Optimal adaptive computations in the Jaffard algebra and localized frames
- Autores: Stephan Dahlke, Massimo Fornasier, Karlheinz Gröchenig
- Localización: Journal of approximation theory, ISSN 0021-9045, Vol. 162, Nº 1, 2010, págs. 153-185
- Idioma: inglés
- DOI: 10.1016/j.jat.2009.04.001
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Resumen
We study the numerical solution of infinite matrix equations Au D 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 Sj¨ostrand 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.
