Search results for "upper and lower semi-frame"

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PIP-Space Valued Reproducing Pairs of Measurable Functions

2019

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.

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
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Reproducing pairs of measurable functions

2017

We analyze the notion of reproducing pair of weakly measurable functions, which generalizes that of continuous frame. We show, in particular, that each reproducing pair generates two Hilbert spaces, conjugate dual to each other. Several examples, both discrete and continuous, are presented.

continuous framesPure mathematicsPartial differential equationMeasurable functionApplied Mathematics010102 general mathematicsBanach spaceupper and lower semi-frames01 natural sciencesDual (category theory)Functional Analysis (math.FA)010101 applied mathematicsMathematics - Functional AnalysisContinuous frameReproducing pairInner product spaceSettore MAT/05 - Analisi MatematicaReproducing pairsUpper and lower semi-frameFOS: Mathematics0101 mathematics41A99 46Bxx 46ExxMathematics
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