Functions with prescribed best linear approximations

Authors:
Patrick L. Combettes;Noli N. Reyes
Affiliations:
UPMC Université Paris 06, Laboratoire Jacques-Louis Lions-UMR CNRS 7598, 75005 Paris, France;University of the Philippines-Diliman, Institute of Mathematics, Quezon City, 1101, Philippines
Venue:
Journal of Approximation Theory
Year:
2010

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Abstract

A common problem in applied mathematics is that of finding 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 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 investigate problems in harmonic analysis, integral equations, signal theory, and wavelet frames.