Let p be a trigonometric polynomial, non-negative on the unit circle T. We say that a measure σ on T belongs to the polynomial Szeg˝ o class, if dσ(e iθ) = σ ′ ac (e iθ)dθ + dσ s (e iθ), σ s is singular, and 2π 0 p(e iθ) log σ ′ ac (e iθ) dθ > −∞ For the associated orthogonal polynomials {ϕ n }, we obtain pointwise asymp-totics inside the unit disc D. Then we show that these asymptotics hold in...

Abstract. In this paper, we study the theory of orthogonal trigonometric polynomials (OTP). We obtain asymptotics of OTP with positive and analytic weight functions by Riemann-Hilbert approach and find they have relations with orthogonal polynomials on the unit circle (OPUC). By the relations and the theory of OPUC, we also get four-terms recurrent formulae, ChristoffelDarboux formula and some ...

We pose and discuss the following conjecture: let s n(z) := ∑n k=0 ( )k k! zk , and for ∈ (0, 1] let ∗( ) be the unique solution ∈ (0, 1] of ∫ ( +1) 0 sin (t − ) t −1 dt = 0. Then for 0< ∗( ) and n ∈ N we have | arg[(1− z) s n(z)]| /2, |z|< 1. We prove this for = 1 2 , and in a somewhat weaker form, for = 3 4 . Far reaching extensions of our conjectures and results to starlike functions of orde...

This research proposal concerns several projects at the interface of analytic number theory and random matrix theory. As such the proposal involves ideas from number theory, combinatorics, and analysis. The proposal is divided into two sections, related by the perhaps surprising use they make of combinatorial representation theory, but also by the following theme: both sections will study a spe...

We prove that if a multiple trigonometric series is spherically Abel summable everywhere to an everywhere finite function f(x) which is bounded below by an integrable function, then the series is the Fourier series of f(x) if the coefficients of the multiple trigonometric series satisfy a mild growth condition. As a consequence, we show that if a multiple trigonometric series is spherically con...

This paper examines the use of general orthonormal bases for system identification from frequency domain data. This idea has been studied in great depth for the particular case of the orthonormal trigonometric basis. Here we show that the accuracy of the estimate can be significantly improved by rejecting the trigonometric basis in favour of a more general orthogonal basis that is able to be ad...

The new integrable systems associated to the space of elliptic branched coverings are constructed. The relationship of these systems with elliptic Schlesinger's system Takasaki [1] is described. For the standard twofold elliptic coverings the integrable system is written explicitly. The trigonometric degeneration of our construction is presented.