Spherical h-Harmonic Analysis and Related Topics

• Author / Creator

  Feng, Han

• This thesis contains the following three parts: \egin{description} \item[Part 1(Chapters 1-5):] Spherical $h$-harmonic analysis. \item[Part 2:] Reverse H\"{o}lder's inequality for spherical harmonics. \item[Part 3:] Multivariate Lagrange and Hermite approximation and pointwise limits of interpolants. \end{description} The main results of Part 1 are included in two journal papers, one long joint paper with Prof. F. Dai submitted to Adv. Math., and one single-authored paper to appear in Bull. Can. Math. Soc. Results of Part 2 are contained in a joint paper with Prof. F. Dai and Prof. S. Tikhonov to appear in Pro. AMS, and results of Part 3 are from a joint paper with Prof. M. Buhmann submitted to J.London Math. Soc. Part 1 consists of 5 chapters and is organized as follows. Chapter 1 is devoted to a brief description of some background information and main results for Part 1. Chapter 2 contains some preliminary materials on the Dunkl spherical h-harmonic analysis. After that in Chapter 3 the analogues of the classical Hardy-Littlewood-Sobolev (HLS) inequality for the spherical h-harmonics with respect to general reflection groups on the sphere is established. A critical index for the validity of the HLS inequality is obtained and is expressed explicitly involving in the multiplicity function and the structure of the reflection grouop, which allows us to compute the critical indexes for most known examples of reflection groups. One of the main difficulties in our proofs lies in the fact that an explicit formula for the Dunkl intertwining operator is unknown in the case of general reflection groups, and therefore, closed forms of the reproducing kernels for the spaces of spherical $h$-harmonics are not available. A novel feature in our argument is to apply weighted Christoffel functions to establish new sharp pointwise estimates of some highly localized kernel functions associated to the spherical $h$-harmonic expansions. In Chapter 4, we introduce Riesz transforms for the spherical $h$-harmonic expansions, which are motivated by a new elegant decomposition of the Dunkl-Laplace-Beltrami operator involving the tangent gradient and the difference operators. These Riesz transforms are shown to have properties similar to those of the classical Riesz means. In particular, the $L^p$ boundedness of these operators is proved. % More importantly, the $L^p$-boundedness of the Riesz transforms is established. The proof of the main result in this chapter uses the Calderon-Zygmund decomposition, but the main difficulty is to establish some sharp kernel estimates related to the Riesz transforms. Finally, it is worthwhile to point out that the decomposition of the Dunkl-Laplace-Beltrami operator, discovered in this thesis, seems to be of independent interest. Indeed, as an application of this decomposition, in the last section of this chapter we establish the uncertainty principle with respect to the spherical $h$-harmonic expansions on the weighted spheres. Finally, we close this part by extending the results in preceding chapters to the corresponding weighted orthogonal expansions on the unit balls and the simplices. These results, in particular, generalize a classical inequality of Muckenhoupt and Stein [{\it Trans. Amer. Math. Soc. }{\f 118}(1965), 17--92] on conjugate ultraspherical polynomial expansions. In Part 2 our aim is to determine the sharp asymptotic order of the reverse H\"{o}lder inequality for spherical harmonics $Y_n$ of degree $n$ on the unit sphere $\sph$ of $\mathcal^d$ as n tends to infinity. It is shown that, in many cases, these sharp estimates are significantly better than the corresponding estimates in the Nikolskii inequality for spherical polynomials. These inequalities allow us to improve a result on the restriction conjecture of Fourier transform, as well as the sharp constant in the Pitt inequalities on $\mathcal{R}^d$. Finally, Part 3 studies various approaches to multivariate interpolation. Precisely, we analyse interpolation and the reproduction of polynomials and other functions by linear combinations of shifts of radial basis functions and cardinal interpolants. We also consider gridded data Hermite interpolation. Of particular interest in practice is a class of radial basis functions which contains the celebrated multiquadrics and inverse multiquadrics for instance. For those, we provide new results on the asymptotic limits of the aforementioned cardinal interpolants when the parameter in the generalised multiquadric function $(r^2+c^2)^\gamma$ diverges.

• Subjects / Keywords

  • Hardy-Littlewood-Sobolev inequalities
  • Uncertainty Principle
  • Nikolskii type inequality
  • Dunkl analysis
  • Spherical harmonic analysis
  • Riesz transforms

• Graduation date

  Spring 2016

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/R3BZ61H01

• License

  This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.