Let p be a trigonometric polynomial, nonnegative 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 this asymptotics holds in...

In this paper, two new subclasses of holomorphic and bi-univalent functions are introduced by using Balancing polynomials. Then, coefficient estmations determined for the first coefficients belonging to these classses. Finally, Fekete-Szeg¨o problem is handled in defined.

The frequency analysis problem is to determine the unknown frequencies of a trigonometric signal with a sample of size N . Recently there has been established a method for determining those frequencies by using the asymptotic behaviour of the zeros of certain Szeg o polynomials. The Szeg o polynomials in question are orthogonal on the unit circle with respect to an inner product de ned by a m...

By making use of $q$-derivative and $q$-integral operators, we define a class analytic bi-univalent functions in the unit disk $|z|<1$. Subsequently, investigate some properties such as early coefficient estimates then obtain Fekete-Szeg\"o inequality for both real complex parameters. Further, interesting corollaries are discussed.

In this paper our primary concern is with the establishment of weighted Hardy inequalities in L(Ω) and Rellich inequalities in L(Ω) depending upon the distance to the boundary of domains Ω ⊂ R with a finite diameter D(Ω). Improved constants are presented in most cases.

Szeg˝ o's procedure to connect orthogonal polynomials on the unit circle and orthogonal polynomials on [−1, 1] is generalized to nonsymmetric measures. It generates the so-called semi-orthogonal functions on the linear space of Laurent polynomials Λ, and leads to a new orthogonality structure in the module Λ × Λ. This structure can be interpreted in terms of a 2 × 2 matrix measure on [−1, 1], a...

We study the properties of stationary G-chains in terms their generating functions. In particular, we prove an analogue Szeg\H{o} limit theorem for symplectic eigenvalues, derive expression entropy rate quantum Gaussian processes, and distribution eigenvalues truncated block Toeplitz matrices. also introduce a concept numerical range, analogous to that some its basic properties, mainly context ...

The zeros of semi-orthogonal functions with respect to a probability measure µ supported on the unit circle can be applied to obtain Szeg˝ o quadrature formulas. The discrete measures generated by these formulas weakly converge to the orthogonality measure µ. In this paper we construct families of semi-orthogonal functions with interlacing zeros, and give a representation of the support of µ in...