Adaptive Quarklet Tree Approximation

TL;DR

This paper introduces an adaptive algorithm for near-optimal function approximation using quarklet frames, leveraging tree structures to achieve rapid convergence rates in $L_2([0,1])$.

Contribution

It develops a novel adaptive method based on tree structures of quarklet frames, inspired by $hp$-approximation, to efficiently approximate functions with near-best error rates.

Findings

Achieves inverse-exponential convergence rates in numerical experiments.

Demonstrates the effectiveness of the adaptive algorithm for function approximation.

Supports theoretical claims with numerical evidence.

Abstract

This paper is concerned with near-optimal approximation of a given function $f \in L_2([0,1])$ with elements of a polynomially enriched wavelet frame, a so-called quarklet frame. Inspired by $hp$-approximation techniques of Binev, we use the underlying tree structure of the frame elements to derive an adaptive algorithm that, under standard assumptions concerning the local errors, can be used to create approximations with an error close to the best tree approximation error for a given cardinality. We support our findings by numerical experiments demonstrating that this approach can be used to achieve inverse-exponential convergence rates.