On polynomials orthogonal to all powers of a Chebyshev polynomial on a segment

In this paper we describe polynomials orthogonal to all powers of a Chebyshev polynomial on a segment.     

Some inequalities for symmetric functions and an application to orthogonal polynomials

We present some sharp inequalities for symmetric functions and give an application to orthogonal polynomials.     

On simple families of cyclic polynomials

We study polynomials giving cyclic extensions over rational function fields with one variable satisfying some conditions. By using them, we construct families of cyclic polynomials over some algebraic number fields. And these families give non-Kummer (or non-Artin-Schreier) cyclic extensions. In this paper, we see that our polynomials have two nice arithmetic properties. One is simplicity: our polynomials and their discriminants have more simple expressions than previous results, e.g. Dentzer (1995), Malle and Mazat (1999) and Smith (1991), etc. The other is a ``systematic' property: if one of our polynomials gives an extension , then for every intermediate field we can easily find polynomials giving from systematically.

     

On the cost of approximating all roots of a complex polynomial

This work examines the computational complexity of a homotopy algorithm in approximating all roots of a complex polynomialf. It is shown that, probabilistically, monotonic convergence to each of the roots occurs after a determined number of steps. Moreover, in all subsequent steps, each rootz is approximated by a complex numberx, where ifx ₀ =x, x _j =x _j–1 –f(x _j–1)/f(x _j–1),j = 1, 2,, then |x _j –z| < (1/|x ₀ –z|)|x _j–1–z|².     

On q-orthogonal polynomials over a collection of complex origin intervals related to little q-Jacobi polynomials

We construct the sequence of orthogonal polynomials with respect to an inner product which is defined by q-integrals over a collection of intervals in the complex plane. We prove that they are connected with little q-Jacobi polynomials. For such polynomials we discuss a few representations, a recurrence relation, a difference equation, a Rodrigues-type formula and a generating function. 2000 Mathematics Subject Classification Primary—33D45, 42C05     

Functional calculus under the Tadmor-Ritt condition, and free interpolation by polynomials of a given degree

For Banach space operators T satisfying the Tadmor-Ritt condition ||(zI−T)⁻¹||?C|z−1|⁻¹, |z|>1, we prove that the best-possible constant C_T(n) bounding the polynomial calculus for T, ||p(T)||?C_T(n)||p||_∞, deg(p)?n, behaves (in the worst case) as as n→∞. This result is based on a new free (Carleson type) interpolation theorem for polynomials of a given degree.     

Construction of polynomials that are orthogonal with respect to a discrete bilinear form

We describe a new algorithm for the computation of recursion coefficients of monic polynomials {p _j } _j ^=0/n that are orthogonal with respect to a discrete bilinear form (f, g) := _k ^=1/m f(x _k )g(x _k )w _k ,m n, with real distinct nodesx _k and real nonvanishing weightsw _k . The algorithm proceeds by applying a judiciously chosen sequence of real or complex Givens rotations to the diagonal matrix diag[x ₁,x ₂, ...,x _m ] in order to determine an orthogonally similar complex symmetric tridiagonal matrixT, from whose entries the recursion coefficients of the monic orthogonal polynomials can easily be computed. Fourier coefficients of given functions can conveniently be computed simultaneously with the recursion coefficients. Our scheme generalizes methods by Elhay et al. [6] based on Givens rotations for updating and downdating polynomials that are orthogonal with respect to a discrete inner product. Our scheme also extends an algorithm for the solution of an inverse eigenvalue problem for real symmetric tridiagonal matrices proposed by Rutishauser [20], Gragg and Harrod [17], and a method for generating orthogonal polynomials based theoron [18]. Computed examples that compare our algorithm with the Stieltjes procedure show the former to generally yield higher accuracy except whenn m. Ifn is sufficiently much smaller thanm, then both the Stieltjes procedure and our algorithm yield accurate results.Research supported in part by the Center for Research on Parallel Computation at Rice University and NSF Grant No. DMS-9002884.     

Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights

The paper has three parts. In the first part we apply the theory of commuting pairs of (pseudo) difference operators to the (formal) asymptotics of orthogonal polynomials: using purely geometrical arguments we show heuristically that the asymptotics, for large degrees, of orthogonal polynomial with respect to varying weights is intimately related to certain spinor bundles on a hyperelliptic algebraic curve reproducing formulae appearing in the works of Deift et al. on the subject.In the second part we show that given an arbitrary nodal hyperelliptic curve satisfying certain conditions of admissibility we can reconstruct a sequence of polynomials orthogonal with respect to semiclassical complex varying weights supported on several curves in the complex plane. The strong asymptotics of these polynomials will be shown to be given by the spinors introduced in the first part using a Riemann-Hilbert analysis.In the third part we use Strebel theory of quadratic differentials and the procedure of welding to reconstruct arbitrary admissible hyperelliptic curves. As a result we can obtain orthogonal polynomials whose zeroes may become dense on a collection of Jordan arcs forming an arbitrary forest of trivalent loop-free trees.     

Derivatives of a finite class of orthogonal polynomials related to inverse Gamma distribution

In this work, we consider derivatives of a finite class of orthogonal polynomials with respect to weight function which is related to the probability density function of the inverse gamma distribution over the positive real line. General properties for this derivative class such as orthogonality, Rodrigues’ formula, recurrence relation, generating function and various other related properties such as self-adjoint form and normal form are indicated. The corresponding Gaussian quadrature formulae are introduced with examples. These examples are provided to support the advantages of considering the derivatives class of the finite class of orthogonal polynomials related to inverse gamma distribution. The orthogonality property related to the Fourier transform of the derivative class under discussion is also given.     

On a New Approach to Frequency Sounding of Layered Media

Frequency sounding of layered media is modeled by a hyperbolic problem. Within the framework of this model, we formulate an inverse problem. Applying the Laplace transform and introducing the impedance function, the latter is first reduced to the inverse boundary value problem for the Riccati equation and then to the Cauchy problem for a first-order quadratic equation. The advantage of such transformations is that the quadratic equation does not contain an unknown coefficient. For a specific class of data, it is shown that the Cauchy problem is uniquely solvable. Based on the asymptotic behavior of solutions to both the Riccati and quadratic equations, a stable reconstruction algorithm is constructed. Its feasibility is demonstrated in computational experiments.     

Existence and scattering of small solutions to a Boussinesq type equation of sixth order

In this paper, we consider the existence and uniqueness of the global small solution as well as the small data scattering result to the Cauchy problem for a Boussinesq type equation of sixth order with the nonlinear term f(u) behaving as as u→0 in . The main method and techniques used in our paper are the Littlewood-Paley dyadic decomposition, the stationary phase estimate and some properties of Bessel function.     

On zeros of polynomials orthogonal with respect to a quasi-definite inner product on the unit circle

In this paper we present some results concerning the zeros of sequences of polynomials orthogonal with respect to a quasi-definite inner product on the unit circle. We study zero general properties, the existence of sequences with prefixed zeros and some situations concerning the polynomials with multiple zeros.     

Analyticity of a Nonlinear Operator Associated to the Conformal Representation in Schauder Spaces. An Integral Equation Approach

We consider the Riemann map g_ζ,w of the complex unit disk to the plane domain 𝕀[ζ] enclosed by the Jordan curve ζ and normalized by the conditions g_ζ,w(0) = w, g′_ζ,w(0) > 0, where w is a point of 𝕀[ζ], and we present a nonlinear singular integral equation approach to prove that the nonlinear operator which takes the pair (ζ, w) to the map g^(–1)_ζ,w ○ ζ is real analytic in Schauder spaces.     

How to recover anL-series from its values at almost all positive integers. Some remarks on a formula of Ramanujan

We define a class of analytic functions which can be obtained from their values at almost all positive integers by a canonical interpolation procedure. All the usualL-functions belong to this class which is interesting in view of the extensive investigations of special values of motivicL-series. A number of classical contour integral formulas appear as particular cases of the interpolation scheme. The paper is based on a formula of Ramanujan and results of Hardy. An approach to the problem via distributions is also presented.     

一类Boussinesq型方程整体解的存在性和唯一性

证明一类6阶Boussinesq型方程Cauchy问题整体广义解和整体古典解的存在性和唯一性,给出解在有限时刻发生爆破的充分条件.     

一类奇异积分和Cauchy型积分关于积分曲线的稳定性

本文讨论了当任意给定的f（τ,t）在某个区域E内属于H类时,奇异积分在封闭或开口光滑曲线E发生光滑扰动时的稳定性,并给出了相应的误差估计．作为应用,我们还讨论了当（t）在E内属于H类时,Cauchy型积分,在封闭光滑曲线E发生光滑扰动时的稳定性及误差估计．