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 ${LS}_n^N:\mathcal{C}\left[-1,1\right]\to {\mathcal{P}}_n$? 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 $n^4/N\to 0$ for $n,N\to \infty$. Furthermore we obtain the following result: Let $f\in \left\{g\in {\mathcal{C}}^1\left[-1,1\right]:g^{\prime }\in \mathcal{B}\mathcal{V}\left[-1,1\right]\right\}$ and let $(N_n{)}_n$ be a sequence of natural numbers with $n^4/N_n\to 0$. Then the least square method ${LS}_n^{N_n}[f]$ converges for each $x\in [-1,1]$.

26 pages, submitted

References in corpus (1)

• Spectral accuracy for the Hahn polynomials

Cited by in corpus (2)

• Spectral accuracy for the Hahn polynomials
• On convergence of discrete methods of least squares on equidistant nodes