Numerical evaluation of a 2-D Cauchy principal value integral based on quasi-interpolating splines

In this paper the author presents a method for the numerical solution of a 2-D Cauchy principal value of the form where S is a domain with a continuous boundary. By using polar coordinates, the integral is reduced to the form where the finite-part of the integral. We construct the relative product rule based on quasi-inter polating splines. Convergence results are proved and numerical examples are given.     

On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals

Summary In a previous paper the authors proposed a modified Gaussian rule _* ^m (wf;t)to compute the integral (wf;t) in the Cauchy principal value sense associated with the weightw, and they proved the convergence in closed sets contained in the integration interval. The main purpose of the present work is to prove uniform convergence of the sequence { _* ^m (wf;t)} on the whole integration interval and to give estimates for the remainder term. The same results are shown for particular subsequences of the Gaussian rules _m (wf;t) for the evaluation of Cauchy principal value integrals. A result on the uniform convergence of the product rules is also discussed and an application to the numerical solution of singular integral equations is made.     

On the convergence of Hunter's quadrature rule for Cauchy principal value integrals

Hunter's (n+1)-point quadrature rule for the approximate evaluation of the Cauchy principal value integralf ¹ _–1 (w(x)f(x)/(x – ))dx, –1<<1, is based on approximatingf by the polynomial which interpolatesf at the point and then zeros of the orthogonal polynomialp _n generated by the weight functionw. Sufficient conditions are given to ensure the convergence of a suitably chosen subsequence of the quadrature rules to the integral, whenf is Hölder continuous on [–1,1].     

Finite part integrals of local bivariate c1 quasi-interpolating splines

A local bivariate C¹ quasi-interpolating spline operator with a four directional mesh is considered and studied. Based on the above operator we present cubature formulas for 2-D singular integrals, defined in the Hadamard finite part sense. Convergence results are obtained for a wide class of functions. Moreover numerical tests are given. Sponsored by M.U.R.S.T. and C.N.R. of Italy.     

Conditions of convergence of boundary values of Cauchy type integrals

In a domain G bounded by a rectifiable Jordan curve γ let there be given a sequence of analytic functions {f_n(z)} representable by Cauchy-Lebesgue type integrals $$f_n (z) = \smallint _y \frac{{\omega _n (\zeta )}}{{\zeta - z}}d\zeta .$$ A theorem is established which enables one to determine from the convergence in measure of {ω_n(ζ)} on a set e ?γ whether or not there is convergence in measure on the same set of {f_n(ζ)}, the angular boundary values of the functionsf_n(z).     

On uniform convergence of measures

Summary A new and simpler proof is given in Section 3 for the sufficiency part of Theorem 3.1 in Ranga Rao [6] and its generalization by Billingsley and TopsØe [1]. Essential for the proof, which does not require the topological space X to be metric, is Lemma 2.1. As examples of possible wider application of this lemma, simple proofs are given for a well known result on uniformity in convergence of distribution functions (Example 2.3) and of Theorem 4.2 in Ranga Rao [6]. The derivation of the latter from the lemma is substantially simpler than the derivation from Theorem 3.1 in Ranga Rao [6]. Another result on uniformity, given in Rubin [7], is closely related to the Ascoli theorem, but outside the scope of applicability of our lemma.Most of the work on the paper was done while the author was with the Institute for Information Theory and Automation of Czechoslovak Academy of Sciences. After August 21, 1968 the author worked on it while a guest of the Forschungsinstitut für Mathematik, Eidgenössische Technische Hochschule, Zürich, and the paper was completed at Michigan State University.     

Some Gauss-type formulae for the evaluation of Cauchy principal values of integrals

A number of formulae are derived for the numerical evaluation of integrals of the form, whereg(x) possesses one or more simple poles in the interval (–1, 1). The formulae are based on Gauss-Legendre quadrature.     

On the variational characterization and convergence of bivariate splines

It is shown that bivariate interpolatory splines defined on a rectangleR can be characterized as being unique solutions to certain variational problems. This variational property is used to prove the uniform convergence of bivariate polynomial splines interpolating moderately smooth functions at data which includes interpolation to values on a rectangular grid. These results are then extended to bivariate splines defined on anL-shaped region.This research was supported by a University of Kansas General Research Grant.     

A class of C2 quasi-interpolating splines free of Gibbs phenomenon

Numerical Algorithms - In many applications, it is useful to use piecewise polynomials that satisfy certain regularity conditions at the joint points. Cubic spline functions emerge as good...     

On the uniform convergence of trigonometric series

In this paper, we obtain sufficient conditions for the uniform convergence of trigonometric series with monotone (in the extended sense) coefficients.     

On proximal convergence in uniform spaces

The paper deals with proximal convergence and Leader's theorem, in the constructive theory of uniform apartness spaces. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the uniform convergence of special sine integrals

We study the uniform convergence of sine integrals of measurable functions on the open half-line. We also extend results of F. Móricz and give examples for appropriate functions.     

On the distributions of boundary values of Cauchy integrals

We use new methods to give short proofs to some known results on the distributions of boundary values of Cauchy integrals. We also indicate some further generalizations.

     

On the uniform convergence of samples of lebesgue integrals

Developments and generalizations of a theorem of Halkin, on the limit of the multiple balayage of vector integrals, are deduced via Riemann integration theory.