Abstract
Let \(A=(a_{n,k})_{n,k\ge 0}\) be a non-negative matrix. Denote by \(L_{\ell _p,F_q}(A)\) the supremum of those \(\ell ,\) satisfying the following inequality:
where \(x\ge 0\), \(x\in \ell _p\) and \(\{f_n\}_{n=0}^\infty \) is the Fibonacci numbers sequence. In this paper, first we introduce the Fibonacci weighted sequence space, \(F_{w,p}~(0< p < 1)\), of non-absolute type which is the p-normed space and is linearly isomorphic to the space \(\ell _p(w)\), where the weight sequence \(w=\{w_n\}_{n=0}^\infty \) is a increasing, non-negative sequence of real numbers. Then we focus on the evaluation of \(L_{\ell _p,F_q}(A^t)\) for a lower triangular matrix A, where \(0<q\le p<1\). A similar result is also established for \(L_{\ell _p,F_q}(H_\mu ^\alpha )\) where \(H^\alpha _\mu \) is the generalized Hausdorff matrix, \(0<q\le p\le 1\) and \(\alpha \ge 0.\) In each case, a lower estimate is obtained which is related to the Fibonacci numbers. As an application of such estimate in frame theory, we present the concept of Fibonacci frames for a separable Hilbert space \(\mathcal {H},\) as a special case of E-frames which were recently introduced by the authors in Talebi and Dehghan (Banach J Math Anal 9(3):43–74, 2015). We study some properties of Fibonacci frames and characterize all Fibonacci orthonormal bases, Fibonacci Riesz bases and Fibonacci frames starting with an arbitrary orthonormal basis for \(\mathcal {H}\). Finally, we characterize all dual Fibonacci frames for a given Fibonacci frame.
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The authors have benefited a lot from the referee’s report. So, we thank the reviewer for his/her careful reading and making some useful comments which improved the presentation of the paper.
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Dedicated to Imam Hassan Mojtaba (peace be upon him)
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Dehghan, M.A., Talebi, G. Lower Bound for Matrices on the Fibonacci Sequence Spaces and its Applications in Frame Theory. Results Math 72, 1087–1107 (2017). https://doi.org/10.1007/s00025-016-0632-x
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DOI: https://doi.org/10.1007/s00025-016-0632-x