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Affine, Quasi-affine and Co-affine Frames

Indian Statistical Institute Series, ISSN: 2523-3122, Page: 131-160
2021
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Book Chapter Description

In this chapter, we show that an affine system X(Ψ ) generated by a finite family of functions Ψ in L(K) is an affine frame for L(K) if and only if the corresponding quasi-affine system X~ (Ψ ) is a quasi-affine frame. Moreover, their exact lower and upper bounds are equal. This result also holds for Bessel families. We characterize the translation invariance of a sesquilinear operator associated with a pair of affine systems. Then we define the affine and quasi-affine dual of Ψ and show that a finite subset Φ of L(K) is an affine dual of Ψ if and only if it is a quasi-affine dual. We discuss briefly about general sesquilinear operators on L(K) × L(K) and characterize the translation invariance of such operators in terms of Fourier multipliers on K. In the last section, we show that L(K) cannot have a co-affine frame and identify some subspaces of L(K) that can admit frames consisting of co-affine systems. We will use the results obtained in this chapter to provide a characterization of wavelets in Chap. 4.

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