Local Legendre Frame Approximation from Equispaced Data

TL;DR

The paper introduces a local Legendre frame (LLF) method for stable function approximation from equispaced data, using local partitioning and regularization to handle Runge phenomenon and oscillatory functions.

Contribution

It presents a novel LLF approach with a shared sampling matrix, enabling efficient offline-online computation and effective handling of various function types.

Findings

LLF achieves high accuracy for smooth and moderately oscillatory functions.

The method remains applicable to highly oscillatory functions with more data.

It provides an effective strategy for detecting and localizing singularities in piecewise smooth functions.

Abstract

We propose a local Legendre frame (LLF) method for function approximation from equispaced data on a finite interval. Motivated by the difficulty of stable high-order polynomial approximation at equispaced points, especially in the presence of the Runge phenomenon, the method partitions the interval into subintervals, maps each subinterval to a common reference interval, and computes local coefficients by a truncated singular value decomposition (TSVD) regularization. Since all subintervals share the same local sampling matrix, the method admits a natural offline--online implementation. We establish a quasi-optimal estimate for the regularized reconstruction and discuss practical parameter selection. Numerical results show that LLF attains high accuracy for relatively smooth and moderately oscillatory functions, while it remains applicable to highly oscillatory functions, although comparable accuracy generally requires more sampling points. For continuous piecewise smooth functions with derivative singularities, the method also provides an effective detect--localize--correct strategy based on one-sided coefficient-energy indicators. These results indicate that LLF provides a stable and flexible local approximation framework for equispaced data.