Linear Combinations of Frames and Frame Packets

Abstract

We find coefficients $c_{mn} (m, n \in \mathbb Z)$ such that for an arbitrary frame $\lbrace f_n \rbrace_{n \in \mathbb Z}$ the set of vectors $\lbrace \phi_m \rbrace_{m \in \mathbb Z} = \lbrace \sum _{n \in \mathbb Z} c_{mn}f_n \rbrace_{m \in \mathbb Z}$ 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 $\lbrace E_{\beta m} T_{\alpha n}g \rbrace_{m, n \in \mathbb Z}$ 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.