Some Problems Concerning the Test Functions in the Szegö and Avram-Parter Theorems

Abstract

The Szegö and Avram-Parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues and singular values of Toeplitz matrices as the matrix dimension increases to infinity. This paper is concerned with some questions that arise when the test functions do not satisfy the known growth restrictions at infinity or when the test function has a logarithmic singularity within the range of the symbol. Several open problems are listed and accompanied by a few new results that illustrate the delicacy of the matter.

The Szegő and Avram–Parter theorems for general test functions

“…Note that Golinskii and Ibragimov [3] proved that the so-called strong Szegő limit theorem for positive generating functions is indeed true whenever it makes sense (see also the books [2] and [6]). In [1] we showed that the answer to our question is negative: ST and APT are proper subsets of C + (R) and C + [0, ∞), respectively.…”

“…The counterexample in [1] is highly oscillating and leaves us with the question whether ST and APT contain at least all monotonous functions. Our first main result tells us that, surprisingly, this is not the case: The proof of this theorem is rather sophisticated.…”

“…Serra Capizzano [4] solved the problem of characterizing the maximal class of test functions for which (1) and (2) hold under the assumption that both sides of (1) and (2) are finite. He showed that this happens for all a ∈ L p (p 1) if and only if G(λ) = O(|λ| p ) as |λ| → ∞ and F (s) = O(s p ) as s → ∞.…”