共查询到20条相似文献,搜索用时 0 毫秒
1.
Karl Deckers Adhemar Bultheel 《Journal of Mathematical Analysis and Applications》2009,356(2):764-768
Let {φn} be a sequence of rational functions with arbitrary complex poles, generated by a certain three-term recurrence relation. In this paper we show that under some mild conditions, the rational functions φn form an orthonormal system with respect to a Hermitian positive-definite inner product. 相似文献
2.
A. V. Abrahamyan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2010,45(1):1-7
The paper is devoted to some properties of orthogonal on the unit circle rational functions with fixed poles. 相似文献
3.
A. V. Abramyan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2010,45(5):253-257
The paper is devoted to investigation of the asymptotic behavior of orthogonal on the unit circle rational functions with fixed poles. 相似文献
4.
K. Pan 《Journal of Computational and Applied Mathematics》1995,60(3):347-355
Let = zn,mm=1n with |zn,m| < 1, n = 1,2,…, be an arbitrary sequence of complex numbers. We generalize the orthogonal rational functions with poles at . We study the weak convergence and the interpolation properties of the orthogonal rational functions. 相似文献
5.
Adhemar Bultheel Pablo González-Vera Erik Hendriksen Olav Njåstad 《Numerical Algorithms》1992,3(1):105-116
In this paper we shall be concerned with the problem of approximating the integralI
{f}=
–
f(ei) d(), by means of the formulaI
n
{f}=
j=1
n
A
j
(n)
f(x
j
(n)
) where is some finite positive measure. We want the approximation to be so thatI
n{f}=I
{f} forf belonging to certain classes of rational functions with prescribed poles which generalize in a certain sense the space of polynomials. In order to get nodes {x
j
(n)
} of modulus 1 and positive weightsA
j
(n)
, it will be fundamental to use rational functions orthogonal on the unit circle analogous to Szeg polynomials.The work of the first author is partially supported by a research grant from the Belgian National Fund for Scientific Research. 相似文献
6.
The paper aims to investigate the convergence of the q -Bernstein polynomials Bn,q(f;x) attached to rational functions in the case q>1. The problem reduces to that for the partial fractions (x−α)−j, j∈N. The already available results deal with cases, where either the pole α is simple or α≠q−m, m∈N0. Consequently, the present work is focused on the polynomials Bn,q(f;x) for the functions of the form f(x)=(x−q−m)−j with j?2. For such functions, it is proved that the interval of convergence of {Bn,q(f;x)} depends not only on the location, but also on the multiplicity of the pole – a phenomenon which has not been considered previously. 相似文献
7.
8.
9.
Adhemar Bultheel Erik Hendriksen Pablo Gonzlez-Vera Olav Njstad 《Journal of Computational and Applied Mathematics》1994,50(1-3):159-170
Quadrature formulas on the unit circle were introduced by Jones in 1989. On the other hand, Bultheel also considered such quadratures by giving results concerning error and convergence. In other recent papers, a more general situation was studied by the authors involving orthogonal rational functions on the unit circle which generalize the well-known Szeg
polynomials. In this paper, these quadratures are again analyzed and results about convergence given. Furthermore, an application to the Poisson integral is also made. 相似文献
10.
A general theory of uniform approximation with rational functions having negative poles is developed. An existence theory is given and local characterization and uniqueness results are developed. Algorithms for computing these approximants are given, together with numerical results. 相似文献
11.
In this paper, a collocation method using a new weighted orthogonal system on the half-line, namely the rational Gegenbauer functions, is introduced to solve numerically the third-order nonlinear differential equation, af?+ff″=0, where a is a constant parameter. This method solves the problems on semi-infinite domain without truncating it to a finite domain and transforming the domain of the problems to a finite domain. For a=2, the equation is the well-known Blasius equation, which is a laminar viscous flow over a semi-infinite flat plate. We solve this equation by considering 1?a?2 and compare the new results with the established results to show the efficiency and accuracy of the new method. 相似文献
12.
Marc Van Barel Adhemar Bultheel 《Journal of Computational and Applied Mathematics》1994,50(1-3):545-563
We already generalized the Rutishauser—Gragg—Harrod—Reichel algorithm for discrete least-squares polynomial approximation on the real axis to the rational case. In this paper, a new method for discrete least-squares linearized rational approximation on the unit circle is presented. It generalizes the algorithms of Reichel—Ammar—Gragg for discrete least-squares polynomial approximation on the unit circle to the rationale case. The algorithm is fast in the sense that it requires order m computation time where m is the number of data points and is the degree of the approximant. We describe how this algorithm can be implemented in parallel. Examples illustrate the numerical behavior of the algorithm. 相似文献
13.
14.
Joris Van Deun 《Numerical Algorithms》2007,45(1-4):89-99
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. 相似文献
15.
Let w() be a positive weight function on the unit circle of the complex plane. For a sequence of points {
k
}
k = 1
included in a compact subset of the unit disk, we consider the orthogonal rational functions
n
that are obtained by orthogonalization of the sequence { 1, z / 1, z
2 / 2, ... } where
, with respect to the inner product
In this paper we discuss the behaviour of
n
(t) for t = 1 and n under certain conditions. The main condition on the weight is that it satisfies a Lipschitz–Dini condition and that it is bounded away from zero. This generalizes a theorem given by Szeg in the polynomial case, that is when all
k
= 0. 相似文献
16.
17.
A matricial computation of quadrature formulas for orthogonal rational functions on the unit circle, is presented in this paper. The nodes of these quadrature formulas are the zeros of the para-orthogonal rational functions with poles in the exterior of the unit circle and the weights are given by the corresponding Christoffel numbers. We show how these nodes can be obtained as the eigenvalues of the operator Möbius transformations of Hessenberg matrices and also as the eigenvalues of the operator Möbius transformations of five-diagonal matrices, recently obtained. We illustrate the preceding results with some numerical examples. 相似文献
18.
Let µ be a positive bounded Borel measure on a subsetI of the real line and = {1, ..., n} a sequence of arbitrary complexpoles outside I. Suppose {1, ..., n} is the sequence of rationalfunctions with poles in orthonormal on I with respect to µ. First, we are concernedwith reducing the number of different coefficients in the three-termrecurrence relation satisfied by these orthonormal rationalfunctions. Next, we consider the case in which I = [–1, 1] and µ satisfies the Erdos–Turán conditionµ' > 0 a.e. on I (where µ' is the Radon–Nikodymderivative of the measure µ with respect to the Lebesguemeasure) to discuss the convergence of n+1(x)/n(x) as n tendsto infinity and to derive asymptotic formulas for the recurrencecoefficients in the three-term recurrence relation. Finally,we give a strong convergence result for n(x) under the morerestrictive condition that µ satisfies the Szeg condition(1 – x2)–1/2 log µ'(x) L1([– 1, 1]). 相似文献
19.
AbstractThe 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. 相似文献