This is a Toeplitz matrix of dimension N with a piecewise continuous symbol. For N sufficiently large, each of the illustrated pseudospectra will include the spectrum of the infinite dimensional operator, but for this class of piecewise continuous symbols, this value of N needs to be especially large. The eigenvalues of this example have been studied by Basor and Morrison [1] and the pseudospectra are illustrated in [2].
[1]: Basor and Morrison "The Fisher-Hartwig conjecture and Toeplitz eigenvalues", Linear Algebra Appl. 202 (1994), pp. 129-142).
[2]: A. Boettcher, M. Embree, and L. N. Trefethen "Piecewise continuous Toeplitz matrices and operators: slow approach to infinity", Oxford University Computing Laboratory Numerical Analysis Technical Report 00/19, August 2000.
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