arXiv subjects

James Tipton

Publications and source records attributed to James Tipton.

5 recordsLinked to original sources

arXiv2021

A Unified Approach to Computing the Zeros of Classical Orthogonal Polynomials

The authors present a unified method for calculating the zeros of the classical orthogonal polynomials based upon the electrostatic interpretation and its connection to the energy minimization problem. Examples are given with error estimates for three cases of the Jacobi polynomials, three cases of the Laguerre polynomials, and the Hermite polynomials. In the case of the Chebyshev polynomials, exact errors are given.

math.CA↗

arXiv2018

Partial Classification of Polynomials and an Orthonormal Basis Construction on the Associated Basin of Attraction

In the paper "Infinite product representations for kernels and iterations of functions", the authors associate certain Fatou subsets with reproducing kernel Hilbert spaces. They also present a method for constructing an orthonormal basis for said Hilbert space, but the method depends on the polynomial of the given Fatou set. We provide a partial classification of those polynomials the method applies to.

math.FA↗

arXiv2017

Dynamics of Orthonormal Bases Associated to Basins of Attraction

In the paper "Infinite Product Represenations for Kernels and Iterations of Functions", a technique was developed which allows for the construction of a reproducing kernel Hilbert space on basins of attraction containing $0$. When the right conditions are met, an explicit orthonormal basis can be constructed using a particular class of operators. It is natural then to consider how the orthonormal basis changes as we let the basin of attraction vary. We will consider this question for the basins of attraction containing $0$ of the family of polynomials $\mathcal{F} = \{az^{2^{n+2}}-2az^{2^{n+1}}:a\neq0\}$, where $n\in\mathbb{N}$.

math.DS↗

arXiv2012

The Kauffman polynomial and trivalent graphs

We construct a state model for the two-variable Kauffman polynomial using planar trivalent graphs. We also use this model to obtain a polynomial invariant for a certain type of trivalent graphs embedded in three-dimensional space.

math.GT↗

arXiv2011

Multiplier Sequences for Simple Sets of Polynomials

In this paper we give a new characterization of simple sets of polynomials B with the property that the set of B-multiplier sequences contains all Q-multiplier sequence for every simple set Q. We characterize sequences of real numbers which are multiplier sequences for every simple set Q, and obtain some results toward the partitioning of the set of classical multiplier sequences.

math.CV↗