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Gibbs Phenomenon For Jacobi Approximations, Riti Bahl

Senior Projects Spring 2021

The classical Gibbs phenomenon is a peculiarity that arises when approximating functions near a jump discontinuity with the Fourier series. Namely, the Fourier series "overshoots" (and "undershoots") the discontinuity by approximately 9% of the total jump. This same phenomenon, with the same value of the overshoot, has been shown to occur when approximating jump-discontinuous functions using specific families of orthogonal polynomials. In this paper, we extend these results and prove that the Gibbs phenomenon exists for approximations of functions with interior jump discontinuities with the two-parameter family of Jacobi polynomials P_n^(a,b)(x). In particular, we show that for …

Orthogonal Polynomials With Respect To The Measure Supported Over The Whole Complex Plane, Meng Yang

USF Tampa Graduate Theses and Dissertations

In chapter 1, we present some background knowledge about random matrices, Coulomb gas, orthogonal polynomials, asymptotics of planar orthogonal polynomials and the Riemann-Hilbert problem. In chapter 2, we consider the monic orthogonal polynomials, $\{P_{n,N}(z)\}_{n=0,1,\cdots},$ that satisfy the orthogonality condition,

\begin{equation}\nonumber \int_\mathbb{C}P_{n,N}(z)\overline{P_{m,N}(z)}e^{-N Q(z)}dA(z)=h_{n,N}\delta_{nm} \quad(n,m=0,1,2,\cdots), \end{equation}

where $h_{n,N}$ is a (positive) norming constant and the external potential is given by

$$Q(z)=|z|^2+ \frac{2c}{N}\log \frac{1}{|z-a|},\quad c>-1,\quad a>0.$$

The orthogonal polynomial is related to the interacting Coulomb particles with charge $+1$ for each, in the presence of an extra particle with charge $+c$ at $a.$ For $N$ large and a fixed ``c'' this …

Rapid Generation Of Jacobi Matrices For Measures Modified By Rational Factors, Amber Sumner

Dissertations

Orthogonal polynomials are important throughout the fields of numerical analysis and numerical linear algebra. The Jacobi matrix J for a family of n orthogonal polynomials is an n x n tridiagonal symmetric matrix constructed from the recursion coefficients for the three-term recurrence satisfied by the family. Every family of polynomials orthogonal with respect to a measure on a real interval [a,b] satisfies such a recurrence. Given a measure that is modified by multiplying by a rational weight function r(t), an important problem is to compute the modified Jacobi matrix Jmod corresponding to the new measure from knowledge of J. There …

Orthogonal Polynomials, George Gevork Antashyan

Theses Digitization Project

This thesis will show work on Orthogonal Polynomials. In mathematics, the type of polynomials that are orthogonal to each other under inner product are called orthogonal polynomials. Jacobi polynomials, Laguerre polynomials, and Hermite polynomials are examples of classical orthogonal polynomials that was invented in the nineteenth century. The theory of rational approximations is one of the most important applications of orthogonal polynomials.

Leonhard Euler's Contribution To Infinite Polynomials, Jack Dean Meekins

Theses Digitization Project

This thesis will focus on Euler's famous method for solving the infinite polynomial. It will show how he manipulated the sine function to find all possible points along the sine function such that the sine A would equal to y; these would be roots of the polynomial. It also shows how Euler set the infinite polynomial equal to the infinite product allowing him to determine which coefficients were equal to which reciprocals of the roots, roots squared, roots cubed, etc.

The Sheffer B-Type 1 Orthogonal Polynomial Sequences, Daniel Galiffa

Electronic Theses and Dissertations

In 1939, I.M. Sheffer proved that every polynomial sequence belongs to one and only one type. Sheffer extensively developed properties of the B-Type 0 polynomial sequences and determined which sets are also orthogonal. He subsequently generalized his classification method to the case of arbitrary B-Type k by constructing the generalized generating function A(t)exp[xH1(t) + · · · + xk+1Hk(t)] = ∑∞n=0 Pn(x)tn, with Hi(t) = hi,iti + hi,i+1t i+1 + · · · , h1,1 ≠ 0. Although extensive research has been done on characterizing polynomial sequences, no analysis has yet been completed on sets of type one or higher …

An Introduction To Hellmann-Feynman Theory, David Wallace

Electronic Theses and Dissertations

The Hellmann-Feynman theorem is presented together with certain allied theorems. The origin of the Hellmann-Feynman theorem in quantum physical chemistry is described. The theorem is stated with proof and with discussion of applicability and reliability. Some adaptations of the theorem to the study of the variation of zeros of special functions and orthogonal polynomials are surveyed. Possible extensions are discussed.