A note on the zeros of Freud–Sobolev orthogonal polynomials

We prove that the zeros of a certain family of Sobolev orthogonal polynomials involving the Freud weight function e^-x4${\mathrm{e}}^{-x^4}$ on R$\mathbb{R}$ are real, simple, and interlace with the zeros of the Freud polynomials, i.e., those polynomials orthogonal with respect to the weight function e^-x4${\mathrm{e}}^{-x^4}$. Some numerical examples are shown.     

The weighted (0,1,…,m−2,m)-interpolation technique based on the roots of the classical orthogonal polynomials and application in deriving new quadrature rules

We investigate the problem of weighted (0,1,…,m?2,m)-interpolation (m≧2, ${m\in \mathbb{N}}$ ) on the roots of the classical orthogonal polynomials. The necessary and sufficient conditions for the existence and uniqueness of this problem are established. Meanwhile, the explicit representation (characterization) of the weighted (0,1,…,m?2,m)-interpolation is given. As an application we obtain a Birkhoff type quadrature formula which is exact for the polynomials of degree at most mn. Also we investigate the error function of the new (0,1,…,m?2,m)-Birkhoff type weighted quadrature formulae using Peano kernel theorem. Some numerical examples are given to support the results.     

Zeros of para–orthogonal polynomials and linear spectral transformations on the unit circle

We study the interlacing properties of zeros of para–orthogonal polynomials associated with a nontrivial probability measure supported on the unit circle dµ and para–orthogonal polynomials associated with a modification of dµ by the addition of a pure mass point, also called Uvarov transformation. Moreover, as a direct consequence of our approach, we present some results related with the Christoffel transformation.     

Convergence of Padé approximants of Stieltjes-type meromorphic functions and the relative asymptotics of orthogonal polynomials on the real line

We obtain results on the convergence of Padé approximants of Stieltjes-type meromorphic functions and the relative asymptotics of orthogonal polynomials on unbounded intervals. These theorems extend some results given by Guillermo López in this direction substituting the Carleman condition in his theorems by the determination of the corresponding moment problem.     

Nonclassical weighted estimates of certain Calderónzygmund operators on the plane

Let F be a compact subset of ? let μ be a Borel measure on F, and let ρ(z) be the distance of z to F. Denote $$A_K (f)(z) = \int\limits_F {K(\varsigma ,z)f(\varsigma ) dm(\varsigma ), z \in \mathbb{C}\backslash F}$$ where K (ζ, z) is either (ζ-z)² or (|ζ-z|(ζ-z))^-1 and m is the Lebesgue measure. Let ψ be a monotone nondecreasing positive function on (0, ∞) and let Φ(z)=Ψ(ρ(z))ρ(z), z ε ?/F. Under some additional assumptions on μ, it is proved that A_K is bounded from L² (μ) to L² (Φm) if and only if $$\int\limits_0^{ + 1} {\tfrac{{\psi (t)}}{t} + \int\limits_1^\infty {\tfrac{{\psi (t)}}{{t^2 }}dt< \infty } }$$ Thus, no interference of values of K of various signs is observed in such a situation. Bibliography: 4 titles.     

Sharp estimates for the convergence rate of “hyperbolic” partial sums of double fourier series in orthogonal polynomials

Two-variable functions f(x, y) from the class L ₂ = L ₂((a, b) × (c, d); p(x)q(y)) with the weight p(x)q(y) and the norm $$\left\| f \right\| = \sqrt {\int\limits_a^b {\int\limits_c^d {p(x)q(x)f^2 (x,y)dxdy} } }$$ are approximated by an orthonormal system of orthogonal P _n (x)Q _n (y), n, m = 0, 1, ..., with weights p(x) and q(y). Let $$E_N (f) = \mathop {\inf }\limits_{P_N } \left\| {f - P_N } \right\|$$ denote the best approximation of f ?? L ₂ by algebraic polynomials of the form $$\begin{array}{*{20}c} {P_N (x,y) = \sum\limits_{0 < n,m < N} {a_{m,n} x^n y^m ,} } \\ {P_1 (x,y) = const.} \\ \end{array}$$ . Consider a double Fourier series of f ?? L ₂ in the polynomials P _n (x)Q _m (y), n, m = 0, 1, ..., and its ??hyperbolic?? partial sums $$\begin{array}{*{20}c} {S_1 (f;x,y) = c_{0,0} (f)P_o (x)Q_o (y),} \\ {S_N (f;x,y) = \sum\limits_{0 < n,m < N} {c_{n,m} (f)P_n (x)Q_m (y), N = 2,3, \ldots .} } \\ \end{array}$$ A generalized shift operator Fh and a kth-order generalized modulus of continuity ?? _k (A, h) of a function f ?? L ₂ are used to prove the following sharp estimate for the convergence rate of the approximation: $\begin{gathered} E_N (f) \leqslant (1 - (1 - h)^{2\sqrt N } )^{ - k} \Omega _k (f;h),h \in (0,1), \hfill \\ N = 4,5,...;k = 1,2,... \hfill \\ \end{gathered} $ . Moreover, for every fixed N = 4, 9, 16, ..., the constant on the right-hand side of this inequality is cannot be reduced.     

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.     

An application of Szegő polynomials to the computation of certain weighted integrals on the real line

In this paper, a new approach in the estimation of weighted integrals of periodic functions on unbounded intervals of the real line is presented by considering an associated weight function on the unit circle and making use of both Szegő and interpolatory type quadrature formulas. Upper bounds for the estimation of the error are considered along with some examples and applications related to the Rogers-Szegő polynomials, the evaluation of the Weierstrass operator, the Poisson kernel and certain strong Stieltjes weight functions. Several numerical experiments are finally carried out.     

Approximation by p -Faber polynomials in the weighted Smirnov class Ep( G,ω ) and the Bieberbach polynomials

Let G\subset C be a finite domain with a regular Jordan boundary L . In this work, the approximation properties of a p -Faber polynomial series of functions in the weighted Smirnov class E ^p (G,ω) are studied and the rate of polynomial approximation, for f∈ E ^p ( G,ω) by the weighted integral modulus of continuity, is estimated. Some application of this result to the uniform convergence of the Bieberbach polynomials π _n in a closed domain \overline G with a smooth boundary L is given. February 25, 1999. Date revised: October 20, 1999. Date accepted: May 26, 2000.     

Multivariate orthogonal polynomials, homogeneous Padé approximants and Gaussian cubature

The connection between orthogonal polynomials, Padé approximants and Gaussian quadrature is well known and will be repeated in section 1. In the past, several generalizations to the multivariate case have been suggested for all three concepts [4,6,9,...], however without reestablishing a fundamental and clear link. In sections 2 and 3 we will elaborate definitions for multivariate Padé and Padé-type approximation, multivariate polynomial orthogonality and multivariate Gaussian integration in order to bridge the gap between these concepts. We will show that the new m-point Gaussian cubature rules allow the exact integration of homogeneous polynomials of degree 2m−1, in any number of variables. A numerical application of the new integration rules can be found in sections 4 and 5. This revised version was published online in June 2006 with corrections to the Cover Date.     

Asymptotics of the Number of Threshold Functions and the Singularity Probability of Random {±1}-Matrices

Doklady Mathematics - Two results concerning the number $$P(2,n)$$ of threshold functions and the singularity probability $${{\mathbb{P}}_{n}}$$ of random ( $$n \times n$$ ) $${\text{\{ }} \pm...     

Divergence of diagonal Padé approximants and the asymptotic behavior of orthogonal polynomials associated with nonpositive measures

In this paper we prove the existence of real- and complex-valued measuresμ on the interval [?1,1] with the property that the diagonal Padé approximants [n/n],n=1,2,..., to the functionf(z)=∫dμ(x)/(x?z) neither converge at any fixed pointz∈C～[?1,1] nor converge in capacity on any open (nonempty) setS inC～[?1,1]. This result is derived from a theorem on the asymptotic behavior of orthogonal polynomials. It will be shown that it is possible to construct measuresμ. on [?1,1] such that for any arbitrarily prescribed asymptotic behavior there exist subsequences of the associated orthogonal polynomialsQ _n that have this behavior.     

Legendre–Fenchel transform of the spectral exponent of polynomials of weighted composition operators

For the spectral radius of weighted composition operators with positive weight e ^φ T _α , \({\varphi\in C(X)}\) , acting in the spaces L ^p (X, μ) the following variational principle holds

$\ln r(e^\varphi T_\alpha)=\max_{\nu\in M^1_\alpha} \left\{\int\limits_X\varphi d\nu-\frac{\tau_\alpha(\nu)}{p}\right\},$

where X is a Hausdorff compact space, \({\alpha:X\mapsto X}\) is a continuous mapping and τ _α some convex and lower semicontinuous functional defined on the set \({M^1_\alpha}\) of all Borel probability and α-invariant measures on X. In other words \({\frac{\tau_\alpha}{p}}\) is the Legendre– Fenchel conjugate of ln r(e ^φ T _α ). In this paper we consider the polynomials with positive coefficients of weighted composition operator of the form \({A_{\varphi, {\bf c}}= \sum_{k=0}^n e^{c_k} (e^{\varphi} T_{\alpha})^k}\) , \({{\bf c}=(c_k)\in {\Bbb R}^{n+1}}\) . We derive two formulas on the Legendre–Fenchel transform of the spectral exponent ln r(A _φ,c) considering it firstly depending on the function φ and the variable c and secondly depending only on the function φ, by fixing c.

     

Construction and implementation of asymptotic expansions for Jacobi–type orthogonal polynomials

We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree n goes to \(\infty \). These are defined on the interval [?1, 1] with weight function

$$w(x)=(1-x)^{\alpha}(1+x)^{\beta}h(x), \quad \alpha,\beta>-1 $$

and h(x) a real, analytic and strictly positive function on [?1, 1]. This information is available in the work of Kuijlaars et al. (Adv. Math. 188, 337–398 2004), where the authors use the Riemann–Hilbert formulation and the Deift–Zhou non-linear steepest descent method. We show that computing higher-order terms can be simplified, leading to their efficient construction. The resulting asymptotic expansions in every region of the complex plane are implemented both symbolically and numerically, and the code is made publicly available. The main advantage of these expansions is that they lead to increasing accuracy for increasing degree of the polynomials, at a computational cost that is actually independent of the degree. In contrast, the typical use of the recurrence relation for orthogonal polynomials in computations leads to a cost that is at least linear in the degree. Furthermore, the expansions may be used to compute Gaussian quadrature rules in \(\mathcal {O}(n)\) operations, rather than \(\mathcal {O}(n^{2})\) based on the recurrence relation.

     

Mean ergodicity and spectrum of the Cesàro operator on weighted $$c_0$$ spaces

A detailed investigation is made of the continuity, the compactness and the spectrum of the Cesàro operator \(\mathsf {C}\) acting on the weighted Banach sequence space \(c_0(w)\) for a bounded, strictly positive weight w. New features arise in the weighted setting (e.g. existence of eigenvalues, compactness, mean ergodicity) which are not present in the classical setting of \(c_0\).     

Minimum degree of the difference of two polynomials over {\mathbb Q}, and weighted plane trees

A weighted bicolored plane tree (or just tree for short) is a bicolored plane tree whose edges are endowed with positive integral weights. The degree of a vertex is defined as the sum of the weights of the edges incident to this vertex. Using the theory of dessins d’enfants, which studies the action of the absolute Galois group on graphs embedded into Riemann surfaces, we show that a weighted plane tree is a graphical representation of a pair of coprime polynomials \(P,Q\in {\mathbb C}\,[x]\) such that: (a)  \(\deg P = \deg Q\) , and \(P\) and \(Q\) have the same leading coefficient; (b) the multiplicities of the roots of  \(P\) (respectively, of  \(Q\) ) are equal to the degrees of the black (respectively, white) vertices of the corresponding tree; (c) the degree of the difference \(P-Q\) attains the minimum which is possible for the given multiplicities of the roots of \(P\)  and  \(Q\) . Moreover, if a tree in question is uniquely determined by the set of its black and white vertex degrees (we call such trees unitrees), then the corresponding polynomials are defined over \({\mathbb Q}\) . The pairs of polynomials \(P,Q\) such that the degree of the difference \(P-Q\) attains the minimum, and especially those defined over \({\mathbb Q}\) , are related to some important questions of number theory. Dozens of papers, from 1965 (Birch et al. in Norske Vid Selsk Forh 38:65–69, 1965) to 2010 (Beukers and Stewart in J Number Theory 130:660–679, 2010), were dedicated to their study. The main result of this paper is a complete classification of the unitrees, which provides us with the most massive class of such pairs defined over  \({\mathbb Q}\) . We also study combinatorial invariants of the Galois action on trees, as well as on the corresponding polynomial pairs, which permit us to find yet more examples defined over  \({\mathbb Q}\) . In a subsequent paper, we compute the polynomials \(P,Q\) corresponding to all the unitrees.