On Hahn polynomial expansion of a continuous function of bounded variation

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 $P_k^{α,β}$ and the Hahn polynomials $Q_k\left(\cdot;α,β,N\right)$. 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 $x\in[-1,1]$.