Kernels of Gramian operators for frames in shift-invariant subspaces

✍ Scribed by A. Askari Hemmat; J.-P. Gabardo

• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 256 KB
• Volume: 435
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2011.01.018

✦ Synopsis

For an invertible n×n matrix B and a finite or countable subset of

Our main objects of interest in this paper are the kernel of the associated Gramian G(.) and dual Gramian G(.) operator-valued functions. We show in particular that the orthogonal complement of M in L 2 (R n ) can be generated by a Parseval frame obtained from a shift-invariant system having m generators where m = dim(Ker( G(.))) ∞ . Furthermore, this Parseval frame can be taken to be an orthonormal basis exactly when dim(Ker( G(.))) = m almost everywhere. Analogous results in terms of dim(Ker(G(.))) are also obtained concerning the existence of a collection of m sequences in the orthogonal complement of the range of analysis operator associated with the frame X whose shifts either form a Parseval frame or an orthonormal basis for that orthogonal complement.