Reconstruction of Functions from Prescribed Proximal Points

TL;DR

This paper develops a unified framework for reconstructing functions in Hilbert spaces from nonlinear, convex constraints using fixed point methods, with applications in signal recovery.

Contribution

It introduces a novel block-iterative algorithm that handles discontinuous prescriptions via firmly nonexpansive operators, extending classical approximation techniques.

Findings

Algorithm successfully reconstructs functions from prescribed points.

Numerical results demonstrate effectiveness in signal recovery.

Framework applies to a broad range of approximation problems.

Abstract

Under investigation is the problem of finding the best approximation of a function in a Hilbert space subject to convex constraints and prescribed nonlinear transformations. We show that in many instances these prescriptions can be represented using firmly nonexpansive operators, even when the original observation process is discontinuous. The proposed framework thus captures a large body of classical and contemporary best approximation problems arising in areas such as harmonic analysis, statistics, interpolation theory, and signal processing. The resulting problem is recast in terms of a common fixed point problem and solved with a new block-iterative algorithm that features approximate projections onto the individual sets as well as an extrapolated relaxation scheme that exploits the possible presence of affine constraints. A numerical application to signal recovery is demonstrated.