Sampling Formulae and Optimal Factorizations of Projections

Abstract

Let H be a Hilbert space, W a closed subspace of H, and Q a (linear bounded) projection from H onto W with null space M⊥. We study decompositions like Qf = ∑n∊ℕ 〈 f, hn〉 fn, where {fn}n∊ℕ and {hn}n∊ℕ are frames for the subspaces W and M, respectively. This type of decompositions corresponds to sampling formulae. By considering the synthesis operator F (resp. H) of the sequence {fn}n∊ℕ (resp. {hn}n∊ℕ), the formula above can be expressed as the factorization Q = FH*. We study different properties of these factorizations and decompositions of oblique and orthogonal projections. Several characterizations of these decompositions are presented. By means of an operator inequality for positive operators, we get a result which minimizes the norm of F — H.