Article

# Structured Parseval Frames in Hilbert \$C^*\$-modules

04/2006;
Source: arXiv

ABSTRACT We investigate the structured frames for Hilbert \$C^{*}\$-modules. In the case that the underlying \$C^{*}\$-algebra is a commutative \$W^*\$-algebra, we prove that the set of the Parseval frame generators for a unitary operator group can be parameterized by the set of all the unitary operators in the double commutant of the group. Similar result holds for the set of all the general frame generators where the unitary operators are replaced by invertible and adjointable operators. Consequently, the set of all the Parseval frame generators is path-connected. We also obtain the existence and uniqueness results for the best Parseval multi-frame approximations for multi-frame generators of unitary operator groups on Hilbert \$C^*\$-modules when the underlying \$C^{*}\$-algebra is commutative.

0 0
·
0 Bookmarks
·
79 Views
• Source
##### Article: Frames in Hilbert C*-modules and C*-algebras
[hide abstract]
ABSTRACT: We present a general approach to a modular frame theory in C*-algebras and Hilbert C*-modules. The investigations rely on the idea of geometric dilation to standard Hilbert C*-modules over unital C*-algebras that possess orthonormal Hilbert bases, and of reconstruction of the frames by projections and by other bounded modular operators with suitable ranges. We obtain frame representations and decomposition theorems, as well as similarity and equivalence results for frames. Hilbert space frames and quasi-bases for conditional expectations of finite index on C*-algebras appear as special cases. Using a canonical categorical equivalence of Hilbert C*-modules over commutative C*-algebras and (F)Hilbert bundles the results find a reintepretation for frames in vector and (F)Hilbert bundles. Fields of applications are investigations on Cuntz-Krieger-Pimsner algebras, on conditional expectations of finite index, on various ranks of C*-algebras, on classical frame theory of Hilbert spaces (wavelet and Gabor frames), and others. 2001: In the introduction we refer to related publications in detail.
11/2000;
• Source
##### Article: Symmetric approximation of frames and bases in Hilbert spaces
[hide abstract]
ABSTRACT: We consider existence and uniqueness of symmetric approximation of frames by normalized tight frames and of symmetric orthogonalization of bases by orthonormal bases in Hilbert spaces H . More precisely, we determine whether a given frame or basis possesses a normalized tight frame or orthonormal basis that is quadratically closest to it, if there exists such frames or bases at all. A crucial role is played by the Hilbert-Schmidt property of the operator (P-|F|), where F is the adjoint operator of the frame transform F*: H --> l_2 of the initial frame or basis and (1-P) is the projection onto the kernel of F. The result is useful in wavelet theory.
01/1999;
• ##### Article: Approximations for Gabor and wavelet frames
Trans. Amer. Math. Soc. 01/2003;