Approximating the Inverse Frame Operator from Localized Frames

Details Approximating the Inverse Frame Operator from Localized Frames Approximating the Inverse Frame Operator from Localized Frames

2012-03-29

arxiv.org/abs/1203.6433v1

Mathematics

Numerical Analysis

23 pages

Scientific paper

This investigation seeks to establish the practicality of numerical frame approximations. Specifically, it develops a new method to approximate the inverse frame operator and analyzes its convergence properties. It is established that sampling with {\em well-localized frames} improves both the accuracy of the numerical frame approximation as well as the robustness and efficiency of the (finite) frame operator inversion. Moreover, in applications such as magnetic resonance imaging, where the given data often may not constitute a well-localized frame, a technique is devised to project the corresponding frame data onto a more suitable frame. As a result, the target function may be approximated as a finite expansion with its asymptotic convergence solely dependent on its smoothness. Numerical examples are provided.

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