Geodesic
Refining some Inequalities for Frames with Specht's Ratio
Kragujevac Journal of Mathematics, Tome 46 (2022) no. 1, p. 39 .

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

We give a new lower bound in some inequalities for frames in a Hilbert space. If $\{f_i\}_{i\in I}$ is a Parseval frame for the Hilbert space $\mathbb{H}$ with frame operator $ S f =\sum_{i\in I} 〈f, f_i〉f_i$, then, for every $J\subset I$ and $f\in \mathbb{H}$, we have \[ eft(\dfrac{1+2lpha}{2+2lpha}\right) \|f\|^{2} eq um_{iı J} |〈f,f_i〉|^2+eft\|um_{iı J^{c}}〈f,f_i〉f_i\right\|^2, \] where $\alpha=\inf \left\lbrace R\left(\frac{\|S_{J^{c}}f\|}{\|S_{J}f\|}\right)\,:\, f\in\mathbb{H} , J\subset I \right\rbrace$ with Specht's ratio $R$. Also we obtain some improvements of the inequalities for general frames and alternate dual frames under suitable conditions. Our results refine the remarkable results obtained by Balan et al. and Gavruta.
Classification : 42C15
Keywords: Specht’s ratio, frame, Parseval frame, inequality 
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     title = {Refining some {Inequalities} for {Frames} with {Specht's} {Ratio}},
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Fahimeh Sultanzadeh; Mahmoud Hassani; Mohsen Erfanian Omidvar; Rajab Ali kamyabi Gol. Refining some Inequalities for Frames with Specht's Ratio. Kragujevac Journal of Mathematics, Tome 46 (2022) no. 1, p. 39 . http://geodesic.mathdoc.fr/item/KJM_2022_46_1_a3/