On Hahn polynomial expansion of a continuous function of bounded variation

TL;DR

This paper investigates the pointwise convergence of least squares polynomial approximations on equidistant grids, relating Jacobi and Hahn polynomial expansions, and establishes conditions for convergence based on the ratio of nodes to polynomial degree.

Contribution

It establishes new convergence criteria for least squares polynomial approximation using Hahn polynomials, linking it to Jacobi polynomial expansions and the ratio of grid points to polynomial degree.

Findings

Pointwise convergence occurs if the Jacobi expansion converges and n^4/N tends to zero.

Least squares approximation converges for functions with bounded variation derivatives under specified conditions.

The relation between Hahn and Jacobi polynomials is key to understanding convergence behavior.

Abstract

We consider the well-known method of least squares on an equidistant grid with $N+1$ nodes on the interval $[-1,1]$. We investigate the following problem: For which ratio $N/n$ and which functions, do we have pointwise convergence of the least square operator ${LS}_n^N:\mathcal{C}\left[-1,1\right]\rightarrow\mathcal{P}_n$? To solve this problem we investigate the relation between the Jacobi polynomials $P_k^{\alpha,\beta}$ and the Hahn polynomials $Q_k\left(\cdot;\alpha,\beta,N\right)$. Thereby we describe the least square operator ${LS}_n^N$ by the expansion of a function by Hahn polynomials. In particular we present the following result: The series expansion $\sum_{k=0}^n{\hat{f} Q_k}$ of a function $f$ by Hahn polynomials $Q_k$ converges pointwise, if the series expansion $\sum_{k=0}^n{\hat{f} P_k}$ of the function $f$ by Jacobi polynomials $P_k$ converges pointwise and if ${n^4}/N\rightarrow 0$ for $n,N\rightarrow\infty$. Furthermore we obtain the following result: Let $f\in\left\{g\in\mathcal{C}^1\left[-1,1\right]:g^\prime\in\mathcal{BV}\left[-1,1\right]\right\}$ and let $(N_n)_{n}$ be a sequence of natural numbers with ${n^4}/{N_n}\rightarrow 0$. Then the least square method ${LS}_n^{N_n}[f]$ converges for each $x\in[-1,1]$.