Polynomials with a root close to an integer

Let us consider the set of polynomials with integer coefficients of a given degree and of bounded height. We prove that among all polynomials in this set with no integer roots the polynomialx ⁿ-H(x ⁿ⁻¹+x ⁿ⁻²+...+1) has a root closest to an integer. Partially supported by the Lithuanian State Science and Studies Foundation. Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 3, pp. 310–316, July–September, 1999.     

Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences

In this paper some decompositions of Cauchy polynomials, Ferrers-Jackson polynomials and polynomials of the form x ²ⁿ + y ²ⁿ , n ∈ ℕ, are studied. These decompositions are used to generate the identities for powers of Fibonacci and Lucas numbers as well as for powers of the so called conjugate recurrence sequences. Also, some new identities for Chebyshev polynomials of the first kind are presented here.     

Recursive generation of the Galerkin–Chebyshev matrix for convolution kernels

The Galerkin–Chebyshev matrix is the coefficient matrix for the Galerkin method (or the degenerate kernel approximation method) using Chebyshev polynomials. Each entry of the matrix is defined by a double integral. For convolution kernels K(x-y) on finite intervals, this paper obtains a general recursion relation connecting the matrix entries. This relation provides a fast generation of the Galerkin–Chebyshev matrix by reducing the construction of a matrix of order N from N ²+O(N) double integral evaluations to 3N+O(1) evaluations. For the special cases (a) K(x-y)=|x-y|^α-1(-ln|x-y|) ^p and (b) K(x-y)=K _ν(σ|x-y|) (modified Bessel functions), the number of double integral evaluations to generate a Galerkin–Chebyshev matrix of arbitrary order can be further reduced to 2p+2 double integral evaluations in case (a) and to 8 double integral evaluations in case (b). This revised version was published online in June 2006 with corrections to the Cover Date.     

Equidistribution of points via energy

We study the asymptotic equidistribution of points with discrete energy close to Robin’s constant of a compact set in the plane. Our main tools are the energy estimates from potential theory. We also consider the quantitative aspects of this equidistribution. Applications include estimates of growth for the Fekete and Leja polynomials associated with large classes of compact sets, convergence rates of the discrete energy approximations to Robin’s constant, and problems on the means of zeros of polynomials with integer coefficients.     

On the number of polynomials of small house

Suppose that ɛ is a positive number, and letd be a sufficiently large positive integer. We consider a set of monic polynomials of degreed with integer coefficients and roots lying in the disk of radiusd ^ɛ/d. We prove that the number of such polynomials is less than exp(d ^2/3+ɛ). Partially supported by the Lithuanian State Science and Studies Foundation. Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp. 214–219, April–June, 1999.     

Shohat-Favard and Chebyshev’s methods in d-orthogonality

This paper is concerned with the Shohat-Favard, Chebyshev and Modified Chebyshev methods for d-orthogonal polynomial sequences d∈ℕ. Shohat-Favard’s method is presented from the concept of dual sequence of a sequence of polynomials. We deduce the recurrence relations for the Chebyshev and the Modified Chebyshev methods for every d∈ℕ. The three methods are implemented in the Mathematica programming language. We show several formal and numerical tests realized with the software developed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Polynomials with integer coefficients and their zeros

We study several related problems on polynomials with integer coefficients. This includes the integer Chebyshev problem, and the Schur problems on means of algebraic numbers. We also discuss interesting applications to the approximation by polynomials with integer coefficients, and to the growth of coefficients for polynomials with roots located in prescribed sets. The distribution of zeros for polynomials with integer coefficients plays an important role in all of these problems.     

Polynomial Interpolation on the Unit Sphere and on the Unit Ball

The problem of interpolation on the unit sphere S ^d by spherical polynomials of degree at most n is shown to be related to the interpolation on the unit ball B ^d by polynomials of degree n. As a consequence several explicit sets of points on S ^d are given for which the interpolation by spherical polynomials has a unique solution. We also discuss interpolation on the unit disc of R ² for which points are located on the circles and each circle has an even number of points. The problem is shown to be related to interpolation on the triangle in a natural way.     

Approximation of multidimensional functions with a given majorant of mixed moduli of continuity

In the paper order-exact upper bounds for the best approximations of classesH _q ^Emphasis>/ω by trigonometric polynomials are obtained. The spectrum of the approximating polynomials lies in sets generated by the level surfaces of the function ω(t). These sets are a generalization of hyperbolic crosses to the case of an arbitrary function ω(t). Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 107–117, January, 1999.     

Continuous and Discrete Tight Frames of Orthogonal Polynomials for a Radially Symmetric Weight

This paper considers tight frame decompositions of the Hilbert space ℘ _n of orthogonal polynomials of degree n for a radially symmetric weight on ℝ ^d , e.g., the multivariate Gegenbauer and Hermite polynomials. We explicitly construct a single zonal polynomial p∈℘ _n with the property that each f∈℘ _n can be reconstructed as a sum of its projections onto the orbit of p under SO(d) (symmetries of the weight), and hence of its projections onto the zonal polynomials p _ξ obtained from p by moving its pole to ξ∈S:={ξ∈ℝ ^d :|ξ|=1}. Furthermore, discrete versions of these integral decompositions also hold where SO(d) is replaced by a suitable finite subgroup, and S by a suitable finite subset. One consequence of our decomposition is a simple closed form for the reproducing kernel for ℘ _n .      

Pólya-Type Polynomial Inequalities in L P Spaces and Best Local Approximation

In this paper, we obtain an extension of the Pólya inequality for univariate real polynomials in L ^p spaces and new estimates for certain class of measurable sets. Inequalities for complex polynomials are also considered. We give an application to a multipoint best local approximation problem for real and complex polynomials.     

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.     

On a Conjecture of E. A. Rakhmanov

It is shown that a conjecture of E. A. Rakhmanov is true concerning the zero distribution of orthogonal polynomials with respect to a measure having a discrete real support. We also discuss the case of extremal polynomials with respect to some discrete L _p -norm, 0 < p ≤∈fty , and give an extension to complex supports. Furthermore, we present properties of weighted Fekete points with respect to discrete complex sets, such as the weighted discrete transfinite diameter and a weighted discrete Bernstein—Walsh-like inequality. August 24, 1998. Date revised: March 26, 1999. Date accepted: April 27, 1999.     

Bivariate fibonacci like p-polynomials

In this article, we study the bivariate Fibonacci and Lucas p-polynomials (p ? 0 is integer) from which, specifying x, y and p, bivariate Fibonacci and Lucas polynomials, bivariate Pell and Pell-Lucas polynomials, Jacobsthal and Jacobsthal-Lucas polynomials, Fibonacci and Lucas p-polynomials, Fibonacci and Lucas p-numbers, Pell and Pell-Lucas p-numbers and Chebyshev polynomials of the first and second kind, are obtained. Afterwards, we obtain some properties of the bivariate Fibonacci and Lucas p-polynomials.     

Invariants and Chebyshev polynomials

On different compact sets from ℝ ⁿ , new multidimensional analogs of algebraic polynomials least deviating from zero (Chebyshev polynomials) are constructed. A brief review of the analogs constructed earlier is given. Estimates of values of the best approximation obtained by using extremal signatures, lattices, and finite groups are presented.     

On Polynomials Sharing Preimages of Compact Sets, and Related Questions

In this paper we give a solution of the following problem: under what conditions on infinite compact sets and polynomials f ₁, f ₂ do the preimages f ₁⁻¹{K ₁} and f ₂⁻¹{K ₂} coincide. Besides, we investigate some related questions. In particular, we show that polynomials sharing an invariant compact set distinct from a point have equal Julia sets. Received: May 2006, Accepted: June 2006     

Bivariate Gonarov polynomials and integer sequences

Univariate Gonarov polynomials arose from the Gonarov interpolation problem in numerical analysis.They provide a natural basis of polynomials for working with u-parking functions,which are integer sequences whose order statistics are bounded by a given sequence u.In this paper,we study multivariate Gonarov polynomials,which form a basis of solutions for multivariate Gonarov interpolation problem.We present algebraic and analytic properties of multivariate Gonarov polynomials and establish a combinatorial relation with integer sequences.Explicitly,we prove that multivariate Gonarov polynomials enumerate k-tuples of integers sequences whose order statistics are bounded by certain weights along lattice paths in Nk.It leads to a higher-dimensional generalization of parking functions,for which many enumerative results can be derived from the theory of multivariate Gonarov polynomials.     

Totally real algebraic integers in small intervals

We show that there are irreducible monic polynomials having all roots in an interval of length close to 4. These are obtained by perturbing the coefficients of the respective Chebyshev polynomials. In particular, we obtain that our earlier lower bound for the house of totally real algebraic integers is sharp up to a logarithmic factor. Partially supported by the Lithuanian State Science and Studies Foundation. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 307–312, July–September, 2000.     

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.     

Asymptotics and weighted estimates of meixner polynomials orthogonal on the gird {0, δ, 2δ, ...}

Suppose that 0<δ≤1,N=1/δ, and α, ga≥0, is an integer. For the classical Meixner polynomials orthonormal on the gird {0, δ, 2δ, ...} with weight ρ(x)=(1-e ^−δ)^αг(Nx+α+ 1)/г(Nx+1), the following asymptotic formula is obtained: . The remainderv _n,N ^α (z) forn≤λN satisfies the estimate