First applications of a formula for the error of finite sinc interpolation

In former articles we have given a formula for the error committed when interpolating a several times differentiable function by the sinc interpolant on a fixed finite interval. In the present work we demonstrate the relevance of the formula through several applications: correction of the interpolant through the insertion of derivatives to increase its order of convergence, improvement of the barycentric formula, rational sinc interpolants (with and without replacement of the (usually unknown) derivatives with finite differences), convergence acceleration through extrapolation and improvement of one-sided interpolant

A formula for the error of finite sinc-interpolation over a finite interval

Sinc-interpolation is a very efficient infinitely differentiable approximation scheme from equidistant data on the infinite line. It, however, requires that the interpolated function decreases rapidly or is periodic. We give an error formula for the case where neither of these conditions is satisfie

A formula for the error of finite sinc interpolation with an even number of nodes

Sinc interpolation is a very efficient infinitely differentiable approximation scheme from equidistant data on the infinite line. We give a formula for the error committed when the function neither decreases rapidly nor is periodic, so that the sinc series must be truncated for practical purposes. To do so, we first complete a previous result for an odd number of points, before deriving a formula for the more involved case of an even number of point

A formula for optimal integration in H2

AbstractThe weights âj of the optimal integration formula Q̂ = Σjâjf(zj) in H2 for given integration points zj are the exact integrals of the cardinal functions in the corresponding formula for optimal evaluation. By writing these cardinal functions as sums of their principal values, we very easily obtain a closed formula for the weights. In the case of real zj's, this formula makes explicit a series formula of Wilf. We compare numerically the accuracy of the optimal formula with that of some well-known integration formulae. For points equidistant on a circle of radius r, the formula allows an alternate derivation of a formula obtained by Golomb. We give also the barycentric formula for optimal evaluation with these points, as well as an experimentally stable sequence of radii r for integrating with an increasing number of points

A circular interpretation of the Euler–Maclaurin formula

The present work makes the case for viewing the Euler–Maclaurin formula as an expression for the effect of a jump on the accuracy of Riemann sums on circles and draws some consequences thereof, e.g., when the integrand has several jumps. On the way we give a construction of the Bernoulli polynomials tailored to the proof of the formula and we show how extra jumps may lead to a smaller quadrature error

Linear rational finite differences from derivatives of barycentric rational interpolants

Derivatives of polynomial interpolants lead in a natural way to approximations of derivatives of the interpolated function, e.g., through finite differences. We extend a study of the approximation of derivatives of linear barycentric rational interpolants and present improved finite difference formulas arising from these interpolants. The formulas contain the classical finite differences as a special case and are more stable for calculating one-sided derivatives as well as derivatives close to boundaries

Linear barycentric rational quadrature

Linear interpolation schemes very naturally lead to quadrature rules. Introduced in the eighties, linear barycentric rational interpolation has recently experienced a boost with the presentation of new weights by Floater and Hormann. The corresponding interpolants converge in principle with arbitrary high order of precision. In the present paper we employ them to construct two linear rational quadrature rules. The weights of the first are obtained through the direct numerical integration of the Lagrange fundamental rational functions; the other rule, based on the solution of a simple boundary value problem, yields an approximation of an antiderivative of the integrand. The convergence order in the first case is shown to be one unit larger than that of the interpolation, under some restrictions. We demonstrate the efficiency of both approaches with numerical test

Barycentric Lagrange Interpolation

Barycentric interpolation is a variant of Lagrange polynomial interpolation that is fast and stable. It deserves to be known as the standard method of polynomial interpolation.\ud \ud Dedicated to the memory of Peter Henrici (1923-1987

The linear barycentric rational method for a class of delay Volterra integro-differential equations

A method for solving delay Volterra integro-differential equations is introduced. It is based on two applications of linear barycentric rational interpolation, barycentric rational quadrature and barycentric rational finite differences. Its zero–stability and convergence are studied. Numerical tests demonstrate the excellent agreement of our implementation with the predicted convergence orders

The linear barycentric rational quadrature method for Volterra integral equations

We introduce two direct quadrature methods based on linear rational interpolation for solving general Volterra integral equations of the second kind. The first, deduced by a direct application of linear barycentric rational quadrature given in former work, is shown to converge at the same rate as the rational quadrature rule but is costly on long integration intervals. The second, based on a composite version of this quadrature rule, loses one order of convergence but is much cheaper. Both require only a sample of the involved functions at equispaced nodes and yield an infinitely smooth solution of most classical examples with machine precision