Approximation of the inverse frame operator and applications to Gabor frames
Journal of Approximation Theory
Adaptive wavelet methods for elliptic operator equations: convergence rates
Mathematics of Computation
Adaptive Wavelet Schemes for Elliptic Problems---Implementation and Numerical Experiments
SIAM Journal on Scientific Computing
Adaptive Wavelet Methods for Saddle Point Problems---Optimal Convergence Rates
SIAM Journal on Numerical Analysis
The finite section method and problems in frame theory
Journal of Approximation Theory
Adaptive Solution of Operator Equations Using Wavelet Frames
SIAM Journal on Numerical Analysis
| Hi-index | 0.00 |
We study the numerical solution of infinite matrix equations Au=f for a matrix A in the Jaffard algebra. These matrices appear naturally via frame discretizations in many applications such as Gabor analysis, sampling theory, and quasi-diagonalization of pseudo-differential operators in the weighted Sjostrand class. The proposed algorithm has two main features: firstly, it converges to the solution with quasi-optimal order and complexity with respect to classes of localized vectors; secondly, in addition to @?^2-convergence, the algorithm converges automatically in some stronger norms of weighted @?^p-spaces. As an application we approximate the canonical dual frame of a localized frame and show that this approximation is again a frame, and even an atomic decomposition for a class of associated Banach spaces. The main tools are taken from adaptive algorithms, from the theory of localized frames, and the special Banach algebra properties of the Jaffard algebra.