Introduction to the Contents of Issue 49:3

Abstract

In this issue, we have happened to get several contributions in the Approximation and Numerical Quadrature area. In the first paper Rafael Campos and Fransisco Meijo Diaz develop quadrature formulas for discrete two sided Laplace transforms. These formulas give convergent approximations for functions that fall off to zero fast enough. An asymptotic formula yields an algorithm that computes a discrete Laplace transform using only exponentials. In another paper, Frances Kuo, Gregorz Wasilkowski and Henryk Woźniakowski investigate the power of algorithms to compute approximate values of a function in a reproducing Hilbert kernel space. They show that standard information, using n function values at randomly distributed points, is nearly as good as generalized information that uses n values of arbitrary linear functionals. In another paper, Paula Lamberti develops cubature formulas based on two dimensional B splines over an irregular triangulation. Two papers deal with how to find an approximation to a function given at irregularly distributed data points. A Lamni, Hamid Mraoui, Dress Sbibih and Ahmed Zidna approximate a function over a sphere by tensor product trigonometric splines or periodic algebraic trigonometric wavelets. The intention is to use such approximations for data compression. Francesca Mazzia and Alessandra Sestini develop a quasi interpolation scheme based on Hermite B splines of arbitrary degree. A small linear system is solved for each knot. For reasonably distributed knots, it can be used as a collocation method for solving a system of ordinary differential equations. In two related papers, it is shown how to develop higher order splitting methods for dissipative evolution equations. Complex time steps with positive real parts are used independently by Francois Castella, Philippe Chartier, Stephane Descombes