Extensions of Bessel sequences to dual pairs of frames
The authors study extensions of Bessel sequences in separable Hilbert spaces. They prove that given two Bessel sequences \(\{f_i\}_{i\in I}\) and \(\{g_i\}_{i\in I}\) in the Hilbert space \(\mathcal{H}\), there exist Bessel sequences \(\{p_j\}_{j\in J}\) and \(\{q_j\}_{j\in J}\) such that the systems \(\{h_t\}_{t\in T}:=\{f_i\}_{i\in I}\cup\{p_j\}_{j\in J}\) and \(\{k_t\}_{t\in T}:=\{g_i\}_{i\in I}\cup\{q_j\}_{j\in J}\) form a pair of dual frames for \(\mathcal{H}\), meaning that \(\{h_t\}_{t\in T}\) and \(\{k_t\}_{t\in T}\) are frames and satisfy the reconstruction formula \[ f=\sum_{t\in T}\langle f,h_t \rangle k_t,\qquad\forall\,\,\,f\in\mathcal{H}. \] The result is more flexible than the one proven by \textit{P. G. Casazza} and \textit{N. Leonhard} [Contemp. Math. 451, 11--31 (2008; Zbl 1210.42047)] and \textit{D. Li} and \textit{W. Sun} [Acta Math. Sin., Engl. Ser. 25, No. 2, 287--292 (2009; Zbl 1177.41030)], which says that any Bessel sequence can be extended to a tight frame. Additionally, in contrast to the extension to a tight frame, the extension to a pair of dual frames does not involve the square root of any operator and therefore it might be more efficient from the calculational point of view. In fact, this is illustrated by the authors with concrete examples. When the initial pair of Bessel sequences has a particular structure it might be convenient to have the extensions of them to be of the same shape. The authors prove that this is possible when having Gabor sequences. More precisely, they show that two Gabor Bessel sequences in \(L^2(\mathbb{R})\) can be extended to a pair of dual frames by adding two Gabor systems. Furthermore, when each of the original Gabor sequences is generated by compactly supported functions, the generators of the Gabor systems one needs to add to have a pair of frame duals can be chosen to be compactly supported as well. Under certain conditions, a result in this spirit is obtained for wavelet systems.
scientific article; zbMATH DE number 6141631