Mosaic approximations of discrete analogs of Calderón-Zygmund operators (Q1272034)

Suppose that \(A\in \mathbb{R}^{m\times n}\) is a matrix of rank \(r\). Ler \(B\) be a submatrix of a matrix \(A\in \mathbb{R}^{m\times n}\). By the symbol \(\Gamma(B)\) we denote the \(m\times n\) matrix coinciding with \(A\) at the entries determined by the submatrix \(B\) and having zeros as the other entries. A set of blocks \(A_i\) is called a cover of \(A\) if \(A= \sum_i\Gamma(A_i)\), and a mosaic partition of \(A\) if \(\Gamma(A_i)\) do not have common nonzero entries. In both cases, the corresponding matrix \(A\) is said to be mosaic. The mosaic rank of \(A\) with respect to a chosen cover is defined as follows \[ \text{mr }A= {\sum_i \text{mem }A_i\over m+n}, \] where \(\text{mem }A_i= \min\{m_in_i,\text{rank }A_i(m+ n)\}\). Let \(A\) be the Calderón-Zygmund operator, let \(\widetilde A\) be the matrix of a Galerkin-Petrov type approximation of \(A\). Then for any given \(\varepsilon\), one can chose a number \(M\) so that estimates \(\|\widetilde A-\widetilde A_M\|\leq \varepsilon\), \(\text{mr }\widetilde A_M= O(\ln N\ln^d\varepsilon^{-1})\) hold for the mosaic matrix \(\widetilde A_M\) obtained, in which all blocks are approximated by matrices of rank \(M\).