Pushing the envelope of the test functions in the Szegö and Avram-Parter theorems
Let \(a\) be a complex-valued function in \(L^1 (0, 2\pi )\) and let \(a_n\) be the \(n^{th}\) Fourier coefficient of \(a\), and denote by \(T_n (a)\) the \(n\times n\) Toeplitz matrix \((a_{ij} ), 1\leq i, j\leq n\). All the matrices \(T_n (a)\) are Hermitian iff \(a\) is real-valued. \textit{G. Szegö} [Math. Z. 6, 167--202 (1920); 167--190 (1921) (1920, 1921; JFM 48.0376.03)] has shown that \[ \lim_{n \rightarrow \infty}\frac1n \sum^{n}_{j=1}F(\lambda_{j} (T_n (a))) =\frac{1}{2\pi} \int^{2\pi}_{0} F(a(\theta )) \,d\theta \] under certain conditions on \(a\) and \(F\) including that \(a\) be real-vaued. Here \(\lambda_{1} (A)\leq\lambda_{2} (A)\leq\ldots \leq \lambda_{n} (A)\) are the eigenvalues of a Hermitian matrix \(A\). On the other hand, Avram-Parter type theorems [\textit{F. Avram}, Probab. Theory Relat. Fields 79, No. 1, 37--45 (1988; Zbl 0648.60043); \textit{S. V. Parter}, Linear Algebra Appl. 80, 115--130 (1986; Zbl 0601.15006)] state that under some assumptions on \(a\) and \(F\) we have \[ \lim_{n \rightarrow\infty}\frac1n \sum^{n}_{j=1}F (s_j (T_n (a))) =\frac{1}{2\pi}\int^{2\pi}_{0} F(| a(\theta| ))\,d\theta \] where \(s_1 (A)\leq s_2 (A)\leq \ldots \leq s_n (A)\) are singular values of an \(n\times n\) matrix \(A\). The authors of the paper under review consider the question ``are these theorems true whenever they make sense?'' and they give a negative answer to it. They show that Szegö and Avram-Parter theorems that give the limit of the arithmetic mean of the values of certain test functions \((F )\) at the eigenvalues of Hermitian Toeplitz matrices in Szegö theorems type and the singular values of arbitrary Toeplitz matrices in Avram-Parter theorems as the matrix dimension goes to infinity are not valid for all continuous, nonegative, and monotonously increasing test functions. They also obtain general results that include all versions known so far.
scientific article; zbMATH DE number 5285782
scientific article