Some science articles are beyond my understanding. This is one of them and here is the abstract. Any input is much appreciated.

PL Combettes and NN Reyes, Functions with prescribed best linear approximations, Journal of Approximation Theory, V162 (2010), 1095-1116. DOI: dx.doi.org/10.1016/j.jat.2009.12.007

Initial statement

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.

Their goal in this paper…

In the present paper we study the question of the existence of solutions to such problems.

and what they did…

….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.

NN Reyes is currently with the Dept. of  Mathematics, UP Diliman.