Stable approximation of functions from equispaced samples via Jacobi frames

TL;DR

This paper investigates the use of Jacobi frames for function approximation from equispaced samples, analyzing error behavior and divergence issues related to parameter choices, with implications for numerical stability.

Contribution

It provides new error estimates for Jacobi frame approximation and reveals how parameter choices affect convergence and divergence in approximation accuracy.

Findings

Error decreases with domain parameter $eta$ for differentiable functions

Larger Jacobi polynomial parameters cause divergence in approximation error

Numerical results confirm theoretical error behavior

Abstract

In this paper, we study the Jacobi frame approximation with equispaced samples and derive an error estimation. We observe numerically that the approximation accuracy gradually decreases as the extended domain parameter $\gamma$ increases in the uniform norm, especially for differentiable functions. In addition, we show that when the indexes of Jacobi polynomials $\alpha$ and $\beta$ are larger (for example $\max\{\alpha,\beta\} > 10$), it leads to a divergence behavior on the frame approximation error decay.