Affine, Quasi-affine and Co-affine Frames
Indian Statistical Institute Series, ISSN: 2523-3122, Page: 131-160
2021
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Example: if you select the 1-year option for an article published in 2019 and a metric category shows 90%, that means that the article or review is performing better than 90% of the other articles/reviews published in that journal in 2019. If you select the 3-year option for the same article published in 2019 and the metric category shows 90%, that means that the article or review is performing better than 90% of the other articles/reviews published in that journal in 2019, 2018 and 2017.
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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.
Bibliographic Details
http://www.scopus.com/inward/record.url?partnerID=HzOxMe3b&scp=85122468626&origin=inward; http://dx.doi.org/10.1007/978-981-16-7881-3_3; https://link.springer.com/10.1007/978-981-16-7881-3_3; https://dx.doi.org/10.1007/978-981-16-7881-3_3; https://link.springer.com/chapter/10.1007/978-981-16-7881-3_3
Springer Science and Business Media LLC
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