PIP-Space Valued Reproducing Pairs of Measurable Functions

We analyze the notion of reproducing pairs of weakly measurable functions, ageneralization of continuous frames. The aim is to represent elements of an abstract space Y assuperpositions of weakly measurable functions belonging to a space Z := Z(X, m), where (X, m) is ameasure space. Three cases are envisaged, with increasing generality: (i) Y and Z are both Hilbertspaces; (ii) Y is a Hilbert space, but Z is a PIP-space; (iii) Y and Z are both PIP-spaces. It is shown, inparticular, that the requirement that a pair of measurable functions be reproducing strongly constrainsthe structure of the initial space Y. Examples are presented for each case.