Stability of the notion of approximating class of sequences and applications

Given an approximating class of sequences {{Bn,m}n}m{{Bn,m}n}m for {An}n{An}n, we prove that {{Bn,m+}n}m (X+X+ being the pseudo-inverse of Moore–Penrose) is an approximating class of sequences for {An+}n, where {An}n{An}n is a sparsely vanishing sequence of matrices AnAn of size dndn with dk>dqdk>dq for k>q,k,q∈Nk>q,k,q∈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.