Extremal polynomials related to Zolotarev polynomials

Algebraic polynomials bounded in absolute value by M > 0 in the interval [–1, 1] and taking a fixed value A at a > 1 are considered. The extremal problem of finding such a polynomial taking a maximum possible value at a given point b < ?1 is solved. The existence and uniqueness of an extremal polynomial and its independence of the point b < ?1 are proved. A characteristic property of the extremal polynomial is determined, which is the presence of an n-point alternance formed by means of active constraints. The dependence of the alternance pattern, the objective function, and the leading coefficient on the parameter A is investigated. A correspondence between the extremal polynomials in the problem under consideration and the Zolotarev polynomials is established.     

A remark on the representation of Zolotarev polynomials

Zolotarev polynomials are the polynomials that have minimaldeviation from zero on [–1, 1] with respect to the norm||xⁿ – x^n–1 + a_n–2 x^n–2 + ... + a₁x+ a_n|| for given and for all a_k . This note complements the paper of F. Pehersforfer [J. LondonMath. Soc. (1) 74 (2006) 143–153] with exact (not asymptotic)construction of the Zolotarev polynomials with respect to thenorm L₁ for || < 1 and with respect to the norm L₂ for || 1 in the form of Bernstein–Szegö orthogonal polynomials.For all in L₁ and L₂ norms, the Zolotarev polynomials satisfyexactly (not asymptotically) the triple recurrence relationof the Chebyshev polynomials.     

A refined Gallai-Edmonds structure theorem for weighted matching polynomials

In this work, we prove a refinement of the Gallai-Edmonds structure theorem for weighted matching polynomials by Ku and Wong. Our proof uses a connection between matching polynomials and branched continued fractions. We also show how this is related to a modification by Sylvester of the classical Sturm's theorem on the number of zeros of a real polynomial in an interval. In addition, we obtain some other results about zeros of matching polynomials.     

On sequences of Hurwitz polynomials related to orthogonal polynomials

ABSTRACT

In this contribution, we explore the well-known connection between Hurwitz and orthogonal polynomials. Namely, given a Hurwitz polynomial, it is shown that it can be decomposed into two parts: a polynomial that is orthogonal with respect to some positive measure supported in the positive real axis and its corresponding second-kind polynomial. Conversely, given a sequence of orthogonal polynomials with respect to a positive measure supported in the positive real axis, a sequence of Hurwitz polynomials can be constructed. Based on that connection, we construct sequences of Hurwitz polynomials that satisfy a recurrence relation, in a similar way as the orthogonal polynomials do. Even more, we present a way to construct families of Hurwitz polynomials using two sequences of parameters and a recurrence relation that constitutes an analogue of Favard's theorem in the theory of orthogonal polynomials.     

A fully discrete numerical scheme for weighted mean curvature flow

Summary. We analyze a fully discrete numerical scheme approximating the evolution of n–dimensional graphs under anisotropic mean curvature. The highly nonlinear problem is discretized by piecewise linear finite elements in space and semi–implicitly in time. The scheme is unconditionally stable und we obtain optimal error estimates in natural norms. We also present numerical examples which confirm our theoretical results. Received October 2, 2000 / Published online July 25, 2001     

Orthogonal polynomials and laurent polynomials related to the Hahn-Extonq-Bessel function

Laurent polynomials related to the Hahn-Extonq-Bessel function, which areq-analogues of the Lommel polynomials, have been introduced by Koelink and Swarttouw. The explicit strong moment functional with respect to which the Laurentq-Lommel polynomials are orthogonal is given. The strong moment functional gives rise to two positive definite moment functionals. For the corresponding sets of orthogonal polynomials, the orthogonality measure is determined using the three-term recurrence relation as a starting point. The relation between Chebyshev polynomials of the second kind and the Laurentq-Lommel polynomials and related functions is used to obtain estimates for the latter.     

Nonlinear recurrences related to Chebyshev polynomials

We use two nonlinear recurrence relations to define the same sequence of polynomials, a sequence resembling the Chebyshev polynomials of the first kind. Among other properties, we obtain results on their irreducibility and zero distribution. We then study the \(2\times 2\) Hankel determinants of these polynomials, which have interesting zero distributions. Furthermore, if these polynomials are split into two halves, then the zeros of one half lie in the interval \((-1,1)\), while those of the other half lie on the unit circle. Some further extensions and generalizations of these results are indicated.     

Matrices related to the Bell polynomials

In this paper, we study the matrices related to the partial exponential Bell polynomials and those related to the Bell polynomials with respect to Ω. As a result, the factorizations of these matrices are obtained, which give unified approaches to the factorizations of many lower triangular matrices. Moreover, some combinatorial identities are also derived from the corresponding matrix representations.     

A numerical method for the computation of Faber polynomials for starlike domains

We describe a simple numerical process for computing approximationsto Faber polynomials for starlike domains. This process is basedon using the Theodorsen integral equation method for computingthe Laurent series coefficients of the associated exterior conformalmapping, and then determining the corresponding Faber polynomialsby means of the well-known recurrence relation which is availablefor this purpose.     

A weighted quasilinear equation related to the mean curvature operator

This paper studies a weighted quasilinear perturbation, through the mean curvature flow operator, of the classical linear heat equation. The mean curvature has the effect of maintaining bounded all classical positive steady states of the model, though their derivatives must be somewhere unbounded. The dynamics of the positive solutions of the model is fully described.     

An extremum problem for polynomials related to codes and designs

We give a solution to Yudin’s extremum problem for algebraic polynomials related to codes and designs. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 508–513, April, 2000.     

A connection between numerical ranges of selfadjoint matrix polynomials

In this paper, a geometric connection between the joint numerical range and the numerical range for selfadjoint matrix polynomial is presented. Moreover, algebraic properties for these ranges are investigated.     

Point vortex equilibria related to Bessel polynomials

The method of polynomials is used to construct two families of stationary point vortex configurations. The vortices are placed at the reciprocals of the zeroes of Bessel polynomials. Configurations that translate uniformly, and configurations that are completely stationary, are obtained in this way.     

Finite sections for singular integral operators by weighted Chebyshev polynomials

A finite section method for the approximate solution of singular integral equations with piecewise continuous coefficients on intervals is considered. The problem is transformed in such a way that results which were previously obtained for singular integral equations on the unit circle using localization methods in Banach algebras are applicable to it. Thus, necessary and sufficient conditions for the stability of the approximation method can be proved.     

A numerical solution of the constrained weighted energy problem

A numerical algorithm is presented to solve the constrained weighted energy problem from potential theory. As one of the possible applications of this algorithm, we study the convergence properties of the rational Lanczos iteration method for the symmetric eigenvalue problem. The constrained weighted energy problem characterizes the region containing those eigenvalues that are well approximated by the Ritz values. The region depends on the distribution of the eigenvalues, on the distribution of the poles, and on the ratio between the size of the matrix and the number of iterations. Our algorithm gives the possibility of finding the boundary of this region in an effective way.We give numerical examples for different distributions of poles and eigenvalues and compare the results of our algorithm with the convergence behavior of the explicitly performed rational Lanczos algorithm.     

Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials II

We present a definition of general Sobolev spaces with respect to arbitrary measures, Wh,p (Ω,μ) for 1 ≤p≤∞. In [RARP] we proved that these spaces are complete under very light conditions. Now we prove that if we consider certain general types of measures, then Cc∞ (R) is dense in thee spaces. As an application to Sobolev orthogonal polynomials, we study the boundedness of the multiplication operator. This gives an estimation of the zeroes of Sobolev orthogonal polynomials.