Properties of spatial cross-periodograms using fixed-domain asymptotics

Cross-periodograms can be used to study a multivariate spatial process observed on a lattice. For spatial data, it is often appropriate to study asymptotic properties of statistical procedures under fixed-domain asymptotics in which the number of observations increases in a fixed region while shrinking distances between neighboring observations. Using fixed-domain asymptotics, we prove relative asymptotic unbiasedness and relative consistency of a smoothed cross-periodogram after appropriate filtering of the data. In addition, we show that smoothed cross-periodograms are asymptotically normal when the process is stationary multivariate Gaussian with appropriate assumptions on high-frequency behavior of the spectral density.     

Suboptimal Kronrod extension formulae for numerical quadrature

Summary We consider cases where the Stieltjes polynomial associated with a Gaussian quadrature formula has complex zeros. In such cases a Kronrod extension of the Gaussian rule does not exist. A method is described for modifying the Stieltjes polynomial so that the resulting polynomial has no complex zeros. The modification is performed in such a way that the Kronrod-type extension rule resulting from the addition of then+1 zeros of the modified Stieltjes polynomial to the original knots of the Gaussian rule has only slightly lower degree of precision than normally achieved when the Kronrod extension rule exists. As examples of the use of the method, we present some new formulae extending the classical Gauss-Hermite quadrature rules. We comment on the limited success of the method in extending Gauss-Laguerre rules and show that several modified extensions of the Gauss Gegenbauer formulae exist in cases where the standard Kronrod extension does not.     

Equiconvergence of some complex interpolatory polynomials

Summary Walsh showed the close relation between the Lagrange interpolant in then ^th roots of unity and the corresponding Taylor expansion for functions belonging to a certain class of analytic functions. Recent extensions of this phenomena to Hermite interpolation and other linear processes of interpolation have been surveyed in [3, 5]. Following a recent idea of L. Yuanren [7], we show how new relations between other linear operators can be derived which exhibit Walsh equiconvergence.Dedicated to R. S. Varga on the occasion of his sixtieth birthdayThese authors were supported by NSERC A3094     

Discretization of the Timoshenko Beam problem by thep and theh-p versions of the finite element method

Summary In this paper the discretization of the Timoshenko Beam problem by thep and theh-p versions of the finite element method is considered. Optimal error estimates are established. The locking phenomenon disappears as the thickness of the beam decreases.     

Domain imbedding methods for the Stokes equations

Summary We study direct and iterative domain imbedding methods for the Stokes equations on certain non-rectangular domains in two space dimensions. We analyze a continuous analog of numerical domain imbedding for bounded, smooth domains, and give an example of a simple numerical algorithm suggested by the continuous analysis. This algorithms is applicable for simply connected domains which can be covered by rectangular grids, with uniformly spaced grid lines in at least one coordinate direction. We also discuss a related FFT-based fast solver for Stokes problems with physical boundary conditions on rectangles, and present some numerical results.     

Equilibria of Runge-Kutta methods

Summary It is known that certain Runge-Kutta methods share the property that, in a constant-step implementation, if a solution trajectory converges to a bounded limit then it must be a fixed point of the underlying differential system. Such methods are calledregular. In the present paper we provide a recursive test to check whether given method is regular. Moreover, by examining solution trajectories of linear equations, we prove that the order of ans-stage regular method may not exceed 2[(s+2)/2] and that the maximal order of regular Runge-Kutta method with an irreducible stability function is 4.     

Numerical solution for exterior problems

Summary We solve the Helmholtz equation in an exterior domain in the plane. A perfect absorption condition is introduced on a circle which contains the obstacle. This boundary condition is given explicitly by Bessel functions. We use a finite element method in the bounded domain. An explicit formula is used to compute the solution out of the circle. We give an error estimate and we present relevant numerical results.     

Vandermonde matrices on the circle: Spectral properties and conditioning

Summary We study Vandermonde matrices whose nodes are given by a Van der Corput sequence on the unit circle. Our primary interest is in the singular values of these matrices and the respective (spectral) condition numbers. Detailed information about multiplicities and eigenvectors, however, is also obtained. Two applications are given to the theory of polynomials.Dedicated to R. S. Varga on the occasion of his sixtieth birthdayResearch of A. C. supported by the Fundación Andes, Chile, and by the German Academic Exchange Service (DAAD), Federal Republic of GermanyResearch of W. G. supported, in part, by the National Science Foundation, USA, (Grant CCR-8704404)Research of S. R. supported by the Fondo Nacional de Desarollo Cientßfico y Tecnológico (FONDECYT), Chile, (Grant 237/89), by the Universidad Técnica F. Santa Marßa, Valparaßso, Chile, (Grant 89.12.06), and by the German Academic Exchange Service (DAAD), Federal Republic of Germany     

Calculating singular integrals as an ill-posed problem

Summary Applying numerical quadrature to singular integrals in a straightforward way leads to uncontrolled instability with respect to data errors. In this paper we describe how to control this instability.Dedicated to Professor G. Hämmerlin on the occasion of his 60th birthday.     

Partial differential equations with differential constraints

A geometric setting for constrained exterior differential systems on fibered manifolds with n-dimensional bases is proposed. Constraints given as submanifolds of jet bundles (locally defined by systems of first-order partial differential equations) are shown to carry a natural geometric structure, called the canonical distribution. Systems of second-order partial differential equations subjected to differential constraints are modeled as exterior differential systems defined on constraint submanifolds. As an important particular case, Lagrangian systems subjected to first-order differential constraints are considered. Different kinds of constraints are introduced and investigated (Lagrangian constraints, constraints adapted to the fibered structure, constraints arising from a (co)distribution, semi-holonomic constraints, holonomic constraints).