Extremal measures for a system of orthogonal polynomials

We characterize the extremal measures of an indeterminate moment problem associated with a system of orthogonal polynomials defined by a three-term recurrence relation.     

Applications of the monotonicity of extremal zeros of orthogonal polynomials in interlacing and optimization problems

We investigate monotonicity properties of extremal zeros of orthogonal polynomials depending on a parameter. Using a functional analysis method we prove the monotonicity of extreme zeros of associated Jacobi, associated Gegenbauer and q-Meixner-Pollaczek polynomials. We show how these results can be applied to prove interlacing of zeros of orthogonal polynomials with shifted parameters and to determine optimally localized polynomials on the unit ball.     

Eigenvalue problems to compute almost optimal points for rational interpolation with prescribed poles

Explicit formulas exist for the (n,m) rational function with monic numerator and prescribed poles that has the smallest possible Chebyshev norm. In this paper we derive two different eigenvalue problems to obtain the zeros of this extremal function. The first one is an ordinary tridiagonal eigenvalue problem based on a representation in terms of Chebyshev polynomials. The second is a generalised tridiagonal eigenvalue problem which we derive using a connection with orthogonal rational functions. In the polynomial case (m = 0) both problems reduce to the tridiagonal eigenvalue problem associated with the Chebyshev polynomials of the first kind. Postdoctoral researcher FWO-Flanders.     

Some extremal properties of orthogonal polynomials

Using a particular way of normalizing the orthogonal polynomials, which is most commonly encountered in the synthesis of filtering networks in communication and electronic engineering, two theorems concerning the extremal properties of orthogonal polynomials are first proved. The results are then applied to find the minimum value and the minimizing function for various definite integrals involving weight functions of classical orthogonal polynomials.     

Zero location and asymptotic behavior for extremal polynomials with non-diagonal Sobolev norms

In this paper we are going to study the zero location and asymptotic behavior of extremal polynomials with respect to a non-diagonal Sobolev norm in the worst case, i.e., when the quadratic form is allowed to degenerate. The orthogonal polynomials with respect to this Sobolev norm are a particular case of those extremal polynomials. The multiplication operator by the independent variable is the main tool in order to obtain our results.     

A Nonlinear Generalization of Singular Value Decomposition and Its Applications to Mathematical Modeling and Chaotic Cryptanalysis

In this paper we are going to study the zero location and asymptotic behavior of extremal polynomials with respect to a generalized non-diagonal Sobolev norm in which the product of the function and its derivative appears. The orthogonal polynomials with respect to this Sobolev norm are a particular case of those extremal polynomials. The multiplication operator by the independent variable is the main tool in order to obtain our results.     

Orthogonal polynomials on the unit circle associated with the Laguerre polynomials

Using the well-known fact that the Fourier transform is unitary, we obtain a class of orthogonal polynomials on the unit circle from the Fourier transform of the Laguerre polynomials (with suitable weights attached). Some related extremal problems which arise naturally in this setting are investigated.

     

Solving reactor problems to determine the multiplication: A new method of accelerating outer iterations

A new cyclic iterative method with variable parameters is proposed for accelerating the outer iterations in a process used to calculate K _eff in multigroup problems. The method is based on the use of special extremal polynomials that are distinct from Chebyshev polynomials and take into account the specific nature of the problem. To accelerate the convergence with respect to K _eff, the use of three orthogonal functionals is proposed. Their values simultaneously determine the three maximal eigenvalues. The proposed method was incorporated in the software for neutron-physics calculations for WWER reactors.     

On a Characterization of Integrability for the Reciprocal Weight of Orthogonal Polynomials on the Circle

We prove a necessary and sufficient condition for integrability of the reciprocal weight function of orthogonal polynomials. The condition is given in terms of the asymptotic behaviour of the norm of extremal polynomials with prescribed coefficients.     

On the exact Markov inequality for k-monotone polynomials in uniform and L 1-norms

We consider the classical extremal problem of estimating norms of higher order derivatives of algebraic polynomials when their norms are given. The corresponding extremal problem for general polynomials in uniform norm was solved by A. A. Markov, while Bernstein found the exact constant in the Markov inequality for monotone polynomials. In this note we give Markov-type inequalities for higher order derivatives in the general class of k-monotone polynomials. In particular, in case of first derivative we find the exact solution of this extremal problem in both uniform and L ₁-norms. This exact solution is given in terms of the largest zeros of certain Jacobi polynomials.     

On upper bounds for code distance and covering radius of designs in polynomial metric spaces

The purpose of this paper is to present new upper bounds for code distance and covering radius of designs in arbitrary polynomial metric spaces. These bounds and the necessary and sufficient conditions of their attainability were obtained as the solution of an extremal problem for systems of orthogonal polynomials. For antipodal spaces the behaviour of the bounds in different asymptotical processes is determined and it is proved that this bound is attained for all tight 2k-design.     

Fourier–Padé Approximants for Nikishin Systems

We study type I Fourier–Padé approximation for certain systems of functions formed by the Cauchy transform of finite Borel measures supported on bounded intervals of the real line. This construction is similar to type I Hermite–Padé approximation. Instead of power series expansions of the functions in the system, we take their development in a series of orthogonal polynomials. We give the exact rate of convergence of the corresponding approximants. The answer is expressed in terms of the extremal solution of an associated vector-valued equilibrium problem for the logarithmic potential.      

The extremal function in the Delsarte problem of finding an upper bound for the kissing number in the three-dimensional space

We consider an extremal problem for continuous functions that are nonpositive on a closed interval and can be represented by series in Legendre polynomials with nonnegative coefficients. This problem arises from the Delsarte method of finding an upper bound for the kissing number in the three-dimensional Euclidean space. We prove that the problem has a unique solution, which is a polynomial of degree 27. This polynomial is a linear combination of Legendre polynomials of degrees 0, 1, 2, 3, 4, 5, 8, 9, 10, 20, and 27 with positive coefficients; it has simple root 1/2 and five double roots in (?1, 1/2). We also consider the dual extremal problem for nonnegative measures on [?1, 1/2] and, in particular, prove that an extremal measure is unique.     

Asymptotics of orthogonal polynomials and the numerical solution of ill-posed problems

The paper reviews the impact of modern orthogonal polynomial theory on the analysis of numerical algorithms for ill-posed problems. Of major importance are uniform bounds for orthogonal polynomials on the support of the weight function, the growth of the extremal zeros, and asymptotics of the Christoffel functions.     

The distribution of zeros of asymptotically extremal polynomials

In this paper, we study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms have the same nth root behavior as the weighted norms for certain extremal polynomials. Our results include as special cases several of the previous results of Erd s, Freud, Jentzsch, Szeg and Blatt, Saff, and Simkani. Applications are given concerning the zeros of orthogonal polynomials over a smooth Jordan curve (in particular, on the unit circle) and the zeros of polynomials of best approximation on R to nonentire functions.     

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.     

Extremal Polynomials on Discrete Sets

We study asymptotics for orthogonal polynomials and other extremalpolynomials on infinite discrete sets, typical examples beingthe Meixner polynomials and the Charlier polynomials. Followingideas of Rakhmanov, Dragnev and Saff, weshow that the asymptoticbehaviour is governed by a constrained extremal energy problemfor logarithmic potentials, which can be solved explicitly.We give formulas for the contracted zero distributions, thenth root asymptotics and the asymptotics of the largest zeros.1991 Mathematics Subject Classification: 42C05, 33C25, 31A15.     

Mixed Type Multiple Orthogonal Polynomials for Two Nikishin Systems

We study the logarithmic and ratio asymptotics of linear forms constructed from a Nikishin system which satisfy orthogonality conditions with respect to a system of measures generated by a second Nikishin system. This construction combines type I and type II multiple orthogonal polynomials. The logarithmic asymptotics of the linear forms is expressed in terms of the extremal solution of an associated vector valued equilibrium problem for the logarithmic potential. The ratio asymptotics is described by means of a conformal representation of an appropriate Riemann surface of genus zero onto the extended complex plane.     

Asymptotically extremal polynomials with respect to varying weights and application to Sobolev orthogonality

We study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms, with respect to a sequence of weight functions, have the same nth root asymptotic behavior as the weighted norms of certain extremal polynomials. This result is applied to obtain the (contracted) weak zero distribution for orthogonal polynomials with respect to a Sobolev inner product with exponential weights of the form e−φ(x), giving a unified treatment for the so-called Freud (i.e., when φ has polynomial growth at infinity) and Erdös (when φ grows faster than any polynomial at infinity) cases. In addition, we provide a new proof for the bound of the distance of the zeros to the convex hull of the support for these Sobolev orthogonal polynomials.     

Extremal quasimodular forms for low-level congruence subgroups

In this paper, we study “extremal” quasimodular forms of depth 1 for the Hecke subgroups of level 2, 3, and 4, and relations to modular differential equations and Atkin?s orthogonal polynomials.