Mi, 06.03.2019 14:30
PIP-space valued reproducing pairs of measurable functions
We analyze the notion of reproducing pair of weakly measurable functions, a generalization of continuous frame. The aim is to represent elements of an abstract space Y as superpositions of weakly measurable functions belonging to a space V = V (X; ), where (X; ) is a measure space. Three cases are envisaged, with increasing generality: (i) Y and V are both Hilbert spaces; (ii) Y is a Hilbert space, but V is a pip-space; (iii) Y and V are both pip-spaces. It is shown, in particular, that the requirement that a pair of measurable functions be reproducing strongly constraints the structure of the initial space Y . Examples are presented for each case.