Linear Combinations of Frames and Frame Packets

Abstract

We find coefficients $c_{mn}(m,n\in \mathbb{Z})$ such that for an arbitrary frame $\{f_n{\}}_{n\in \mathbb{Z}}$ the set of vectors $\{{\varphi }_m{\}}_{m\in \mathbb{Z}}=\{{\sum }_{n\in \mathbb{Z}}{c}_{mn}{f}_n{\}}_{m\in \mathbb{Z}}$ will again be a frame. Appropriate coefficients can always be chosen as function values $c_{mn}=g(\frac{n}{\beta }–m\alpha )$), where $g$ belongs to a broad class of functions generating a Gabor frame $\{E_{\beta m}{T}_{\alpha n}g{\}}_{m,n\in \mathbb{Z}}$ for $L^2(\mathbb{R})$. We also prove a version of the splitting trick, which allows to construct a large family of frames based on a single (wavelet or Gabor) frame.