Construction of asymptotically best quadrature formulas

Summary. We show that, for integrals with arbitrary integrable weight functions, asymptotically best quadrature formulas with equidistant nodes can be obtained by applying a certain scheme of piecewise polynomial interpolation to the function to be integrated, and then integrating this interpolant. Received August 7, 1991     

Interpolatory product quadratures for Cauchy principal value integrals with Freud weights

Summary. We prove convergence results and error estimates for interpolatory product quadrature formulas for Cauchy principal value integrals on the real line with Freud–type weight functions. The formulas are based on polynomial interpolation at the zeros of orthogonal polynomials associated with the weight function under consideration. As a by–product, we obtain new bounds for the derivative of the functions of the second kind for these weight functions. Received July 15, 1997 / Revised version received August 25, 1998     

Sequences of orthogonal Laurent polynomials,bi-orthogonality and quadrature formulas on the unit circle

In this paper, the construction of orthogonal bases in the space of Laurent polynomials on the unit circle is considered. As an application, a connection with the so-called bi-orthogonal systems of trigonometric polynomials is established and quadrature formulas on the unit circle based on Laurent polynomials are studied.     

Determinantal formulas for zonal spherical functions on hyperboloids

Abstract. We construct determinantal expressions for the zonal spherical functions on the hyperboloids with p,q odd (and larger than 1). This gives rise to explicit evaluation formulas for hypergeometric series representing half-integer parameter families of Jacobi functions and (via specialization) Jacobi polynomials. Received November 18, 1999 / Published online October 30, 2000     

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     

Estimating quadrature errors for analytic functions using kernel representations and biorthogonal systems

Summary. For analytic functions the remainder term of quadrature rules can be represented as a contour integral with a complex kernel function. From this representation different remainder term estimates involving the kernel are obtained. It is studied in detail how polynomial biorthogonal systems can be applied to derive sharp bounds for the kernel function. It is shown that these bounds are practical to use and can easily be computed. Finally, various numerical examples are presented. Received March 11, 1998 / Revised version January 22, 1999/ Published online November 17, 1999     

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     

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     

Gaussian interval quadrature formula

Summary. We prove the existence of a Gaussian quadrature formula for Tchebycheff systems, based on integrals over non-overlapping subintervals of arbitrary fixed lengths and the uniqueness of this formula in the case the subintervals have equal lengths. Received July 6, 1999 / Published online August 24, 2000     

On the orthogonality of residual polynomials\newline of minimax polynomial preconditioning

Summary. This paper studies polynomials used in polynomial preconditioning for solving linear systems of equations. Optimum preconditioning polynomials are obtained by solving some constrained minimax approximation problems. The resulting residual polynomials are referred to as the de Boor-Rice and Grcar polynomials. It will be shown in this paper that the de Boor-Rice and Grcar polynomials are orthogonal polynomials over several intervals. More specifically, each de Boor-Rice or Grcar polynomial belongs to an orthogonal family, but the orthogonal family varies with the polynomial. This orthogonality property is important, because it enables one to generate the minimax preconditioning polynomials by three-term recursive relations. Some results on the convergence properties of certain preconditioning polynomials are also presented. Received February 1, 1992/Revised version received July 7, 1993     

Peano kernels and bounds for the error constants of Gaussian and related quadrature rules for Cauchy principal value integrals

Summary. We show that, if (), the error term of every modified positive interpolatory quadrature rule for Cauchy principal value integrals of the type , , fulfills uniformly for all , and hence it is of optimal order of magnitude in the classes (). Here, is a weight function with the property . We give explicit upper bounds for the Peano-type error constants of such rules. This improves and completes earlier results by Criscuolo and Mastroianni (Calcolo 22 (1985), 391–441 and Numer. Math. 54 (1989), 445–461) and Ioakimidis (Math. Comp. 44 (1985), 191–198). For the special case of the Gaussian rule, we show that the restriction can be dropped. The results are based on a new representation of the Peano kernels of these formulae via the Peano kernels of the underlying classical quadrature formulae. This representation may also be useful in connection with some different problems. Received November 21, 1994     

Numerical integration over a disc. A new Gaussian quadrature formula

Summary. We construct a quadrature formula for integration on the unit disc which is based on line integrals over distinct chords in the disc and integrates exactly all polynomials in two variables of total degree . Received August 8, 1996 / Revised version received July 2, 1997     

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     

On the analyticity properties of infeasible-interior-point paths for monotone linear complementarity problems

Infeasible-interior-point paths , a positive vector, for a horizontal linear complementarity problem are defined as the solution of () If the path converges for , then it converges to a solution of . This paper deals with the analyticity properties of and its derivatives with respect to r near r = 0 for solvable monotone complementarity problems . It is shown for with a strictly complementary solution that the path , , has an extension to which is analytic also at . If has no strictly complementary solution, then , , has an extension to that is analytic at . Received May 24, 1996 / Revised version received February 25, 1998     

Pointwise Remez- and Nikolskii-type inequalities for exponential sums

Let So is the collection of all n + 1 term exponential sums with constant first term. We prove the following two theorems. Theorem 1 (Remez-type inequality for $E_n$ at 0). Let $s \in \left( 0, \frac 12 \right]\,.$ There are absolute constants $c_1 > 0$ and $c_2 > 0$ such that where the supremum is taken for all $f \in E_n$ satisfying Theorem 2 (Nikolskii-type inequality for $E_n$ ). There are absolute constants $c_1 > 0$ and $c_2 > 0$ such that for every $a < y < b$ and $q > 0\,.$ It is quite remarkable that, in the above Remez- and Nikolskii-type inequalities, behaves like , where denotes the collection of all algebraic polynomials of degree at most n with real coefficients. Received: 4 November 1998 / in final form: 2 March 1999     

Polynomials on the Cauchy circle

Summary. Let be a complex polynomial of degree with and Cauchy radius 1 about the origin. We discuss the order of magnitude of the minimal number such that Previous estimates of are improved to . Some other related properties of these polynomials are also exhibited. Received March 3, 1993     

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     

Optimality of the double exponential formula – functional analysis approach –

Summary. In the light of the functional analysis theory we establish the optimality of the double exponential formula. The argument consists of the following three ingredients: (1) introduction of a number of spaces of functions analytic in a strip region about the real axis, each space being characterized by the decay rate of their elements (functions) in the neighborhood of the infinity; (2) proof of the (near-) optimality of the trapezoidal formula in each space introduced in (1) by showing the (near-) equality between an upper estimate for the error norm of the trapezoidal formula and a lower estimate for the minimum error norm of quadratures; (3) nonexistence theorem for the spaces, the characterizing decay rate of which is more rapid than the double exponential. Received September 15, 1995 / Accepted December 14, 1995     

Constructing cubature formulae by the method of reproducing kernel

Summary. We examine the method of reproducing kernel for constructing cubature formulae on the unit ball and on the triangle in light of the compact formulae of the reproducing kernels that are discovered recently. Several new cubature formulae are derived. Received April 15, 1998 / Revised version received November 24, 1998 / Published online January 27, 2000