Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Gibbs phenomenon for the Krawtchouk polynomials

John Cullinan, Elisabeth Young

Abstract

We study the Fourier approximation $\mathcal{F}_N$ of the sign function by the Krawtchouk polynomials. We give numerical evidence that the Gibbs phenomenon of the approximation differs from the classical Gibbs constant; this is in contrast to other families of orthogonal polynomials. We also show that the steepness $\mathcal{F}_N'(0)$ of the approximation is bounded by explicitly proving $\lim_{N \to \infty} \mathcal{F}_N'(0) = \log 4$. This is also in contrast to approximations by classical orthogonal polynomials, where the steepness has been shown to be unbounded as the degree increases.

The Gibbs phenomenon for the Krawtchouk polynomials

Abstract

We study the Fourier approximation of the sign function by the Krawtchouk polynomials. We give numerical evidence that the Gibbs phenomenon of the approximation differs from the classical Gibbs constant; this is in contrast to other families of orthogonal polynomials. We also show that the steepness of the approximation is bounded by explicitly proving . This is also in contrast to approximations by classical orthogonal polynomials, where the steepness has been shown to be unbounded as the degree increases.
Paper Structure (14 sections, 11 theorems, 103 equations, 1 figure)

This paper contains 14 sections, 11 theorems, 103 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Krawtchouk Polynomials
  4. Main Result
  5. Overview of the Paper
  6. Acknowledgment
  7. Polynomial Interpolation and Independence of $p$
  8. The Approximation
  9. Steepness
  10. The Gibbs Phenomenon and Connection to Classical Families
  11. Super Catalan Numbers
  12. Declarations
  13. Funding
  14. Competing Interests

Key Result

Theorem 4

With all notation as above, we have

Figures (1)

  • Figure 1: Krawtchouk-vs-Trigonometric approximation

Theorems & Definitions (24)

  • Example 1
  • Example 2
  • Theorem 4
  • Proposition 6
  • proof
  • Proposition 7
  • proof
  • Remark 8
  • Proposition 12
  • proof
  • ...and 14 more