|
Applied Math Seminar
The Unreasonable Effectiveness of Banach Algebras in Numerical Analysis
|
|
I will show that several problems arising at the interface of operator theory and numerical analysis (ranging from work by Gohberg et alii, Iserles, as well as Golub) can be solved in an elegant manner by using concepts from Banach algebra theory. In the first part of my talk I will consider the approximate solution of an infinite system of linear equations via the classical finite section methods. Using Banach algebra methods we are able to derive quantitative convergence estimates for the finite section method. We also derive the first finite section method that can be applied to a large class of non-hermitian (and non-Toeplitz-type) matrices. In the second part of my talk I will investigate the interplay of matrix factorizations and Banach algebras. An important noncommutative generalization of the famous Wiener's Lemma states that under certain conditions the inverse $A^{-1}$ of a matrix $A$ will inherit the off-diagonal decay properties from $A$. I will show that this Wiener property also extends to certain matrix factorizations (such as QR or LU factorization) and matrix functions (such as the matrix exponential). I will discuss applications of the above results in signal processing and communications. |