Stability of the notion of approximating class of sequences and applications

作者：Serra Capizzano Stefano^*; Sundqvist Per

来源：Journal of Computational and Applied Mathematics, 2008, 219(2): 518-536.

DOI：10.1016/j.cam.2007.03.027

• Toeplitz (and generalized locally toeplitz) sequence
• algebra of sequences
• generating function
• sparsely vanishing
• cross-validation
• spatially invariant and variant blurring

Given an approximating class of sequences {{B-n,m}(n)}(m) for {A(n)}(n), we prove that {{B-n(+),(m)}(n)}(m) (X+ being the pseudo-inverse of Mobre-Penrose) is an approximating class of sequences for (A(n)(+)}(n), where {A(n)}(n) is a sparsely vanishing sequence of matrices A(n) of size d(n) with d(k) > d(q) for k > q, k, q epsilon N. As a consequence, we extend distributional spectral results on the algebra generated by Toeplitz sequences, by including the (pseudo) inversion operation, in the case where the sequences that are (pseudo) inverted are distributed as sparsely vanishing symbols. Applications to preconditioning and a potential use in image/signal restoration problems are presented.

• 出版日期2008-10-1