction we refer to [13], [16], [17].…”

Section: Introductionmentioning

confidence: 99%