We analyze the notion of reproducing pairs of weakly measurable functions, a generalization of continuous frames. The aim is to represent elements of an abstract space Y as superpositions of weakly measurable functions belonging to a space Z := Z(X, m), where (X, m) is a measure space. Three cases are envisaged, with increasing generality: (i) Y and Z are both Hilbert spaces; (ii) Y is a Hilbert space, but Z is a PIP-space; (iii) Y and Z are both PIP-spaces. It is shown, in particular, that the requirement that a pair of measurable functions be reproducing strongly constrains the structure of the initial space Y. Examples are presented for each case.
|Titolo:||PIP-Space Valued Reproducing Pairs of Measurable Functions|
|Data di pubblicazione:||2019|
|Citazione:||Antoine, J., & Trapani, C. (2019). PIP-Space Valued Reproducing Pairs of Measurable Functions. AXIOMS, 8(2), 52.|
|Appare nelle tipologie:||1.01 Articolo in rivista|