Approximation Theory and Harmonic Analysis on Spheres and Balls

Dai, Chang‐Feng; Xu, Yuan

doi:10.1007/978-1-4614-6660-4

Cited by 372 publications

(352 citation statements)

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“…They enter the study of Calogero-Moser-Sutherland models [28], they play a central role in the theory of multivariate orthogonal polynomials associated to reflection groups [11], they give rise to families of stochastic processes [20,25], and they can be used to construct quantum superintegrable systems involving reflections [13,14]. Dunkl operators also find applications in harmonic analysis and integral transforms [7,24], as they naturally lead to the Laplace-Dunkl operators, which are second-order differential/difference operator that generalize the standard Laplace operator.…”

Section: Introductionmentioning

confidence: 99%

A Dirac–Dunkl Equation on S 2 and the Bannai–Ito Algebra

Bie

Genest

Vinet

2016

Commun. Math. Phys.

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The Dirac-Dunkl operator on the 2-sphere associated to the Z 3 2 reflection group is considered. Its symmetries are found and are shown to generate the Bannai-Ito algebra. Representations of the Bannai-Ito algebra are constructed using ladder operators. Eigenfunctions of the spherical Dirac-Dunkl operator are obtained using a Cauchy-Kovalevskaia extension theorem. These eigenfunctions, which correspond to Dunkl monogenics, are seen to support finite-dimensional irreducible representations of the Bannai-Ito algebra.

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Section: Introductionmentioning

confidence: 99%

A Dirac–Dunkl Equation on S 2 and the Bannai–Ito Algebra

Bie

Genest

Vinet

2016

Commun. Math. Phys.

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“…For the proof of (1)(a) and (1)(b), see Proposition 1.8.7 in [2]. And (1)(c) follows from (1)(a) by applying componentwise.…”

Section: Some Calculations Related To Spherical Harmonicsmentioning

confidence: 95%

“…Then it can be shown that the functions ψ m,n (θ)Y n,j (ω), m − n ≥ 0 are eigenfunctions of ∆ S 3 with eigenvalues m(m + 2). In fact, these functions are the spherical harmonics of degree m on S 3 in the coordinate system (θ, ω), see (1.5.6) in [2] and also [8]. Hence if g = Φ(θ, ω) then the functions…”

Section: Jacobi Expansions and Jacobi-riesz Transformsmentioning

confidence: 99%

“…In order to compute T ζ (ρ(k)f ) we make use of the following facts: (i) since ρ(k) acts on the ω-variable, it commutes with ∆ = ∆ S 3 ; (ii) the spherical gradient ∇ S 2 is related to the gradient ∇ on R 3 via the equation ∇ S 2 h(ω) = ∇h(ω) − ω(ω · ∇ S 2 h(ω)) (see equation 1.8.12 in [2]) and (iii) for any x, u ∈ R 3 and k ∈ K, k u · ∇h(kx) = u · ∇(ρ(k)h)(x) which follows by direct calculation. In order to deal with the term N f we also need the relation ∇ = 1 r ∇ S 2 + ω ∂ ∂r and the fact that for k ∈ K, k(u × v) = k u × k v for any two vectors.…”

Section: Mixed Norm Estimates For the Riesz Transformsmentioning

confidence: 99%

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Mixed norm estimates for the Riesz transforms on $SU(2)$

Boggarapu¹,

Thangavelu²

2016

Publicacions Matemàtiques

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“…More information on this subject can be found, e.g., in [2,6,21,32,33,52,57,60]; see also [5,10,58] for the group representation approach. …”

Section: Elements Of Analysis On the Spherementioning

confidence: 99%

The λ-cosine transforms with odd kernel and the hemispherical transform

Rubin

2014

Fract Calc Appl Anal

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We review some basic facts about the λ-cosine transforms with odd kernel on the unit sphere S n−1 in R n . These transforms are represented by the spherical fractional integrals arising as a result of evaluation of the Fourier transform of homogeneous functions. The related topic is the hemispherical transform which assigns to every finite Borel measure on S n−1 its values for all hemispheres. We revisit the known facts about this transform and obtain new results. In particular, we show that the classical FunkRadon-Helgason inversion method of spherical means is applicable to the hemispherical transform of L p -functions.MSC 2010 : Primary 44A12; Secondary 47G10 Key Words and Phrases: fractional integrals, hemispherical transform, λ-cosine transforms with odd kernel, spherical means IntroductionThe following analytic families of fractional integrals arise as a result of evaluation of the Fourier transform of homogeneous functions on R n \ {0}. Here S n−1 is the unit sphere in R n , λ is a complex parameter, γ n,λ andγ n,λ are normalizing coefficients which will be specified later, d * σ stands for the normalized surface element on S n−1 . We call these operators the λ-cosine transforms with even and odd kernel, respectively. The term cosine transform is borrowed from integral geometry and reflects the fact that θ · σ is nothing but the cosine of the angle between the unit vectors θ and σ. Operators of this kind occur in different branches of mathematics without naming or under different names. In the following, we often skip the prefix λ, for short.The main concern of the present article is the operator family (1.2). Regarding (1.1), extensive information, including generalizations and applications, can be found in [16,26,36,47,48,49].Our next topic is the hemispherical transform which assigns to every finite Borel measure on S n−1 its values for all hemispheres in S n−1 . This transform is intimately connected withC λ . It was introduced by Funk [13] for n = 3 and studied in [4,20,21,43,16,54]. A similar transform for half-spaces in R n was studied in [23,28,41] in connection with Kolmogorov's problem [27, Problem No. 16]. Unlike the half-space transform, the hemispherical transform is non-injective and its kernel is composed by even measures with the mean value zero. The latter means that our consideration can be focused on odd measures or functions.In the present paper we review basic facts about operatorsC λ and the hemispherical transform. We update our previous proofs given in [43] and obtain new results. In particular, in Section 5.4 we show that the Funk-Radon-Helgason inversion method of spherical means, which plays an important role in the theory of Radon transforms [22,50], is applicable to the hemispherical transform.The paper is organized as follows. In Section 2 we recall some facts from Fractional Calculus and analysis on the sphere. In Section 3 we show how (1.1) and (1.2) arise in the framework of the Fourier analysis on R n and discuss properties of these operators, including action in ...

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Approximation Theory and Harmonic Analysis on Spheres and Balls

Cited by 372 publications

References 0 publications

A Dirac–Dunkl Equation on S 2 and the Bannai–Ito Algebra

A Dirac–Dunkl Equation on S 2 and the Bannai–Ito Algebra

Mixed norm estimates for the Riesz transforms on $SU(2)$

The λ-cosine transforms with odd kernel and the hemispherical transform

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