Approximation of the frame coefficients using finite dimensional methods

Peter G. Casazza; Ole Christensen

doi:10.1117/12.276847

1 October 1997 Approximation of the frame coefficients using finite dimensional methods

Peter G. Casazza, Ole Christensen

Author Affiliations +

Journal of Electronic Imaging, Vol. 6, Issue 4, (October 1997). https://doi.org/10.1117/12.276847

Abstract

A frame is a family {f_i}^∞_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 {f_i}ⁿ_i=1 of the frame and the orthogonal projection P_n onto span {f_i}ⁿ_i=1. For f ⊂ H, P_nf has a representationas a linear combination of f_i, 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 theinversion problem. In the same spirit, we approximate the solution to a moment problem. It turns out that the class of "well-behavingframes" are identical for the two problems we consider.

Citation Download Citation

Peter G. Casazza and 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

Published: 1 October 1997

ACCESS THE FULL ARTICLE

INSTITUTIONAL
Select your institution to access the SPIE Digital Library.

SELECT YOUR INSTITUTION

PERSONAL
Sign in with your SPIE account to access your personal subscriptions or to use specific features such as save to my library, sign up for alerts, save searches, etc.

PERSONAL SIGN IN

No SPIE Account? Create one

PURCHASE THIS CONTENT

SUBSCRIBE TO DIGITAL LIBRARY

50 downloads per 1-year subscription

Members: $195

Non-members: $335 ADD TO CART

25 downloads per 1 - year subscription

Members: $145

Non-members: $250 ADD TO CART

PURCHASE SINGLE ARTICLE

Includes PDF, HTML & Video, when available