Estimation of sparse polynomial approximation error to continuous function

TL;DR

This paper develops methods to accurately estimate the approximation error of sparse polynomial approximations for continuous functions, using l1-minimization and orthonormal polynomial bases, with implications for exact recovery and efficiency.

Contribution

It introduces a novel error estimation framework for sparse polynomial approximation using l1-minimization and extends analysis to weighted cases without boundedness constraints.

Findings

Exact recovery of sparse polynomials when the function is sparse.

Fewer polynomial degrees needed for smoother functions.

Error bounds expressed in terms of sparsity and approximation error.

Abstract

The sparse polynomial approximation of continuous functions has emerged as a prominent area of interest in function approximation theory in recent years. A key challenge within this domain is the accurate estimation of approximation errors. This paper focuses on continuous functions, characterizing their sampled values as a combination of the values of their best approximation polynomials within a finite-dimensional polynomial space and the associated remainder terms. Consequently, the sampled values of a function can be interpreted as noisy samples of the values of its best approximation polynomial, with the noise equivalent to the remainder term's values at those points. By selecting a uniformly bounded orthonormal polynomial system as the basis for this finite-dimensional space, it becomes feasible to formulate noise constraint inequalities and l1-minimization problems or their weighted l1-minimization variants. This paper provides estimations for the approximation error of the sparse polynomial derived from the l1-minimization method, characterizing the error in terms of the quasi-norm of the sampled function or its best uniform approximation polynomial, the sparsity, and the best approximation error. The analysis reveals that if the sampled function is a sparse polynomial from a finite-dimensional space, it can be reconstructed exactly. Moreover, it is observed that the smoother the sampled function, the fewer degrees of the sparse polynomial are required to attain a given approximation accuracy. The paper also extends this analysis to estimate the L2-norm approximation error for the sparse polynomial obtained via the weighted l1-minimization method, noting that in this context, the orthonormal polynomial system does not need to be uniformly bounded for the conclusions to hold.