Vittoria Demichelis

Numerical integration based on quasi-interpolating splines

In this paper product quadrature rules based on quasi-interpolating splines are proposed and convergence results are proved for bounded integrands. Convergence results are also proved for sequences of Cauchy principal value integrals of these quasiinterpolating splines. Some comparisons with other methods and numerical examples are given.ZusammenfassungDie vorliegende Arbeit behandelt auf quasiinterpolierenden Splines basierende Produktquadraturformeln und beweist Konvergenzresultate für limitierte Integranden. Konvergenzresultate mit diesen quasi-interpolierenden Splines werden auch für Cauchysche Hauptwertintegralsequenzen bewiesen. Vergleiche mit anderen Methoden und numerische Beispiele werden angegeben.

An algorithm for numerical integration based on quasi-interpolating splines

In this paper product quadratures based on quasi-interpolating splines are proposed for the numerical evaluation of integrals with anL1-kernel and of Cauchy Principal Value integrals.

Quasi-interpolatory splines based on Schoenberg points

By using the Schoenberg points as quasi-interpolatory points, we achieve both generality and economy in contrast to previous sets, which achieve either generality or economy, but not both. The price we pay is a more complicated theory and weaker error bounds, although the order of convergence is unchanged. Applications to numerical integration are given and numerical examples show that the accuracy achieved, using the Schoenberg points, is comparable to that using other sets.

The use of modified quasi-interpolatory splines for the solution of the Prandtl equation

Abstract Modified quasi-interpolatory splines are used for the numerical solution of the generalized Prandtl equation. A Nystrom type method is applied, based on inserting in the integral equation a modified quasi-interpolatory spline instead of the unknown function. The integral equation can then be solved by collocation and evaluation of suitable Cauchy principal value integrals. Necessary conditions have been established for demonstrating that the approximate solution of the equation converges to the true solution.

Uniform convergence for cauchy principal value integrals of modified quasi-interpolatory splines ∗

A sequence of modified quasi-interpolatory splines is introduced for numerical evaluation of Cauchy principal value integrals. Uniform convergence is proved, for a sequence of locally uniform partitions of the integration interval.

Cubature rule associated with a discrete blending sum of quadratic spline quasi-interpolants

A new cubature rule for a parallelepiped domain is defined by integrating a discrete blending sum of C^1 quadratic spline quasi-interpolants in one and two variables. We give the weights and the nodes of this cubature rule and we study the associated error estimates for smooth functions. We compare our method with cubature rules based on the tensor products of spline quadratures and classical composite Simpsons rules.

Spline approximation with interpolation constraints for numerical integration

By using a Walsh-type theorem, we impose a finite number of interpolation constraints to a polynomial spline approximation operator. These constrained approximating splines reproduce polynomials and give orders of convergence identical to those of the unconstrained ones. We give an application in numerical integration by using the constrained splines for evaluation of Cauchy principal value integrals.

A method for uniform approximation under interpolation constraints

A method for uniform approximation of a real function f(x) under a finite number of auxiliary interpolation conditions is proposed. The method is obtained by introducing the Shepard basis functions in a well-known Walsh theorem. The approximation properties for f(x) and for its derivatives, whenever they exist, are given. Two meaningful applications are provided: the first is in the area of numerical methods for evaluating Cauchy principal value intergals; the second is obtained by imposing a finite number of interpolation constraints to a Bernstein polynomial.

Martensen splines and finite-part integrals

We state a uniform convergence theorem for finite-part integrals which are derivatives of weighted Cauchy principal value integrals. We prove that a sequence of Martensen splines, based on locally uniform meshes, satisfies the sufficient conditions required by the theorem. We construct the quadrature rules based on such splines and illustrate their behaviour by presenting some numerical results and comparisons with composite midpoint, Simpson and Newton-Cotes rules.

Smoothness and error bounds of Martensen splines

Martensen splines M f of degree n interpolate f and its derivatives up to the order n - 1 at a subset of the knots of the spline space, have local support and exactly reproduce both polynomials and splines of degree ? n . An approximation error estimate has been provided for f ? C n + 1 .This paper aims to clarify how well the Martensen splines M f approximate smooth functions on compact intervals. Assuming that f ? C n - 1 , approximation error estimates are provided for D j f , j = 0 , 1 , ? , n - 1 , where D j is the j th derivative operator. Moreover, a set of sufficient conditions on the sequence of meshes are derived for the uniform convergence of D j M f to D j f , for j = 0 , 1 , ? , n - 1 .