Wiener’s lemma for infinite matrices

Abstract:

The classical Wiener 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 lemma, with an equivalent formulation suitable for our generalization involving commutative algebra of infinite matrices ${\mathcal W}:=\{(a(j-j’))_{j,j’\in \mathbf {Z}^d}: \ \sum _{j\in \mathbf {Z}^d} |a(j)|<\infty \}$. In the study of spline approximation, (diffusion) wavelets and affine frames, Gabor frames on non-uniform grid, and non-uniform sampling and reconstruction, the associated algebras of infinite matrices are extremely non-commutative, but we expect those non-commutative algebras to have a similar property to Wiener’s lemma for the commutative algebra ${\mathcal W}$. In this paper, we consider two non-commutative algebras of infinite matrices, the Schur class and the Sjöstrand class, and establish Wiener’s lemmas for those matrix algebras.