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Open AccessArticle

PIP-Space Valued Reproducing Pairs of Measurable Functions

by Jean-Pierre Antoine 1,*,† and Camillo Trapani 2,†
1
Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium
2
Dipartimento di Matematica e Informatica, Università di Palermo, I-90123 Palermo, Italy
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Axioms 2019, 8(2), 52; https://doi.org/10.3390/axioms8020052
Received: 16 January 2019 / Revised: 16 April 2019 / Accepted: 18 April 2019 / Published: 30 April 2019
(This article belongs to the Special Issue Harmonic Analysis and Applications)
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 , μ ), where ( X , μ ) 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-Text
Keywords: reproducing pairs; continuous frames; upper and lower semi-frames; partial inner product spaces reproducing pairs; continuous frames; upper and lower semi-frames; partial inner product spaces
MDPI and ACS Style

Antoine, J.-P.; Trapani, C. PIP-Space Valued Reproducing Pairs of Measurable Functions. Axioms 2019, 8, 52.

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