Functions with Prescribed Best Linear Approximations

TL;DR

This paper investigates the existence of functions with prescribed best approximations in Hilbert spaces, characterizing the inverse best approximation property and linking it to convergence rates in projection algorithms.

Contribution

It introduces the IBAP concept, provides geometric characterizations, and connects these to convergence in algorithms, with applications across harmonic analysis and signal processing.

Findings

Characterization of the IBAP in terms of subspace geometry

Connections between IBAP and linear convergence rates

Applications to harmonic analysis, integral equations, and wavelet frames

Abstract

A common problem in applied mathematics is to find a function in a Hilbert space with prescribed best approximations from a finite number of closed vector subspaces. In the present paper we study the question of the existence of solutions to such problems. A finite family of subspaces is said to satisfy the \emph{Inverse Best Approximation Property (IBAP)} if there exists a point that admits any selection of points from these subspaces as best approximations. We provide various characterizations of the IBAP in terms of the geometry of the subspaces. Connections between the IBAP and the linear convergence rate of the periodic projection algorithm for solving the underlying affine feasibility problem are also established. The results are applied to problems in harmonic analysis, integral equations, signal theory, and wavelet frames.