Generalized Jacobi polynomials are orthogonal polynomials related to a weight function which is smooth and positive on the whole interval of orthogonality up to a finite number of points, where algebraic singularities occur. The influence of these singular points on the asymptotic behaviour of the recurrence coefficients is investigated. AMS(MOS) subject classification. 42C05.

In this chapter we study cones in the real Hilbert spaces of Hermitian matrices and real valued trigonometric polynomials. Based on an approach using such cones and their duals, we establish various extension results for positive semidefinite matrices and nonnegative trigonometric polynomials. In addition, we show the connection with semidefinite programming and include some numerical experiments.

A problem which often arises while fitting implicit polynomials to 2D and 3D data sets is the following: Although the data set is simple, the fit exhibits undesired phenomena, such as loops, holes, extraneous components, etc. Previous work tackled these problems by optimizing heuristic cost functions, which penalize some of these topological problems in the fit. This paper suggests a different ...

We consider the problem of fixed-polynomial lower bounds on the size of arithmetic circuits computing uniform families of polynomials. Assuming the generalised Riemann hypothesis (GRH), we show that for all k, there exist polynomials with coefficients in MA having no arithmetic circuits of size O(n) over C (allowing any complex constant). We also build a family of polynomials that can be evalua...

We investigate the completely positive semidefinite cone CS+, a new matrix cone consisting of all n×n matrices that admit a Gram representation by positive semidefinite matrices (of any size). In particular, we study relationships between this cone and the completely positive and the doubly nonnegative cone, and between its dual cone and trace positive non-commutative polynomials. We use this n...

We employ the fact that certain divided differences can be written as weighted means of B-splines and hence are positive. These include complete homogeneous symmetric polynomials even degree 2p, positivity which is a classical result by D.B. Hunter. extend Hunter's to fractional degree, defined via Jacobi's bialternant formula. show in particular these have positive real part for degrees ? with...

We present a method for the accurate numerical evaluation of a family of trigonometric series arising in the design of special-purpose quadrature rules for boundary element methods. The series converge rather slowly, but can be expressed in terms of Fourier-Chebyshev series that converge rapidly. Let r be a positive integer, and define the functions Gr and Hr by

Let n = p1 1 · · · pr r be the prime factorization of a positive integer n. Define the excess of n to be (α1−1)+ · · ·+(αr −1), which is the difference between the total multiplicity α1 + · · · + αr and the number of distinct primes in the factorization. An integer with excess 0 is also said to be square-free. Let Ek denote the set of positive integers of excess k, k = 0, 1, 2, . . .. Rényi pro...

Take a linear ordinary differential operator d(z) = Pk i=1 Qi(z) d dzi with polynomial coefficients and set r = maxi=1,...,k(degQi(z) − i). If d(z) satisfies the conditions: i) r ≥ 0 and ii) degQk(z) = k + r we call it a nondegenerate higher Lamé operator. Following the classical examples of E. Heine and T. Stieltjes we initiated in [6] the study of the following multiparameter spectral problem...

In 1892, D. Hilbert began what is now called Inverse Galois Theory by showing that for each positive integer m, there exists a polynomial of degree m with rational coefficients and associated Galois group Sm, the symmetric group on m letters, and there exists a polynomial of degree m with rational coefficients and associated Galois group Am, the alternating group on m letters. In the late 1920’...