Optimal adaptive computations in the Jaffard algebra and localized frames

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• 作者：Stephan Dahlke ; Massimo Fornasier ; Karlheinz Grö ; chenig
• 关键词：Adaptive scheme ; Jaffard algebra ; Frames in Banach spaces ; Best approximation ; Localization of frames ; Sparse matrix
• 刊名：Journal of Approximation Theory
• 出版年：2010
• 出版时间：January 2010
• 年：2010
• 卷：162
• 期：1
• 页码：153-185
• 全文大小：1212 K

文摘

We study the numerical solution of infinite matrix equations for a matrix 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 Sjöstrand 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 ℓ²-convergence, the algorithm converges automatically in some stronger norms of weighted ℓ^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.