Richardson extrapolation of iterated discrete projection methods for eigenvalue approximation

In this paper, the eigenvalue approximation of a compact integral operator with a smooth kernel is discussed. We propose asymptotic error expansions of the iterated discrete Galerkin and iterated discrete collocation methods, and asymptotic error expansion of approximate eigenvalues. We then apply Richardson extrapolation to obtain higher order super-convergence of eigenvalue approximations. Numerical examples are presented to illustrate the theoretical estimate.     

LU implementation of the modified minimal polynomial extrapolation method for solving linear and nonlinear systems

We give an efficient implementation of the modified minimalpolynomial extrapolation (MMPE) method for solving linear andnonlinear systems. We will show how to choose the auxiliaryvectors in the MMPE method such that the resulting approximationsare always defined. This new implementation, which is basedon an LU factorization with a pivoting strategy, is inexpensiveboth in time and storage as compared with other extrapolationmethods.     

Optimum extrapolation and interpolation designs,I

Summary For regression problems where observations may be taken at points in a set X which does not coincide with the set Y on which the regression function is of interest, we consider the problem of finding a design (allocation of observations) which minimizes the maximum over Y of the variance function (of estimated regression). Specific examples are calculated for one-dimensional polynomial regression when Y is much smaller than or much larger than X. A related problem of optimum estimation of two regression coefficients is studied. This paper contains proofs of results first announced at the 1962 Minneapolis Meeting of the Institute of Mathematical Statistics. No prior knowledge of design theory is needed to read this paper. John Simon Guggenheim Memorial Foundation Fellow. Research supported in part by the office of Naval Research under Contract No. Nonr 266(04) (NR 047-005). The research of this author was supported in part by the U.S. Air Force under Contract No. AF 18(600)-685.     

A general extrapolation procedure revisited

TheE-algorithm is the most general extrapolation algorithm actually known. The aim of this paper is to provide a new approach to this algorithm. This approach gives a deeper insight into theE-algorithm, and allows one to obtain new properties and to relate it to other algorithms. Some extensions of the procedure are discussed.     

A note on extrapolation theory

We show some explicit computations on extrapolation of L^p spaces.     

On extrapolation of Jagerman and Stetter rules

In this paper quadrature rules introduced by Jagerman [1] and Stetter [2] are considered and asymptotic expansions for the error given. This allows to make use of the Romberg extrapolation process. Such rules can be viewed as generalizations of the well-known mid-point rule. Thus, numerical examples comparing these rules are finally presented.     

The Appell interpolation problem

A general linear interpolation problem is considered. We will call it the Appell interpolation problem because the solution can be expressed by a basis of Appell polynomials. Some classical and non-classical examples are also considered. Finally, numerical calculations are given.     

Linear interpolation and Sobolev orthogonality

There is a strong connection between Sobolev orthogonality and Simultaneous Best Approximation and Interpolation. In particular, we consider very general interpolatory constraints , defined by

where f belongs to a certain Sobolev space, a_ij() are piecewise continuous functions over [a,b], b_ijk are real numbers, and the points t_k belong to [a,b] (the nonnegative integer m depends on each concrete interpolation scheme). For each f in this Sobolev space and for each integer l greater than or equal to the number of constraints considered, we compute the unique best approximation of f in , denoted by p_f, which fulfills the interpolatory data , and also the condition that best approximates f⁽ⁿ⁾ in (with respect to the norm induced by the continuous part of the original discrete–continuous bilinear form considered).     

Asymptotic expansion and extrapolation for Bernstein polynomials with applications

Given a real functionf C ^2k [0,1],k 1 and the corresponding Bernstein polynomials {B _n (f)} _n we derive an asymptotic expansion formula forB _n (f). Then, by applying well-known extrapolation algorithms, we obtain new sequences of polynomials which have a faster convergence thanB _n (f). As a subclass of these sequences we recognize the linear combinations of Bernstein polynomials considered by Butzer, Frentiu, and May [2, 6, 9]. In addition we prove approximation theorems which extend previous results of Butzer and May. Finally we consider some applications to numerical differentiation and quadrature and we perform numerical experiments showing the effectiveness of the considered technique.This work was partially supported by a grant from MURST 40.     

Holomorphic semigroups in interpolation and extrapolation spaces

Communicated by R. Nagel     

Meromorphic extendibility and rigidity of interpolation

Let T be the unit circle, f be an α-Hölder continuous function on T, α>1/2, and A be the algebra of continuous function in the closed unit disk that are holomorphic in D. Then f extends to a meromorphic function in D with at most m poles if and only if the winding number of f+h on T is bigger or equal to −m for any h∈A such that f+h≠0 on T.     

Convergence of local variational spline interpolation

In this paper we first revisit a classical problem of computing variational splines. We propose to compute local variational splines in the sense that they are interpolatory splines which minimize the energy norm over a subinterval. We shall show that the error between local and global variational spline interpolants decays exponentially over a fixed subinterval as the support of the local variational spline increases. By piecing together these locally defined splines, one can obtain a very good C⁰ approximation of the global variational spline. Finally we generalize this idea to approximate global tensor product B-spline interpolatory surfaces.     

Index-doubling in sequences by Aitken extrapolation

Aitken extrapolation, applied to certain sequences, yields the even-numbered subsequence of the original. We prove that this is true for sequences generated by iterating a linear fractional transformation, and for some sequences of convergents of the regular continued fractions of certain quadratic irrational numbers.     

Smooth-curve interpolation: A generalized spline-fit procedure

A method is presented for finding the smoothest curve through a set of data points. Smoothest refers to the equilibrium, or minimum-energy configuration of an ideal elastic beam constrained to pass through the data points. The formulation of the smoothest curve is seen to involve a multivariable boundary-value minimization problem which makes use of a numerical solution of the beam non-linear differential equation. The method is shown to offer considerable improvement over the spline technique for smooth-curve interpolation.     

On the use of trigonometric approximations in extrapolation methods

Standard trigonometric interpolation formulae are modified for use in extrapolation methods. Some applications are indicated.     

Modularity results for interpolation,amalgamation and superamalgamation

Wolter in [38] proved that the Craig interpolation property transfers to fusion of normal modal logics. It is well-known [21] that for such logics Craig interpolation corresponds to an algebraic property called superamalgamability. In this paper, we develop model-theoretic techniques at the level of first-order theories in order to obtain general combination results transferring quantifier-free interpolation to unions of theories over non-disjoint signatures. Such results, once applied to equational theories sharing a common Boolean reduct, can be used to prove that superamalgamability is modular also in the non-normal case. We also state that, in this non-normal context, superamalgamability corresponds to a strong form of interpolation that we call “comprehensive interpolation property” (which consequently transfers to fusions).