Partial sums of orthonormal bases preserving positivity -- and martingales (Q1271482)

The paper under review is motivated by issues in medical and astronomical image reconstruction [\textit{H. H. Barret}, \textit{J. L. Denny}, \textit{R. F. Wagner} and \textit{K. J. Myers}: ``Objective assessment of image quality. II: Fisher information, Fourier crosstalk, and figures of merit for task performance'', J. Opt. Soc. Am. A 12, 834-852 (1995)]. The article deals with a characterization, for \(L^2\)-spaces associated to finite measure spaces \((X,{\mathcal A},P)\), of those orthonormal bases with the following positivity preserving property: If \(f\in L^2(X)\) denotes a nonnegative function, then the partial sums in the expansion of \(f\) are nonnegative. The authors establish that the bases are necessarily generalized Haar functions, and the partial sums are a martingale closed on the right by the function \(f\).