On convergence of discrete methods of least squares on equidistant nodes
René Goertz
TL;DR
The paper analyzes the convergence of the discrete least-squares operator on an equidistant grid with $N+1$ nodes for approximating a function $f$ by a polynomial of degree $n$, using a discrete Jacobi-type weighting via Hahn polynomials. In the ultraspherical case $\alpha=\beta$, it proves uniform convergence under a smoothness class for $f$ and a growth condition linking $N$ and $n$, specifically when $N_n\ge2n(n+1)$. The key contribution is an explicit worst-case error bound $D_{n,N}$, expressed through gamma-function ratios, and a rigorous comparison to Brass’s continuous Jacobi-bound $C_n$, showing $D_{n,N}\le C_n$. The results yield practical guidelines for choosing the node-count sequence $N_n$ to guarantee uniform convergence, and establish that the discrete method can perform at least as well as the continuous counterpart in the worst-case sense, with quantifiable guarantees.
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 ${LS}_n^N:\mathcal{C}\left[-1,1\right]\rightarrow\mathcal{P}_n$? 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 $Q_k\left(\cdot;α,β,N\right)$. Without additional assumptions to functions $f\in\mathcal{C}\left[-1,1\right]$ it can not be guaranteed uniform convergence. But with $α=β$ and additional assumptions to $f$ and $\left(N_n\right)_{n\in\mathbb{N}}$ we obtain convergence and prove the following results: For an $α\geq0$ let $f\in\left\{g\in\mathcal{C}^\infty\left[-1,1\right]:\ \lim\limits_{n\to\infty}{\sup\limits_{x\in[-1,1]}{\left\lvert g^{(n)}(x)\right\rvert}\frac{n^{α+1/2}}{2^nn!}}=0\right\}$ and let $(N_n)_{n}$ be a sequence of natural numbers with $N_n\geq2n(n+1)$. 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 $\mathcal{K}_{n+1}:=\left\{f\in\mathcal{C}^{n+1}\left[-1,1\right]:\ \sup\limits_{x\in[-1,1]}{\left\lvert f^{(n+1)}(x)\right\rvert\leq1}\right\}$.