Approximation with Bernstein–Szegö polynomials

We present approximation kernels for orthogonal expansions with respect to Bernstein–Szegö polynomials. Theconstruction is derived from known results for Chebyshev polynomials of the first kind and does not pose any restrictions on the Bernstein–Szegö polynomials.     

Schröder iteration functions associated with a one-parameter family of biquadratic polynomials

Schröder iteration functions, a generalization of the Newton–Raphson method to determine roots of equations, are generally rational functions which possess some critical points, free to converge to attracting cycles. These free critical points, however, satisfy some higher-degree polynomial equations which we solve analytically. Then, with the help of microcomputer plots, we examine the Julia sets of the Schröder functions and the orbits of all their free critical points associated with a particular one-parameter family of quartic polynomials, by walking in their dynamic and parameter spaces. This examination takes place in the complex plane as well as on the Riemann sphere.     

Szeg? type polynomials and para-orthogonal polynomials

Szeg? type polynomials with respect to a linear functional M for which the moments M[tn]=μ−n are all complex, μ−n=μn and Dn≠0 for n?0, are considered. Here, Dn are the associated Toeplitz determinants. Para-orthogonal polynomials are also studied without relying on any integral representation. Relation between the Toeplitz determinants of two different types of moment functionals are given. Starting from the existence of polynomials similar to para-orthogonal polynomials, sufficient conditions for the existence of Szeg? type polynomials are also given. Examples are provided to justify the results.     

The Szeg? curve and Laguerre polynomials with large negative parameters

We study the asymptotic zero distribution of the rescaled Laguerre polynomials, , with the parameter αn varying in such a way that . The connection with the so-called Szeg? curve is shown.     

A moment problem associated to rational Szegő functions

Leta ₁,...,a _p be distinct points in the finite complex plane ?, such that |a _j|>1,j=1,..., p and let \(b_j = 1/\bar \alpha _j ,\) j=1,..., p. Let μ₀, μ _π ^(j) , ν _π ^(j) j=1,..., p;n=1, 2,... be given complex numbers. We consider the following moment problem. Find a distribution ψ on [?π, π], with infinitely many points of increase, such that $$\begin{array}{l} \int_{ - \pi }^\pi {d\psi (\theta ) = \mu _0 ,} \\ \int_{ - \pi }^\pi {\frac{{d\psi (\theta )}}{{(e^{i\theta } - a_j )^n }} = \mu _n^{(j)} ,} \int_{ - \pi }^\pi {\frac{{d\psi (\theta )}}{{(e^{i\theta } - b_j )^n }} = v_n^{(j)} ,} j = 1,...,p;n = 1,2,.... \\ \end{array}$$ It will be shown that this problem has a unique solution if the moments generate a positive-definite Hermitian inner product on the linear space of rational functions with no poles in the extended complex plane ?^* outside {a ₁,...,a _p,b ₁,...,b _p}.     

Extensions of Szegö's theory of orthogonal polynomials,II

Let {? _n (dμ)} be a system of orthonormal polynomials on the unit circle with respect to a measuredμ. Szegö's theory is concerned with the asymptotic behavior of? _n (dμ) when logμ'∈L ¹. In what follows we will discuss the asymptotic behavior of the ratio φ_n(dμ ₁)/φ_n(dμ ₂) off the unit circle in casedμ ₁ anddμ ₂ are close in a sense (e.g.,dμ ₂=g dμ ₁ whereg≥0 is such thatQ(e ^it )g(t) andQ(e ^it )/g(t) are bounded for a suitable polynomialQ) and μ ₁ ^′ >0 almost everywhere or (a somewhat weaker requirement) lim _n→∞Φ _n (dμ ₁,0)=0, for the monic polynomials Φ _n . The consequences for orthogonal polynomials on the real line are also discussed.     

Maximal functions associated with Fourier multipliers of Mikhlin-Hörmander type

We show that maximal operators formed by dilations of Mikhlin- Hörmander multipliers are typically not bounded on L^p(^d). We also give rather weak conditions in terms of the decay of such multipliers under which L^p boundedness of the maximal operators holds.Christ, Grafakos and Seeger were supported in part by NSF grants. Honzík was supported by 201/03/0931 Grant Agency of the Czech Republic     

Extensions of Szegö's theory of orthogonal polynomials,III

Let {ø _n (dμ)} be a system of orthonormal polynomials on the unit circle with respect to a measuredμ. Szegö's theory is concerned with the asymptotic behavior ofø _n (dμ) when logμ′∈L ¹. In what follows we will discuss the asymptotic behavior of the ratioø _n (dμ ₂)/ø _n (dμ ₁) on the unit circle whendμ ₁ anddμ ₂ are close in a sense (e.g.,dμ ₂=gdμ ₁, where g≥0 is such thatQ(e ^it )g(t) andQ(e ^it )/g(t) are bounded for a suitable polynomialQ) and μ ₁ ^′ >0 almost everywhere or (a somewhat weaker requirement) lim _n→∞Φ _n (dμ ₁,0)=0 for the monic polynomial Φ _n . The asymptotic behavior of the same fraction outside the unit circle was discussed in an earlier paper.     

Sensitivity analysis for Szegő polynomials

Szegő polynomials are orthogonal with respect to an inner product on the unit circle. Numerical methods for weighted least-squares approximation by trigonometric polynomials conveniently can be derived and expressed with the aid of Szegő polynomials. This paper discusses the conditioning of several mappings involving Szegő polynomials and, thereby, sheds light on the sensitivity of some approximation problems involving trigonometric polynomials. This Research supported in part by NSF grant DMS-0107858.     

Narayana polynomials and Hall–Littlewood symmetric functions

We show that Narayana polynomials are a specialization of row Hall–Littlewood symmetric functions. Using λ-ring calculus, we generalize to Narayana polynomials the formulas of Koshy and Jonah for Catalan numbers.     

Convergence of Padé approximants of partial theta functions and the Rogers-Szegö polynomials

We investigate the convergence of sequences of Padé approximants for the partial theta function $$h_q (z): = \sum\limits_{j = 0}^\infty { q^{j(j - 1)/2_{Z^j } } } , q = e^{i\theta } , \theta \in [0,2\pi ).$$ Whenθ/(2π) is irrational, this function has the unit circle as its natural boundary. We determine subrogions of ¦z¦ < 1 in which sequences of Padé approximants converge uniformly, and subrogions in which they converge in capacity, but not uniformly. In particular, we show that only a proper subsequence of the diagonal sequence {[n/n]} _n=1 ^∞ converges locally uniformly in all of ¦z¦< l; in contrast, no subsequence of any Padé row {[m/n]} _m=1 ^∞ (withn ≥ 2 fixed) can converge locally uniformly in all of ¦z¦ < 1. Further, we obtain the zero and pole distributions of sequences of Padé approximants by analyzing the zero distribution of the Rogers-Szegö polynomials $$G_n (z): = \sum\limits_{j = 0}^n {\left[ {\begin{array}{*{20}c} n \\ j \\ \end{array} } \right]} z^j , n = 0,1,2,....$$      

Polynomial zerofinders based on Szegő polynomials

The computation of zeros of polynomials is a classical computational problem. This paper presents two new zerofinders that are based on the observation that, after a suitable change of variable, any polynomial can be considered a member of a family of Szegő polynomials. Numerical experiments indicate that these methods generally give higher accuracy than computing the eigenvalues of the companion matrix associated with the polynomial.     

Orthogonal systems of functions associated with the solutions of Löwner's equation

This paper is devoted to LSwner's well-known method in the theory of univalent functions. Let 0<=t<, be the solution of Löwner's equation under the initial condition, and let. Assume that the coefficients are defined by the expansionOne proves the theorem: the functions form an orthogonal system of functions on [0, ). One gives several corollaries of this theorem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 24–35, 1983.