%0 Journal Article
%T PIP-Space Valued Reproducing Pairs of Measurable Functions
%A Camillo Trapani
%A Jean-Pierre Antoine
%J -
%D 2019
%R https://doi.org/10.3390/axioms8020052
%X Abstract 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 , ¦Ì ), where ( X , ¦Ì ) 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. View Full-Tex
%K reproducing pairs
%K continuous frames
%K upper and lower semi-frames
%K partial inner product spaces
%U https://www.mdpi.com/2075-1680/8/2/52