Space Of Real Polynomials Research Articles

Polynomials associated with a functional product of the Hermite type

We have found the motivation for this paper in the research of a quantized closed Friedmann cosmological model. There, the second‐order linear ordinary differential equation emerges as a wave equation for the physical state functions. Studying the polynomial solutions of this equation, we define a new functional product in the space of real polynomials. This product includes the indexed weight functions which depend on the degrees of participating polynomials. Although it does not have all of the properties of an inner product, a unique sequence of polynomials can be associated with it by an additional condition. In the special case presented here, we consider the Hermite‐type weight functions and prove that the associated polynomial sequence can be expressed in the closed form via the Hermite polynomials. Also, we find their Rodrigues‐type formula and a four‐term recurrence relation. In contrast to the zeros of Hermite polynomials, which are symmetrically located with respect to the origin, the zeros of the new polynomial sequence are all positive. Copyright © 2015 John Wiley & Sons, Ltd.

Orthogonal polynomials with varying weight of Laguerre type

In this paper, we define and examine a new functional product in the space of real polynomials. This product includes the weight function which depends on degrees of the participants. In spite of it does not have all properties of an inner product, we construct the sequence of orthogonal polynomials. These polynomials can be eigenfunctions of a differential equation what was used in some considerations in the theoretical physics. In special, we consider Laguerre type weight function and prove that the corresponding orthogonal polynomial sequence is connected with Laguerre polynomials. We study their differential properties and orthogonal properties of some related rational and exponential functions.

Linear optimization with cones of moments and nonnegative polynomials

Let A be a finite subset of N^n and R[x]_A be the space of real polynomials whose monomial powers are from A. Let K be a compact basic semialgebraic set of R^n such that R[x]_A contains a polynomial that is positive on K. Denote by P_A(K) the cone of polynomials in R[x]_A that are nonnegative on K. The dual cone of P_A(K) is R_A(K), the set of all A-truncated moment sequences in R^A that admit representing measures supported in K. Our main results are: i) We study the properties of P_A(K) and R_A(K) (like interiors, closeness, duality, memberships), and construct a convergent hierarchy of semidefinite relaxations for each of them. ii) We propose a semidefinite algorithm for solving linear optimization problems with the cones P_A(K) and R_A(K), and prove its asymptotic and finite convergence; a stopping criterion is also given. iii) We show how to check whether P_A(K) and R_A(K) intersect affine subspaces; if they do, we show to get get a point in the intersections; if they do not, we prove certificates for the non-intersecting.

Optimizing Gaussian Quadrature for Positive Definite Strong Moment Functionals

Classical Gaussian quadrature is anchored in the space of real polynomials at polynomial degree 0 and is monotonically directed by successively increasing polynomial degree. The results of recent research, investigating weighing anchor and charting various courses in search of the best currents in the space of real Laurent polynomials, are presented. Standard error bound minimization anchoring and steering algorithms which utilize the moments of positive definite strong moment functionals to guide the construction of ordered orthogonal Laurent polynomial sequences are introduced.

What do real ideal projectors interpolate?

A characterization of interpolation properties of ideal projectors on the space of complex polynomials in several variables was given in [H.M. Möller, Hermite interpolation in several variables using ideal-theoretic methods, in: W. Schempp, K. Zeller (Eds.), Constructive Theory of Functions of Several Variables, in: Lecture Notes in Mathematics, Springer, 1977, pp. 155–163] (cf. also [C. de Boor, A. Ron, On polynomial ideals of finite codimension with applications to box spline theory, J. Math. Anal. Appl. 158 (1991) 168–193]). The purpose of this article is to provide a similar characterization for ideal projectors on the space of real polynomials. Equivalently, we give a description of finite-dimensional D -invariant subspaces of the space R [ [ x 1 , … , x d ] ] of formal power series in d indeterminants with real coefficients.

The arithmetic of certain semigroups of positive operators

Some time ago, S. Bochner gave an interesting analysis of certain positive operators which are associated with the ultraspherical polynomials (1,2). Let {Pn(x)} denote these polynomials, which are orthogonal on [ − 1, 1 ] with respect to the measureand which are normalized by settigng Pn(1) = 1. (The fixed parameter γ will not be explicitly shown.) A sequence t = {tn} of real numbers is said to be ‘positive definite’, which we will indicate by writing , provided thatHere the coefficients an are real, and the prime on the summation sign means that only a finite number of terms are different from 0. This condition can be rephrased by considering the set of linear operators on the space of real polynomials which have diagonal matrices with respect to the basis {Pn(x)}, and noting that

Extreme Points of the Unit Ball in a Space of Real Polynomials

Extreme Points of the Unit Ball in a Space of Real Polynomials