On convergence of discrete methods of least squares on equidistant nodes

Abstract

We consider the well-known method of least squares on an equidistant grid with $N+1$ nodes on the interval $[-1,1]$ with the goal to approximate a function $f\in \mathcal{C}\left[-1,1\right]$ by a polynomial of degree $n$. We investigate the following problem: For which ratio $N/n$ and which functions do we have uniform convergence of the least square operator ${LS}_n^N:\mathcal{C}\left[-1,1\right]\to {\mathcal{P}}_n$? We investigate this problem with a discrete weighting of the Jacobi-type. Thereby we describe the least square operator ${LS}_n^N$ by the expansion of a function by Hahn polynomials $Q_k\left(\cdot ;\alpha ,\beta ,N\right)$. Without additional assumptions to functions $f\in \mathcal{C}\left[-1,1\right]$ it can not be guaranteed uniform convergence. But with $\alpha =\beta$ and additional assumptions to $f$ and ${\left(N_n\right)}_{n\in \mathbb{N}}$ we obtain convergence and prove the following results: For an $\alpha \ge 0$ let $f\in \left\{g\in {\mathcal{C}}^{\infty }\left[-1,1\right]:\underset{n\to \infty }{\lim }\underset{x\in [-1,1]}{\sup }\left|g^{(n)}(x)\right|\frac{n^{\alpha +1/2}}{2^{n}n!}=0\right\}$ and let $(N_n{)}_n$ be a sequence of natural numbers with $N_n\ge 2n(n+1)$. Then the method of least squares ${LS}_n^{N_n}[f]$ converges uniform on $[-1,1]$. Before we determine the maximum error ("worst case") with respect to the sup norm on the classes ${\mathcal{K}}_{n+1}:=\left\{f\in {\mathcal{C}}^{n+1}\left[-1,1\right]:\underset{x\in [-1,1]}{\sup }\left|f^{(n+1)}(x)\right|\le 1\right\}$.

17 pages

References in corpus (1)

• On Hahn polynomial expansion of a continuous function of bounded variation