Lower Bound for Matrices on the Fibonacci Sequence Spaces and its Applications in Frame Theory

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: $$\begin{aligned} {\left\{ {\sum \limits _{n = 0}^\infty {{{\left( {\sum \limits _{k = 0}^n {\frac{{f_k^2}}{{{f_n}{f_{n + 1}}}}\sum \limits _{i = 0}^\infty {{a_{k,i}}{x_i}} } } \right) }^q}} } \right\} ^{1/q}} \ge \ell {\left( {\sum \limits _{n = 0}^\infty {x_n^p} } \right) ^{1/p}}, \end{aligned}$$ 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.