Wiener’s Lemma for Infinite Matrices

The classical Wiener’s lemma and its various generalizations are important and have numerous applications in numerical analysis, wavelet theory, frame theory, and sampling theory. There are many different equivalent formulations for the classical Wiener’s lemma, with an equivalent formulation suitable for our generalization involving commutative algebra of infinite matrices W := {(a(j − j))j,j′∈Zd : ∑ j∈Zd |a(j)| < ∞}. In the study of spline approximation, (diffusion) wavelets and affine frames, Gabor frames on nonuniform grid, and non-uniform sampling and reconstruction, the associated algebras of infinite matrices are extremely non-commutative but we expect those noncommutative algebras have similar property to the Wiener’s lemma for the commutative algebra W. In this paper, we consider two noncommutative algebras of infinite matrices, the Schur class and the Sjöstrand class, and establish Wiener’s lemmas for those matrix algebras. Transactions of the American Mathematical Society, To appear