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.     

Interpolation lattices in several variables

Principal lattices are classical simplicial configurations of nodes suitable for multivariate polynomial interpolation in n dimensions. A principal lattice can be described as the set of intersection points of n + 1 pencils of parallel hyperplanes. Using a projective point of view, Lee and Phillips extended this situation to n + 1 linear pencils of hyperplanes. In two recent papers, two of us have introduced generalized principal lattices in the plane using cubic pencils. In this paper we analyze the problem in n dimensions, considering polynomial, exponential and trigonometric pencils, which can be combined in different ways to obtain generalized principal lattices.We also consider the case of coincident pencils. An error formula for generalized principal lattices is discussed. Partially supported by the Spanish Research Grant BFM2003-03510, by Gobierno de Aragón and Fondo Social Europeo.     

Quantitative mixing results and inner functions

We study in this paper estimates on the size of the sets of points which are well approximated by orbits of other points under certain dynamical systems. We apply the results obtained to the particular case of the dynamical system generated by inner functions in the unit disk of the complex plane.D. Pestana was supported by Grants BFM2003-04780 and BFM-2003-06335-C03-02, Ministerio de Ciencia y Tecnología, Spain.J. L. Fernández and M. V. Melián were supported by Grant BFM2003-04780 from Ministerio de Ciencia y Tecnología, Spain.     

Numerical Properties of Shifted Tridiagonal LU Factorizations

In this paper, we consider shifted tridiagonal matrices. We prove that the standard algorithm to compute the LU factorization in this situation is mixed forward-backward stable and, therefore, componentwise forward stable. Moreover, we give a formula to compute the corresponding condition number in O(n) flops. This research has been partially supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain through grants BFM2003-06335-C03-02 and MTM2006-06671 as well as by the Postdoctoral Fellowship EX2004-0658 provided by Ministerio de Educación y Ciencia of Spain.     

Convergence and computation of simultaneous rational quadrature formulas

We discuss the convergence and numerical evaluation of simultaneous quadrature formulas which are exact for rational functions. The problem consists in integrating a single function with respect to different measures using a common set of quadrature nodes. Given a multi-index n, the nodes of the integration rule are the zeros of the multi-orthogonal Hermite–Padé polynomial with respect to (S, α, n), where S is a collection of measures, and α is a polynomial which modifies the measures in S. The theory is based on the connection between Gauss-type simultaneous quadrature formulas of rational type and multipoint Hermite–Padé approximation. The numerical treatment relies on the technique of modifying the integrand by means of a change of variable when it has real poles close to the integration interval. The output of some tests show the power of this approach in comparison with other ones in use.     

Conformal geometry of surfaces in Lorentzian space forms

We study the conformal geometry of an oriented space-like surface in three-dimensional Lorentzian space forms. After introducing the conformal compactification of the Lorentzian space forms, we define the conformal Gauss map which is a conformally invariant two parameter family of oriented spheres. We use the area of the conformal Gauss map to define the Willmore functional and derive a Bernstein type theorem for parabolic Willmore surfaces. Finally, we study the stability of maximal surfaces for the Willmore functional.Dedicated to Professor T.J. WillmoreSupported by an FPPI Postdoctoral Grant from DGICYT Ministerio de Educación y Ciencia, Spain 1994 and by a DGICYT Grant No. PB94-0750-C02-02     

Cyclicity of coefficient multipliers: Linear structure

We characterize various kinds of cyclicity of sequences of coefficient multipliers, which are operators defined on spaces of holomorphic functions. In the case of a single coefficient multiplier we characterize its cyclicity, which contrasts with the fact that such operators are never supercyclic. Moreover, it is proved that for each cyclic function there is a dense part of the linear span of its orbit all of whose vectors are cyclic. The authors have been partially supported by MCYT Grants BFM2003-03893-C02-01, MTM2004-21420-E and Plan Andaluz de Investigación Junta de Andalucía FQM-127.     

A new numerical quadrature formula on the unit circle

In this paper we study a quadrature formula for Bernstein–Szegő measures on the unit circle with a fixed number of nodes and unlimited exactness. Taking into account that the Bernstein–Szegő measures are very suitable for approximating an important class of measures we also present a quadrature formula for this type of measures such that the error can be controlled with a well-bounded formula. This work was supported by Ministerio de Educación y Ciencia under grants number MTM2005-01320 (E. B. and A. C.) and MTM2006-13000-C03-02 (F. M.).     

A note on saturated formations

This work is part of the Proyecto PS 87-0055-C02-02 of CAICYT (Ministerio de Educación y Ciencia, Spain) and has been done during a visit of the author to the Fachbereich Mathematik of the Johannes Gutenberg-Universität Mainz (West Germany). The author wants to thank this Institution. Special thanks are due to Prof. Dr. K. Doerk for the discussions which led to this paper.     

Quadrature rules based on partial fraction expansions

Quadrature rules are typically derived by requiring that all polynomials of a certain degree be integrated exactly. The nonstandard issue discussed here is the requirement that, in addition to the polynomials, the rule also integrates a set of prescribed rational functions exactly. Recurrence formulas for computing such quadrature rules are derived. In addition, Fejér's first rule, which is based on polynomial interpolation at Chebyshev nodes, is extended to integrate also rational functions with pre-assigned poles exactly. Numerical results, showing a favorable comparison with similar rules that have been proposed in the literature, are presented. An error analysis of a representative test problem is given. This revised version was published online in June 2006 with corrections to the Cover Date.     

Rational interpolation with a single variable pole

In this paper the necessary and sufficient conditions for given data to admit a rational interpolant in _k,1 with no poles in the convex hull of the interpolation points is studied. A method for computing the interpolant is also provided.Partially supported by DGICYT-0121.     

On a conjecture of D. B. Hunter

We consider errors of positive quadrature formulas applied to Chebyshev polynomials. These errors play an important role in the error analysis for many function classes. Hunter conjectured that the supremum of all errors in Gaussian quadrature of Chebyshev polynomials equals the norm of the quadrature formula. We give examples, for which Hunter's conjecture does not hold. However, we prove that the conjecture is valid for all positive quadratures if the supremum is replaced by the limit superior. Considering a fixed positive quadrature formula and the sequence of all Chebyshev polynomials, we show that large errors are rare.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>–dimension free boundedness for Riesz transforms associated to Hermite functions<Superscript>⋆</Superscript>

Riesz transforms associated to Hermite functions were introduced by S. Thangavelu, who proved that they are bounded operators on 1<p< . In this paper we give a different proof that allows us to show that the L^p–norms of these operators are bounded by a constant not depending on the dimension d. Moreover, we define Riesz transforms of higher order and free dimensional estimates of the L^p–bounds of these operators are obtained. In order to prove the mentioned results we give an extension of the Littlewood-Paley theory that we believe of independent interest.Mathematical Subject Classification (2000):42B20, 42B25, 42C10Partially supported by Instituto Argentino de Matemática CONICET, Convenio Universidad Autónoma de Madrid-Universidad Nacional del Litoral, UBACYT 2000-2002 and Ministerio de Ciencia y Tecnologí BFM2002-04013-C02-02     

A Sharp Form of the Cramér–Wold Theorem

The Cramér–Wold theorem states that a Borel probability measure P on ℝ ^d is uniquely determined by its one-dimensional projections. We prove a sharp form of this result, addressing the problem of how large a subset of these projections is really needed to determine P. We also consider extensions of our results to measures on a separable Hilbert space. First author partially supported by the Spanish Ministerio de Ciencia y Tecnología, grant BFM2002-04430-C02-02. Second author partially supported by Instituto de Cooperación Iberoamericana, Programa de Cooperación Interuniversitaria AL-E 2003. Third author partially supported by grants from NSERC and the Canada research chairs program.     

On higher order Padé-type approximants with some prescribed coefficients in the numerator

In this work, we consider the construction of higher order rational approximants to a formal power series, with some prescribed coefficients in their numerators, precisely those of the higher order powers. The denominators of such approximants are related to the so-called Sobolev-type orthogonal polynomials. The elementary properties of these orthogonal polynomials are studied in the regular case.This research was partially supported by Junta de Andalucía, Grupo de Investigación 1107.     

Strong Hardy–Littlewood theorems for analytic functions and mappings of finite distortion

A formula is pointed out that explains why an analytic function often enjoys the same smoothness properties as its modulus. This is extended to quasiregular mappings and, mutatis mutandis, to mappings of finite distortion.Mathematics Subject Classification (1991): 30C65, 30D50, 30D55Supported in part by Grant 02-01-00267 from the Russian Foundation for Fundamental Research, DGICYT Grant BFM2002-04072-C02-01, CIRIT Grant 2001-SGR-00172, by the Ramón y Cajal program (Spain) and by the European Communitys Human Potential Program under contract HPRN-CT-2000-00116 (Analysis and Operators).     

Generation of lattices of points for bivariate interpolation

Principal lattices in the plane are distributions of points particularly simple to use Lagrange, Newton or Aitken–Neville interpolation formulae. Principal lattices were generalized by Lee and Phillips, introducing three-pencil lattices, that is, points which are the intersection of three lines, each one belonging to a different pencil. In this contribution, a semicubical parabola is used to construct lattices of points with similar properties. For the construction of new lattices we use cubic pencils of lines and an addition of lines on them. AMS subject classification 41A05, 65D05, 41A63Research partially supported by the Spanish Research Grant BFM2003-03510, by Gobierno de Aragón and Fondo Social Europeo.     

Gelfand-Kirillov Dimension and Local Finiteness of Symmetrizations of Associative Superalgebras

In this work we extend to superalgebras a result of Skosyrskii [Algebra and Logic, 18 (1) (1979) 49–57, Lemma 2] relating associative and Jordan structures. As an application, we show that the Gelfand-Kirillov dimension of an associative superalgebra coincides with that of its symmetrization, and that local finiteness is equivalent in associative superalgebras and in their symmetrizations. In this situation we obtain that having zero Gelfand-Kirillov dimension is equivalent to being locally finite.Partially supported by MCYT and Fondos FEDER BFM2001-1938-C02-02, and MEC and Fondos FEDER MTM2004-06580-C02-01.Partially supported by a F.P.I. Grant (Ministerio de Ciencia y Tecnología).     

Stability and Sensitivity of Tridiagonal LU Factorization without Pivoting

In this paper the accuracy of LU factorization of tridiagonal matrices without pivoting is considered. Two types of componentwise condition numbers for the L and U factors of tridiadonal matrices are presented and compared. One type is a condition number with respect to small relative perturbations of each entry of the matrix. The other type is a condition number with respect to small componentwise perturbations of the kind appearing in the backward error analysis of the usual algorithm for the LU factorization. We show that both condition numbers are of similar magnitude. This means that the algorithm is componentwise forward stable, i.e., the forward errors are of similar magnitude to those produced by a componentwise backward stable method. Moreover the presented condition numbers can be computed in O(n) flops, which allows to estimate with low cost the forward errors. AMS subject classification (2000) 65F35, 65F50, 15A12, 15A23, 65G50.Received October 2003. Accepted August 2004. Communicated by Per Christian Hansen.Froilán M. Dopico: This research has been partially supported by the Ministerio de Ciencia y Tecnología of Spain through grants BFM2003-06335-C03-02 (M. I. Bueno) and BFM2000-0008 (F. M. Dopico).     

Wall Rational Functions and Khrushchev’s Formula for Orthogonal Rational Functions

The Nevalinna–Pick algorithm yields a continued fraction expansion of every Schur function, whose approximants are identified. These approximants are quotients of rational functions which can be understood as the rational analogs of the Wall polynomials. The properties of these Wall rational functions and the corresponding approximants permit us to obtain a Khrushchev’s formula for orthogonal rational functions. An introduction to the convergence of the Wall approximants in the indeterminate case is presented. This work was partially realized during two stays of the second author at the Norwegian University of Science and Technology (NTNU) financed respectively by Secretaría de Estado de Universidades e Investigación from the Ministry of Education and Science of Spain and by the Department of Mathematical Sciences of NTNU. The work of the second author was also partially supported by the Spanish grants from the Ministry of Education and Science, project code MTM2005-08648-C02-01, and the Ministry of Science and Innovation, project code MTM2008-06689-C02-01, and by Project E-64 of Diputación General de Aragón (Spain).