Piecewise continuous Toeplitz matrices and operators:: Slow approach to infinity

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.