1 October 1997 Approximation of the frame coefficients using finite dimensional methods
Author Affiliations +
A frame is a family {fi}i=1 of elements in a Hilbert space H with the property that every element in H can be written as a (infinite) linear combination of the frame elements. Frame theory describes how one can choose the corresponding coefficients, which are called frame coefficients. From the mathematical point of view this is gratifying, but for applications, it is a problem that the calculation requires inversion of an operator on H. The projection method is introduced to avoid this problem. The basic idea is to consider finite subfamilies {fi}ni=1 of the frame and the orthogonal projection Pn onto span {fi}ni=1. For f ⊂ H, Pnf has a representation as a linear combination of fi, i = 1,2,...,n, and the corresponding coefficients can be calculated using finite dimensional methods. We find conditions implying that those coefficients converge to the correct frame coefficients as n→∞, in which case, we have avoided the inversion problem. In the same spirit, we approximate the solution to a moment problem. It turns out that the class of "well-behaving frames" are identical for the two problems we consider.
Peter G. Casazza, Ole Christensen, "Approximation of the frame coefficients using finite dimensional methods," Journal of Electronic Imaging 6(4), (1 October 1997). https://doi.org/10.1117/12.276847


Riesz wavelets and multiresolution structures
Proceedings of SPIE (December 05 2001)
Computation of the density of weighted wavelet systems
Proceedings of SPIE (November 13 2003)
Biorthogonality and multiwavelets in Hilbert spaces
Proceedings of SPIE (December 04 2000)
Sampling for shift-invariant and wavelet subspaces
Proceedings of SPIE (December 04 2000)
Tight frame approximations for Gabor and wavelet frames
Proceedings of SPIE (December 05 2001)

Back to Top