Rational exponential approximation with real poles

This paper shows that, in the set of rational functions with real poles there exists a best minimax approximation to the exponential function over the non-negative real axis. This minimax approximation has an equal-ripple property similar to the classical Chebyshev approximation and, under certain conditions, it has a form that could be gainfully exploited in the numerical solutions of heat-conduction type problems.     

The barycentric weights of rational interpolation with prescribed poles

We introduce a method for calculating rational interpolants when some (but not necessarily all) of their poles are prescribed. The algorithm determines the weights in the barycentric representation of the rationals; it simply consists in multiplying each interpolated value by a certain number, computing the weights of a rational interpolant without poles, and finally multiplying the weights by those same numbers. The supplementary cost in comparison with interpolation without poles is about (v + 2)N, where v is the number of poles and N the number of interpolation points. We also give a condition under which the computed rational interpolation really shows the desired poles.     

On the structure invariants of proper rational matrices with prescribed finite poles

Abstract

The algebraic structure of matrices defined over arbitrary fields whose elements are rational functions with no poles at infinity and prescribed finite poles is studied. Under certain very general conditions, they are shown to be matrices over an Euclidean domain that can be classified according to the corresponding invariant factors. The relationship between these invariants and the local Wiener–Hopf factorization indices will be clarified. This result can be seen as an extension of the classical theorem on pole placement by Rosenbrock in control theory.     

Rational matrix and operator functions with prescribed singularities

In this paper it is shown how a rational matrix function may be reconstructed when complete information about its zeros and poles is given. The analogous problem for infinite dimensional operator functions is also solved.     

A Turán type inequality for rational functions with prescribed poles

Summary By employing a novel idea and simple techniques, we substantially generalize the Turán type inequality for rational functions with real zeros and prescribed poles established by Min [5] to include L^p spaces for 1≤ p ≤ ∞ while loosing the restriction ρ > 2 at the same time.     

A Turán type inequality for rational functions with prescribed poles

Summary By employing a novel idea and simple techniques, we substantially generalize the Turán type inequality for rational functions with real zeros and prescribed poles established by Min [5] to include L^pspaces for 1≤p≤∞<span style='font-size:10.0pt'>while loosing the restriction ρ > 2 at the same time.     

Rational quadrature formulae on the unit circle with arbitrary poles

Interpolatory quadrature rules exactly integrating rational functions on the unit circle are considered. The poles are prescribed under the only restriction of not lying on the unit circle. A computable upper bound of the error is obtained which is valid for any choice of poles, arbitrary weight functions and any degree of exactness provided that the integrand is analytic on a neighborhood of the unit circle. A number of numerical examples are given which show the advantages of using such rules as well as the sharpness of the error bound. Also, a comparison is made with other error bounds appearing in the literature. The work of the first author was supported by the Dirección General de Investigación, Ministerio de Educación y Ciencia, under grants MTM2006-13000-C03-02 and MTM2006-07186 and by UPM and Comunidad de Madrid under grant CCG06-UPM/MTM-539. The work of the second author was partially supported by the Dirección General de Investigación, Ministerio de Educación y Ciencia, under grant MTM2005-08571.     

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.     

Majoration principles and some inequalities for polynomials and rational functions with prescribed poles

The paper considers the equality cases in the rnajoration principle for meromorphic functions established earlier by V. N. Dubinin and S. I. Kalmykov [Mat. Sb., 198, No. 12, 37–46 (2007)]. As corollaries of this principle, new inequalities for the coefficients and derivatives of polynomials satisfying certain conditions on two intervals are obtained. Simple proofs of some Lukashov’s theorems on the derivatives of rational functions on several intervals are provided. Bibliography: 13 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 143–157.     

Differential inequalities for entire functions of finite degree and rational functions with prescribed poles

We establish new differential inequalities for the entire functions of finite degree with a majorant an entire function without zeros in the lower half-plane, for the entire functions with constraints on zeros and, as a consequence, for the rational functions with prescribed poles. All cases of equality in the main results are found. The estimates obtained generalize and strengthen some inequalities by Bernstein, Gardner, and Govil for entire functions of finite degree; by Smirnov, Aziz, and Shah for algebraic polynomials; and by Borwein and Erdelyi, Aziz and Shah, and the others for rational functions.     

B-splines with homogenized knots

The use of homogenized knots for manipulating univariate polynomials by blossoming algorithms is extended to piecewise polynomials. A generalization of the B-spline to homogenized knots is studied. The new B-spline retains the triangular blossoming algorithms for evaluation, differentiation and knot insertion. Moreover, the B-spline is locally supported and a Marsden’s identity exists. Spaces of natural splines and certain polynomial spline spaces with more general continuity properties than ordinary splines have bases of B-splines over homogenized knots. Applications to nonpolynomial splines such as trigonometric and hyperbolic splines are made.     

B-splines with homogenized knots

The use of homogenized knots for manipulating univariate polynomials by blossoming algorithms is extended to piecewise polynomials. A generalization of the B-spline to homogenized knots is studied. The new B-spline retains the triangular blossoming algorithms for evaluation, differentiation and knot insertion. Moreover, the B-spline is locally supported and a Marsden’s identity exists. Spaces of natural splines and certain polynomial spline spaces with more general continuity properties than ordinary splines have bases of B-splines over homogenized knots. Applications to nonpolynomial splines such as trigonometric and hyperbolic splines are made.     

B-splines with Birkhoff knots

A natural extension of the Curry-SchoenbergB-splines is given, which preserves such critical properties as variation diminishing and total positivity. Using this tool we give a characterization of the Birkhoff interpolation problem for spline functions.Communicated by Dietrich Braess.     

B-splines with geometric knot spacings

In this paper we study B-splines when the intervals between consecutive knots are in geometric progression and obtain generalizations of the particularly simple properties of the uniform B-splines, where the knots are equally spaced.