On quadrature formulae with maximal trigonometric degree of precision

Summary. In [DD] the problem of existence and uniqueness of a quadrature formula (QF) with maximal trigonometric degree of precision (MTDP) with a fixed number of free nodes and fixed different multiplicities at each node is considered. Even the affirmative answer to the question of existence and uniqueness is useless from a practical point of view if the QF is not explicitly found or if a complete characterization for the nodes and for the coefficients of the QF is not given. On the other hand the problem of the complete constructive characterization of the QF with MTDP is one of the main problems in the theory of numerical integration. In this paper we give a complete constructive characterization for the QF with MTDP in the case of a special type of periodic multiplicities. The results can be considered as a natural generalization of the previous results, which are given in [GO] (one-periodic case of multiplicities) and [DD] (two-periodic case of multiplicities). We evaluate the practical usefulness of the optimal numerical methods, which are obtained. Received June 16, 1995 / Revised version received April 3, 1996     

Quadrature rules for rational functions

Summary. It is shown how recent ideas on rational Gauss-type quadrature rules can be extended to Gauss-Kronrod, Gauss-Turán, and Cauchy principal value quadrature rules. Numerical examples illustrate the advantages in accuracy thus achievable. Received June 21, 1999 / Revised version received September 14, 1999 / Published online June 21, 2000     

Another quadrature rule of highest algebraic degree of precision

Summary. We consider certain quadrature rules of highest algebraic degree of precision that involve strong Stieltjes distributions (i.e., strong distributions on the positive real axis). The behavior of the parameters of these quadrature rules, when the distributions are strong -inversive Stieltjes distributions, is given. A quadrature rule whose parameters have explicit expressions for their determination is presented. An application of this quadrature rule for the evaluation of a certain type of integrals is also given. Received April 17, 1991 / Revised version received July 16, 1993     

Quadrature formulae with free nodes for periodic functions

Summary. The problem of existence and uniqueness of a quadrature formula with maximal trignonometric degree of precision for 2-periodic functions with fixed number of free nodes of fixed different multiplicities at each node is considered. Our approach is based on some properties of the topological degree of a mapping with respect to an open bounded set and a given point. The explicit expression for the quadrature formulae with maximal trignometric degree of precision in the 2-periodic case of multiplicities is obtained. An error analysis for the quadrature with maximal trigonometric degree of precision is given. Received April 16, 1992/Revised version received June 21, 1993     

A class of bivariate orthogonal polynomials and cubature formula

Summary. The existence of Gaussian cubature for a given measure depends on whether the corresponding multivariate orthogonal polynomials have enough common zeros. We examine a class of orthogonal polynomials of two variables generated from that of one variable. Received February 9, 1993 / Revised version received January 18, 1994     

Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations

Summary. We prove numerical stability of a class of piecewise polynomial collocation methods on nonuniform meshes for computing asymptotically stable and unstable periodic solutions of the linear delay differential equation by a (periodic) boundary value approach. This equation arises, e.g., in the study of the numerical stability of collocation methods for computing periodic solutions of nonlinear delay equations. We obtain convergence results for the standard collocation algorithm and for two variants. In particular, estimates of the difference between the collocation solution and the true solution are derived. For the standard collocation scheme the convergence results are “unconditional”, that is, they do not require mesh-ratio restrictions. Numerical results that support the theoretical findings are also given. Received June 9, 2000 / Revised version received December 14, 2000 / Published online October 17, 2001     

Is the polynomial so perfidious?

Summary. Wilkinson, in [1], has given a comprehensive account of the numerical difficulties of working with polynomials on a floating point computer. The object of this note is to attempt to rehabilitate the polynomial to a certain extent. In particular it is shown here that polynomial deflation can be performed satisfactorily by a method akin to `backward recursion'. Error analyses and examples are given to illustrate the stability of the process. Received October 4, 1993 / Revised version received July 14, 1993     

Error bounds for minimal energy bivariate polynomial splines

Summary. We derive error bounds for bivariate spline interpolants which are calculated by minimizing certain natural energy norms. Received March 28, 2000 / Revised version received June 23, 2000 / Published online March 8, 2002 RID="*" ID="*" Supported by the National Science Foundation under grant DMS-9870187 RID="**" ID="**" Supported by the National Science Foundation under grant DMS-9803340 and by the Army Research Office under grant DAAD-19-99-1-0160     

Asymptotic error bounds of product formulas for functions with interior singularity

Summary. This paper is concerned with the convergence of product quadrature formulas of interpolatory type based on the zeros of Jacobi polynomials for the approximation of integrals of the type is supposed to be of the form not an integer, . The kernel can be a smooth one or it can contain an algebraic or a logarithmic singularity. Received January 20, 1995     

Holomorphic hulls in terms of curves of nonnegative divisors and generalized polynomial hulls

This paper is devoted to the study of the Basner hulls of compact subsets of ℂ ^N . The principal result is a characterization of these hulls in terms of continuous families of varieties analogous to a well known characterization of polynomially convex hulls. This result is based on recent work of the authors concerning continuous families of divisors. Received: 30 March 1998 / Revised version: 27 September 1998     

Error bounds for Romberg quadrature

Denote by the error of a Romberg quadrature rule applied to the function f. We determine approximately the constants in the bounds of the types and for all classical Romberg rules. By a comparison with the corresponding constants of the Gaussian rule we give the statement “The Gaussian quadrature rule is better than the Romberg method” a precise meaning. Received September 10, 1997 / Revised version received February 16, 1998     

Tight linear envelopes for splines

Summary. A sharp bound on the distance between a spline and its B-spline control polygon is derived. The bound yields a piecewise linear envelope enclosing spline and polygon. This envelope is particularly simple for uniform splines and splines in Bernstein-Bézier form and shrinks by a factor of 4 for each uniform subdivision step. The envelope can be easily and efficiently implemented due to its explicit and constructive nature. Received February 12, 1999 / Revised version received October 15, 1999 / Published online May 4, 2001     

Asymptotic formula for the error in cardinal interpolation

Summary. We give the asymptotic formula for the error in cardinal interpolation. We generalize the Mazur Orlicz Theorem for periodic function. Received February 22, 1999 / Revised version received October 15, 1999 / Published online March 20, 2001     

Macro-elements and stable local bases for splines on Clough-Tocher triangulations

Summary. Macro-elements of arbitrary smoothness are constructed on Clough-Tocher triangle splits. These elements can be used for solving boundary-value problems or for interpolation of Hermite data, and are shown to be optimal with respect to spline degree. We conjecture they are also optimal with respect to the number of degrees of freedom. The construction provides local bases for certain superspline spaces defined over Clough-Tocher refinements of arbitrary triangulations. These bases are shown to be stable as a function of the smallest angle in the triangulation, which in turn implies that the associated spline spaces have optimal order approximation power. Received November 18, 1999 / Published online October 16, 2000     

Barycentric formulae for cardinal (SINC-) interpolants by Jean-Paul Berrut

Summary. A formula for the efficient evaluation of the (truncated) cardinal series is known to be numerically unstable near the interpolation abscissae. Here it is shown how the series can be evaluated in an entirely stable manner. Received February 14, 2000 / Published online October 16, 2000     

A posteriori error estimation with the finite element method of lines for a nonlinear parabolic equation in one space dimension

Summary. Convergence of a posteriori error estimates to the true error for the semidiscrete finite element method of lines is shown for a nonlinear parabolic initial-boundary value problem. Received June 15, 1997 / Revised version received May 15, 1998 / Published online: June 29, 1999     

On a class of Gauss-like quadrature rules

Summary. We consider a problem that arises in the evaluation of computer graphics illumination models. In particular, there is a need to find a finite set of wavelengths at which the illumination model should be evaluated. The result of evaluating the illumination model at these points is a sampled representation of the spectral power density of light emanating from a point in the scene. These values are then used to determine the RGB coordinates of the light by evaluating three definite integrals, each with a common integrand (the SPD) and interval of integration but with distinct weight functions. We develop a method for selecting the sample wavelengths in an optimal manner. More abstractly, we examine the problem of numerically evaluating a set of definite integrals taken with respect to distinct weight functions but related by a common integrand and interval of integration. It is shown that when it is not efficient to use a set of Gauss rules because valuable information is wasted. We go on to extend the notions used in Gaussian quadrature to find an optimal set of shared abcissas that maximize precision in a well-defined sense. The classical Gauss rules come out as the special case and some analysis is given concerning the existence of these rules when . In particular, we give conditions on the weight functions that are sufficient to guarantee that the shared abcissas are real, distinct, and lie in the interval of integration. Finally, we examine some computational strategies for constructing these rules. Received July 15, 1991     

Approximation by translates of refinable functions

Summary. The functions are refinable if they are combinations of the rescaled and translated functions . This is very common in scientific computing on a regular mesh. The space of approximating functions with meshwidth is a subspace of with meshwidth . These refinable spaces have refinable basis functions. The accuracy of the computations depends on , the order of approximation, which is determined by the degree of polynomials that lie in . Most refinable functions (such as scaling functions in the theory of wavelets) have no simple formulas. The functions are known only through the coefficients in the refinement equation – scalars in the traditional case, matrices for multiwavelets. The scalar "sum rules" that determine are well known. We find the conditions on the matrices that yield approximation of order from . These are equivalent to the Strang–Fix conditions on the Fourier transforms , but for refinable functions they can be explicitly verified from the . Received August 31, 1994 / Revised version received May 2, 1995     

Uniform asymptotic expansion of $J_\nu(\nu a)$ via a difference equation

Summary. There are two ways of deriving the asymptotic expansion of , as , which holds uniformly for . One way starts with the Bessel equation and makes use of the turning point theory for second-order differential equations, and the other is based on a contour integral representation and applies the theory of two coalescing saddle points. In this paper, we shall derive the same result by using the three term recurrence relation . Our approach will lead to a satisfactory development of a turning point theory for second-order linear difference equations. Received December 15, 2000 / Published online September 19, 2001