Available at: https://digitalcommons.calpoly.edu/theses/3292
Date of Award
6-2026
Degree Name
MS in Mathematics
Department/Program
Mathematics
College
College of Science and Mathematics
Advisor
Ryan Tully-Doyle
Advisor Department
Mathematics
Advisor College
College of Science and Mathematics
Abstract
Wavelets and wavelet analysis are used in the study of signal processing, quantum field theory, functional analysis, multifractal analysis, and various other areas of mathematics. Multiresolution analysis provides a framework for building a wavelet basis of $\mathcal{L}^{2}(\mathbb{R})$ from a scaling function $\phi$, whose dyadic dilations and translations, $\{2^{j /2}\phi(2^{j}x-k):j,k\in \mathbb{Z}\}$, approximate $\mathcal{L}^{2}(\mathbb{R})$. One of the key properties of $\phi$ is that it must satisfy $\phi(x)=\sum_{k\in \mathbb{Z}}{p_{k}2^{j /2}\phi(2^{j}x-k)}$ with respect to the norm on $\mathcal{L}^{2}(\mathbb{R})$. This equation is called a two-scale difference equation. Such equations enforce a regularity on the ordinary generating function $2^{-1 /2}\sum_{k\in \mathbb{Z}}{p_{k}z^{k}}$, known as the quadrature condition. Conversely, if a series $P(z)=2^{-1 /2}\sum_{k\in \mathbb{Z}}{p_{k}z^{k}}$ satisfies the quadrature condition and sufficient regularity is assumed, it is possible to construct a scaling function for a multiresolution analysis, satisfying the two-scale equation corresponding to the series $P(z)$. This thesis serves as an introduction to multiresolution analysis through the lens of the Haar wavelet and the construction of scaling functions from series satisfying the quadrature condition.
Included in
Analysis Commons, Harmonic Analysis and Representation Commons, Other Mathematics Commons, Signal Processing Commons