Affiliation: | aFakultä t für Mathematik, TU Chemnitz, 09107 Chemnitz, Germany bDepartamento de Matemáticas, CINVESTAV del I.P.N., Apartado Postal 14-740, 07000 México, D.F., Mexico cDepartment of Mathematics, Mechanics and Computer Science, Southern Federal University, 344006 Rostov-on-Don, Russia |
Abstract: | The Szegö and Avram–Parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues of Hermitian Toeplitz matrices and the singular values of arbitrary Toeplitz matrices, respectively, as the matrix dimension goes to infinity. The question on whether these theorems are true whenever they make sense is essentially the question on whether they are valid for all continuous, nonnegative, and monotonously increasing test functions. We show that, surprisingly, the answer to this question is negative. On the other hand, we prove the two theorems in a general form which includes all versions known so far. |