The essence of invertible frame multipliers in scalability

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 liminfn∥φn∥>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\liminf _{n}\parallel \varphi _{n}\parallel >0$\end{document} 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.