Linear Combinations of Frames and Frame Packets

  • Ole Christensen

    Technical University of Denmark, Lyngby, Denmark


We find coefficients cmn(m,nZ)c_{mn} (m, n \in \mathbb Z) such that for an arbitrary frame {fn}nZ\lbrace f_n \rbrace_{n \in \mathbb Z} the set of vectors {ϕm}mZ={nZcmnfn}mZ\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 cmn=g(nβmα)c_{mn} = g(\frac {n}{\beta} – m\alpha)), where gg belongs to a broad class of functions generating a Gabor frame {EβmTαng}m,nZ\lbrace E_{\beta m} T_{\alpha n}g \rbrace_{m, n \in \mathbb Z} for L2(R)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.

Cite this article

Ole Christensen, Linear Combinations of Frames and Frame Packets. Z. Anal. Anwend. 20 (2001), no. 4, pp. 805–815

DOI 10.4171/ZAA/1046