Publication Date

Summer 7-29-2025

Abstract

Chebyshev Polynomials, those that minimize the maximal error on a compact set, are one of the most practical tools for approximating smooth functions. The classical results are on the set [-1, 1]; in this paper, we extend to more complicated subsets of the real line. We demonstrate some classical results and then take the result from [2] on regular Parreau-Widom Sets and extend it to semi-regular sets, defined as sets whose regular part is closed. We introduce the Regularity Coefficient as a series formed by evaluating the Green’s Function at irregular points. This new machinery is applied to the lower and upper bounds of both Chebyhsev and Residual Polynomials. We find new bounds of the Widom Factors on semi-regular sets, given by 2 times the exponential of the Regularity Coefficient and 2 times the exponential of the sum of the Regularity Coefficient and the Parreau-Widom Constant, respectively.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Maxim Zinchenko

Second Committee Member

Maria Cristina Pereyra

Third Committee Member

Matthew Blair

Fourth Committee Member

Jacob Stordal Christiansen

Language

English

Keywords

Approximation, Chebyshev Polynomials, Residual Polynomials, Potential Theory, Green's Functions, Logarithmic Potential, Regular for Potential Theory, Harmonic Functions

Document Type

Dissertation

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