PIP space valued reproducing pairs of measurable functions
(2019) Axioms: Mathematical Logic and Mathematical Physics — (2019)
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- Authors
Antoine, Jean-Pierre 
- Abstract
- 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, μ), where (X, μ) 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.
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Citations
Antoine, J.-P., & Trapani, C. (2019). PIP space valued reproducing pairs of measurable functions. Axioms: Mathematical Logic and Mathematical Physics. Submitted. https://hdl.handle.net/2078.5/23294 (Original work published 2019)