Piecewise Continuous Toeplitz Matrices and Operators: Slow Approach to Infinity
Albrecht Böttcher, Mark Embree, Lloyd N. Trefethen · SIAM Journal on Matrix Analysis and Applications · 2002
The pseudospectra of banded finite dimensional Toeplitz matrices rapidly converge to the pseudospectra of the corresponding infinite dimensional operator. This exponential convergence makes a compelling case for analyzing pseudospectra of such Toeplitz matrices---not just eigenvalues. What if the matrix is dense and its symbol has a jump discontinuity? The pseudospectra of the finite matrices still converge, but it is shown here that the rate is no longer exponential in the matrix dimension---only algebraic.