A Limit Relation for Dunkl-Bessel Functions of Type A and B

Rösler, Margit; Voit, Michael

doi:10.3842/sigma.2008.083

Cited by 7 publications

(6 citation statements)

References 17 publications

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“…Moreover, for real spectral variables λ it is possible to combine this result with Theorems 5.4 and 4.2(2), in order to obtain a convergence result for the Dunkl-type Bessel functions ϕ p λ to the functions ψ λ for p → ∞ with explicit error bounds, similar to Theorem 4.2(2). However, these results will be weaker than those which were derived directly in [RV2].…”

Section: Remarkscontrasting

confidence: 58%

“…Corresponding results for q = 1, i.e., for Jacobi functions, can be found in [V2]. We also mention that our convergence results are related to further limits, e.g., to limits in [D] and [SK] for multivariate polynomials as well as to the convergence of (multivariable) Bessel functions of type B to those of type A and related results for matrix Bessel functions in [RV2], [RV3]. We point out that our convergence results with error bounds may serve as a basis to derive central limit theorems for random walks on the Grassmannians G p,q (F) when for fixed rank q, the time parameter of the random walks as well as the dimension parameter p tend to infinity in a coupled way.…”

Section: Introductionsupporting

confidence: 55%

“…(3) There are similar limit results to those of Theorem 4.2 for Dunkl-type Bessel functions of types A and B, and for Bessel functions on matrix cones with applications in probability; see [RV2], [RV3].…”

Section: Remarksmentioning

confidence: 56%

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Integral representation and uniform limits for some Heckman-Opdam hypergeometric functions of type \textsl{BC}

Rösler

Voit

2015

Trans. Amer. Math. Soc.

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The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi functions in one variable and include the spherical functions of non-compact Grassmann manifolds over the real, complex or quaternionic numbers. There are various limit transitions known for such hypergeometric functions, see e.g.[dJ], [RKV]. In the present paper, we use an explicit form of the Harish-Chandra integral representation as well as an interpolated variant, in order to obtain limit results for three continuous classes of hypergeometric functions of type BC which are distinguished by explicit, sharp and uniform error bounds. The first limit realizes the approximation of the spherical functions of infinite dimensional Grassmannians of fixed rank; here hypergeometric functions of type A appear as limits. The second limit is a contraction limit towards Bessel functions of Dunkl type.

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Section: Remarkscontrasting

confidence: 58%

Section: Introductionsupporting

confidence: 55%

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Integral representation and uniform limits for some Heckman-Opdam hypergeometric functions of type \textsl{BC}

Rösler

Voit

2015

Trans. Amer. Math. Soc.

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“…This is work in progress. We also remark that our limit transition for hypergeometric functions has a counterpart in the Euclidean case, namely the convergence of (suitably scaled) Dunkl-Bessel functions of type B to such of type A, which was obtained in [RV1] by completely different methods.…”

Section: Limit Transition For Spherical Functions Of Noncompactmentioning

confidence: 91%

Limit transition between hypergeometric functions of type BC and type A

Rösler¹,

Koornwinder²,

Voit³

2013

Compositio Math.

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Abstract. Let F BC (λ, k; t) be the Heckman-Opdam hypergeometric function of type BC with multiplicities k = (k 1 , k 2 , k 3 ) and weighted half sum ρ(k) of positive roots. We prove that F BC (λ + ρ(k), k; t) converges for k 1 + k 2 → ∞ and k 1 /k 2 → ∞ to a function of type A for t ∈ R n and λ ∈ C n . This limit is obtained from a corresponding result for Jacobi polynomials of type BC, which is proven for a slightly more general limit behavior of the multiplicities, using an explicit representation of Jacobi polynomials in terms of Jack polynomials.Our limits include limit transitions for the spherical functions of noncompact Grassmann manifolds over one of the fields F = R, C, H when the rank is fixed and the dimension tends to infinity. The limit functions turn out to be exactly the spherical functions of the corresponding infinite dimensional Grassmann manifold in the sense of Olshanski.

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“…We notice that Lemma 4.1 was derived for x, y ∈ i • R N with precise estimates for the rate of convergence in [RV3].…”

Section: A Central Limit Theorem For the Root System B Nmentioning

confidence: 99%

Central limit theorems for multivariate Bessel processes in the freezing regime

Voit

2019

Journal of Approximation Theory

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Multivariate Bessel processes (X t,k ) t≥0 describe interacting particle systems of Calogero-Moser-Sutherland type and are related with β-Hermite and β-Laguerre ensembles. They depend on a root system and a multiplicity k. Recently, several limit theorems were derived for k → ∞ with fixed starting point. Moreover, the SDEs of (X t,k ) t≥0 were used to derive strong laws of large numbers for k → ∞ with starting points of the form √ k · x with x in the interior of the Weyl chambers. Here we provide associated almost sure functional central limit theorems which are locally uniform in t. The Gaussian limit processes admit explicit representations in terms of the solutions of associated deterministic ODEs.

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A Limit Relation for Dunkl-Bessel Functions of Type A and B

Cited by 7 publications

References 17 publications

Integral representation and uniform limits for some Heckman-Opdam hypergeometric functions of type \textsl{BC}

Integral representation and uniform limits for some Heckman-Opdam hypergeometric functions of type \textsl{BC}

Limit transition between hypergeometric functions of type BC and type A

Central limit theorems for multivariate Bessel processes in the freezing regime

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