The essence of invertible frame multipliers in scalability

Norm retrieval algorithms: A new frame theory approach

“…We observe that, this method leads to constructing NRF in R n from a given Riesz basis. To this end, we consider frames with finite excess in n-dimensional real Hilbert space, and we note that such frames contain a Riesz basis; hence, without losing the generality, by changing the index set if necessary, we use an analogous notation applied in [22] and write Φ = {𝜙 i } 0 i=−k ∪ {𝜙 i } i∈I n where 𝜙 = {𝜙 i } i∈I n denotes the Riesz basis of Φ and {𝜙 i } 0 i=−k is the redundant vectors. The strategy applied in this section for characterizing NRFs is based on the size of the following set:…”

“…Also recall that, two frames $$$ \Phi $$$ and $$$ \Psi $$$ are equivalent if there exists an invertible operator $$$ U $$$ on $$$ \mathcal{H} $$$ so that $$$ \Psi =U\Phi $$$. See [14–20, 22] for more detailed information on frames theory and the importance of duality principle.…”

“…We observe that, this method leads to constructing NRF in $$$ {\mathrm{\mathbb{R}}}^n $$$ from a given Riesz basis. To this end, we consider frames with finite excess in $$$ n $$$‐dimensional real Hilbert space, and we note that such frames contain a Riesz basis; hence, without losing the generality, by changing the index set if necessary, we use an analogous notation applied in [22] and write $$$ \Phi ={\left\{{\phi}_i\right\}}_{i=-k}^0\cup {\left\{{\phi}_i\right\}}_{i\in {I}_n} $$$ where $$$ \phi ={\left\{{\phi}_i\right\}}_{i\in {I}_n} $$$ denotes the Riesz basis of $$$ \Phi $$$ and $$$ {\left\{{\phi}_i\right\}}_{i=-k}^0 $$$ is the redundant vectors. The strategy applied in this section for characterizing NRFs is based on the size of the following set: $$$$ \Lambda =\left\{i\in {I}_n:\kern0.30em {S}_{\phi}^{-1}{\phi}_i\perp {\phi}_l, for\ all\kern2.56804pt \hspace{0.1em}l\in \left\{-k,\dots, 0\right\}\right\}.$$…”

“…Also recall that, two frames Φ and Ψ are equivalent if there exists an invertible operator U on  so that Ψ = UΦ. See [14][15][16][17][18][19][20]22] for more detailed information on frames theory and the importance of duality principle.…”

Estimate of the spectral radii of Bessel multipliers and consequences

“…Bessel multipliers, as introduced in [2], are operators in a Hilbert space which have been extensively studied [5,11,24,25], occur in various fields of applications [4,14,21] and include the class of frame multipliers [9,10,12,19,22]. Recently, in [10], given a frame multiplier some regions of the complex plane containing the spectrum have been identified.…”