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​This book provides an introduction to h-harmonics and Dunkl transforms. These are extensions of the ordinary spherical harmonics and Fourier transforms, in which the usual Lebesgue measure is replaced by a reflection-invariant weighted measure. The authors’ focus is on the analysis side of both h-harmonics and Dunkl transforms. Graduate students and researchers working in approximation theory, harmonic analysis, and functional analysis will benefit from this book.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction: Spherical Harmonics and Fourier Transform

Abstract
The purpose of these lecture notes is to provide an introduction to two related topics: h-harmonics and the Dunkl transform. These are extensions of the classical spherical harmonics and the Fourier transform, in which the underlying rotation group is replaced by a finite reflection group. This chapter serves as an introduction, in which we briefly recall classical results on the spherical harmonics and the Fourier transform. Since all results are classical, no proof will be given.
Feng Dai, Yuan Xu

Chapter 2. Dunkl Operators Associated with Reflection Groups

Abstract
In this chapter we introduce the essential ingredient in the Dunkl theory of harmonic analysis. Since our purpose is to study harmonic analysis in weighted spaces, we start with the definition of a family of weight functions invariant under a reflection group in Section 2.1. Dunkl operators are a family of commuting first-order differential and difference operators associated with a reflection group, and are introduced in Section 2.2. The intertwining operator between the Dunkl operators and ordinary derivatives is discussed in Section 2.3.
Feng Dai, Yuan Xu

Chapter 3. h-Harmonics and Analysis on the Sphere

Abstract
Dunkl h-harmonics are defined as homogeneous polynomials satisfying the Dunkl Laplacian equation. They are defined and studied in Section 3.1. Projection operators and orthogonal expansions in spherical h-harmonics are studied in Section 3.2, which includes a concise expression for the reproducing kernel of the spherical h-harmonics. This expression is an analog of the zonal harmonics, which suggests a definition of a convolution operator, defined in Section 3.3 and it helps us to study various summability methods for spherical h-harmonic expansions.
Feng Dai, Yuan Xu

Chapter 4. Littlewood–Paley Theory and the Multiplier Theorem

Abstract
The main result of this chapter is a Marcinkiewitcz multiplier theorem for h-harmonic expansions. Its proof uses general Littlewood–Paley theory for a symmetric diffusion semi-group. Several Littlewood–Paley type g-functions are introduced and studied via the Cesàro means for h-harmonic expansions.
Feng Dai, Yuan Xu

Chapter 5. Sharp Jackson and Sharp Marchaud Inequalities

Abstract
The goal of this chapter is to prove two inequalities, the sharp Jackson inequality and the sharp Marchaud inequality, for the h-harmonic expansions on the sphere \(\mathbb{S}^{d-1}\), which are useful in the embedding theory of function spaces. The multiplier theorem and the Littlewood–Paley inequality established in the prior chapter play crucial roles in their proofs.
Feng Dai, Yuan Xu

Chapter 6. Dunkl Transform

Abstract
The Dunkl transform is a generalization of the Fourier transform and is an isometry in \( {L}^{2}(\mathbb{R}^{d},{{h}^{2}_{\kappa}})\) with \( {{h}_{\kappa}}\) being a reflection invariant weight function. In this chapter we study the Dunkl transform from the point of view of harmonic analysis. In Section 6.1 we show that the Dunkl transform is an isometry in \( {L}^{2}\) space with respect to the measure \( {{h}^{2}_{\kappa}}(x)dx\) on \( \mathbb{R}^{d} \) and it preserves Schwartz class of functions.
Feng Dai, Yuan Xu

Chapter 7. Multiplier Theorems for the Dunkl Transform

Abstract
For a family of weight functions invariant under a finite reflection group, we prove a transference theorem between the L p multiplier of h-harmonic expansions on \(\mathbb{S}^{d}\) and that of the Dunkl transform. This theorem is stated together with some related definitions and notations in Section 7.1. The proof of this transference theorem is, however, rather long, so we split it into three parts, which are given in the Sections 7.2, 7.3, and 7.4, respectively.
Feng Dai, Yuan Xu

Backmatter

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