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.
|Number of pages||22|
|Publication status||Published - 2019|
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Mathematical Physics
- Geometry and Topology