ORCID

0009-0000-4492-203X

Keywords

Graph, Integral transforms, Invariant, Positivity, Radial basis functions, Zero localization

Abstract

This dissertation investigates the positivity problem and zero localization of special functions or integral transforms, with particular emphasis on their implications in analysis, approximation theory, and signal processing. The first half centers on the characterization of positive definite radial functions via Hankel transforms involving Bessel functions. Motivated by classical results such as Bochner's theorem and Schoenberg’s work, we examine analytic conditions under which these transforms remain nonnegative, thereby ensuring the positive definiteness of radial basis functions (RBFs). We then explore the transition beyond positivity by focusing on the reality of zeros in Hankel (or Fourier) transforms. This leads to two complementary approaches: one based on the Laguerre–P{\'o}lya class, involving structural criteria and positivity of the Wronskian, and the other grounded in moment-based techniques using Hankel determinants and orthogonal polynomials to detect non-real zeros. Applications include sharp uniform bounds for regular Coulomb wave functions and and insights into zero distribution problems.

The second half turns to shift-invariant spaces on graphs and their applications in signal recovery. Extending classical shift-invariant theory to graph settings, we introduce graph shift-invariant spaces (GSIS) defined via commutative shift operators and investigate their analytic and spectral structure. These spaces are shown to admit reproducing kernel representations and band-limited property. The final chapter develops a graph-theoretic analogue of Barron spaces, motivated by the approximation theory of shallow neural networks. We establish function space characterizations, universal approximation and learnability results for graph convolutional neural networks (GCNNs), demonstrating that functions in this graph Barron space can be efficiently approximated, independent of the input dimension.

Completion Date

2025

Semester

Summer

Committee Chair

Ismail, Mourad

Degree

Doctor of Philosophy (Ph.D.)

College

College of Sciences

Department

Mathematics

Format

PDF

Identifier

DP0029523

Language

English

Document Type

Thesis

Campus Location

Orlando (Main) Campus

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