We consider Hermitian Toeplitz matrices Tn(An) generated by symbols An that depend on the matrix size n. These symbols take the form (Formula presented.) where An and a0 belong to the simple-loop class SLα for some α⩾2. Under suitable assumptions on the decay of the coefficients βn,k and the regularity of the functions ak, we prove that the eigenvalues of Tn(An) admit a full asymptotic expansion uniformly in the bulk of the spectrum. This work generalizes previous results for fixed and finitely perturbed simple-loop symbols.
EIGENVALUE SUPERPOSITION FOR SERIES OF TOEPLITZ MATRICES WITH MATRIX-SIZE DEPENDENT GENERATING FUNCTIONS
Serra-Capizzano S.
2026-01-01
Abstract
We consider Hermitian Toeplitz matrices Tn(An) generated by symbols An that depend on the matrix size n. These symbols take the form (Formula presented.) where An and a0 belong to the simple-loop class SLα for some α⩾2. Under suitable assumptions on the decay of the coefficients βn,k and the regularity of the functions ak, we prove that the eigenvalues of Tn(An) admit a full asymptotic expansion uniformly in the bulk of the spectrum. This work generalizes previous results for fixed and finitely perturbed simple-loop symbols.
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