Date of Defense
2-6-2026 2:30 PM
Location
Room F1-2119
Document Type
Thesis Defense
Degree Name
Master of Science in Mathematics
College
College of Science
Department
Mathematical Sciences
First Advisor
Salem Ben Said
Keywords
Harmonic Analysis, Heat Equation, Dunkl Theory, Fractional Laplacian, Extension Method
Abstract
In this thesis, we study analytical structures arising from Dunkl theory and their applications to harmonic analysis and fractional Laplacian operators. Dunkl operators are differential-difference operators associated with finite reflection groups, providing a natural generalization of the classical Fourier analysis through the introduction of root systems and multiplicity functions. Within this framework, several classical transforms appear as special cases of the (k,a)-generalized Fourier transform. We study the generalized Fourier transform, its kernel, and the associated translation operator and convolution structures. Using these tools, we construct the corresponding heat kernel and analyze the associated heat semigroup. Our main contribution concerns the study of fractional powers of Dunkl-type Laplacian, for 0 < σ < 1, associated with the (k,a)-generalized Fourier transform. In particular, we focus on the case of arbitrary a > 0, aiming to extend the previously studied case a = 1. These results extend classical fractional analysis to the Dunkl framework and provide new tools for the study of nonlocal operators associated with reflection groups.
Included in
On the Fractional Laplacian Type Operator
Room F1-2119
In this thesis, we study analytical structures arising from Dunkl theory and their applications to harmonic analysis and fractional Laplacian operators. Dunkl operators are differential-difference operators associated with finite reflection groups, providing a natural generalization of the classical Fourier analysis through the introduction of root systems and multiplicity functions. Within this framework, several classical transforms appear as special cases of the (k,a)-generalized Fourier transform. We study the generalized Fourier transform, its kernel, and the associated translation operator and convolution structures. Using these tools, we construct the corresponding heat kernel and analyze the associated heat semigroup. Our main contribution concerns the study of fractional powers of Dunkl-type Laplacian, for 0 < σ < 1, associated with the (k,a)-generalized Fourier transform. In particular, we focus on the case of arbitrary a > 0, aiming to extend the previously studied case a = 1. These results extend classical fractional analysis to the Dunkl framework and provide new tools for the study of nonlocal operators associated with reflection groups.