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The essence of invertible frame multipliers in scalability

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Abstract

The purpose of this paper is twofold. The first is to give some new structural results for the invertibility of Bessel multipliers. Secondly, as applications of these results, we provide some conditions regarding the scaling sequence c = {cn}n which can be used in the role of the scalability of a given frame, a notion which has found more and more applications in the last decade. More precisely, we show that positive and strict scalability coincides for all frames Φ = {φn}n with \(\liminf _{n}\parallel \varphi _{n}\parallel >0\) which in particular provides some equivalent conditions for positive scalability of certain frames. Moreover, it is our objective to consider the effect of optimal frame bounds on the choice of scalings sequence. Along the way, the scalability of Riesz frames and Riesz bases are completely characterized and some necessary conditions for scalability of a near-Riesz basis are determined depending on its norm properties. Next, we turn our attention to the (c-)scalable bounded frame Φ and our results with the aid of the Feichtinger Conjecture give α and β, depending on the optimal frame bounds of Φ, such that the elements of the scaling sequence c should be chosen from the interval [α,β] for all but finitely many n. Finally, we introduce an explicit construction algorithm to produce desired invertible multiplier from the given one which is of interest in its own right.

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Acknowledgements

We are indebted to the referees for the careful reading of the paper, the detailed criticism and their numerous remarks and suggestions which greatly improved the presentation of the paper. Also, the authors especially thank one referee for pointing out an error in the proof of Theorems 2.6 and 2.7 of an earlier version of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah, Iran

    Hossein Javanshiri & Mohammad Abolghasemi

  2. Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran

    Ali Akbar Arefijamaal

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  1. Hossein Javanshiri
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  2. Mohammad Abolghasemi
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  3. Ali Akbar Arefijamaal
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Correspondence to Hossein Javanshiri.

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Communicated by: Gitta Kutyniok

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Javanshiri, H., Abolghasemi, M. & Arefijamaal, A.A. The essence of invertible frame multipliers in scalability. Adv Comput Math 48, 19 (2022). https://doi.org/10.1007/s10444-022-09940-8

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  • DOI: https://doi.org/10.1007/s10444-022-09940-8

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