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چكيده انگليسي :

Pseudo Frame Pseudo Dual of Frame and Their Application Mohsen Karimi mohsen3758 iut@yahoo com March 3 2012 Master of Science Thesis in Farsi Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr R Nasr Isfahani Isfahani@cc iut ac iradvisor 2000 MSC 42C15 Key words Frame Frame Dual Frame Operate Multiresolution Analysis Pseudo Dual of Frame PseudoFrame Pseudo Frame Dual Wavelet Abstract Frame decompositions can be delicate In a separable Hilbert spaces frames stand fora sequence xn H such that x H A x 2 x xn 2 B x 2 where constantsA B 0 are the frame bounds The upper bound of the frame is also known as the Besselcondition for xn This frame is tight if A B and is exact if it ceases to be a framewhen any of its elements is removed There are generally in nitely many duals for a givennonexact frame Let xn be frame for H Then there is dual x such that for every x n in H x n x xn xn n x xn xn When H be a separable Hilbert space we say xn and x form a pair of pseudoframes for H if for every x y in H x y n x x xn y n nthis equation can be particularly convenient for arguments of weak convergence that maybenecessary in general cases Frames are all pseudoframes However a pseudoframe pair neednot be the usual frames In this dissertation a notion of pseudoframes for subspaces PFFS is de ned and characterized in a separable Hilbert space H PFFS functions in a manner of aframe for a subspace X in H Yet none of the pair of sequences xn and x is necessarily ncontained in X This gives rise to attractive properties that center around the exibility A necessary and su cient characterization of PFFSs is provided Analytical formulae forthe constructions of PFFSs in two directions are derived Examples are considered Someinsight relationships of an PFFS for X and a frame of X are also observed thanks to a privatecommunication with Casazza So we show that there are duals x to a given nonexact nframe xn that are not usual frames In particular these duals are not Bessel sequences We call them pseudo duals A characterization and a constructions of pseudo dual are given In section 4 one example of pseudo dual of a wavelet frame is carefully studied This wasour motivation for the study We hope it will demonstrate the roles of pseudo duals intheoretical and practical application To characterize pseudo duals we introduce a notionof pseudoframes for separable Hilbert spaces in section 5 Basic properties of pseudoframesare discussed A characterization and a construction of pseudo duals are given Resultsare constructive Example and potential applications of pseudo frames and pseudo dualstheory in General Deconvolution Problems and Signal Restorations and Noise Reductionsare discussed We show that if H be a Hilbert space and X be a closed subspace of Hcontaining signals of interests where f is a original signal function and g is a observed signalfunction with noise n we can restorated signal with exibility properties of pseudo frames In fact we hope that f g and n is removed 1

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