6533b86efe1ef96bd12cca95

RESEARCH PRODUCT

PIP-Space Valued Reproducing Pairs of Measurable Functions

Camillo TrapaniJean-pierre Antoine

subject

Pure mathematicspartial inner product spacesMeasurable functionLogicGeneralizationreproducing pairs; continuous frames; upper and lower semi-frames; partial inner product spacesStructure (category theory)upper and lower semi-framecontinuous frameAbstract spaceSpace (mathematics)01 natural sciencesMeasure (mathematics)symbols.namesakeSettore MAT/05 - Analisi Matematica0103 physical sciences0101 mathematics010306 general physicsreproducing pairMathematical PhysicsMathematicscontinuous framesAlgebra and Number Theorylcsh:Mathematics010102 general mathematicsHilbert spaceupper and lower semi-frameslcsh:QA1-939reproducing pairssymbolsGeometry and TopologyAnalysis

description

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.

yearjournalcountryeditionlanguage
2019-04-30
10.3390/axioms8020052http://hdl.handle.net/10447/355144