The linear inequality method for rational approximation with interpolation

The linear inequality method for rational minimax approximation is extended to cover approximation under Lagrange-type interpolatory constraints.     

On rational approximations with denominators from thin sets

Given a thin setA we construct a Liouville number α such that for |α −u/υ|, υ ∈A, a nontrivial lower bound can be given.     

The Rounded Cubic Spline

A modification to the conventional spline interpolation methodsis suggested. The new method results in a function which iscontinuous in all higher-order derivatives. This results usuallyin a higher precision as is shown in many examples. The disadvantages are, however, the need of a high precisioncomputer and also the extra computer time involved.     

Pair correlation of roots of rational functions with rational generating functions and quadratic denominators

For any rational functions with complex coefficients A(z),B(z), and C(z), where A(z), C(z) are not identically zero, we consider the sequence of rational functions H _m (z) with generating function ∑H _m (z)t ^m =1/(A(z)t ²+B(z)t+C(z)). We provide an explicit formula for the limiting pair correlation function of the roots of $\prod_{m=0}^{n}H_{m}(z)$ , as n→∞, counting multiplicities, on certain closed subarcs J of a curve $\mathcal{C}$ where the roots lie. We give an example where the limiting pair correlation function does not exist if J contains the endpoints of $\mathcal{C}$ .     

Positive interpolation with rational splines

A method is presented for the construction of positive rational splines of continuity classC ².     

On rational approximations with denominators from thin sets, ii

In the general case a real irrational number cannot be approximated by infinitely many rationalsp/q involving error terms less than q^-2 when the denominatorsq are taken from a given thin set of positive integers. The distribution of irrationals which are situated in close neighborhoods of infinitely many fractionsp/q, whereq is restricted to the elements of a thin set, depends on the asymptotic behaviour of theq’s and on their arithmetic properties.     

Range restricted interpolation using Gregory's rational cubic splines

The construction of range restricted univariate and bivariate interpolants to gridded data is considered. We apply Gregory's rational cubic C¹ splines as well as related rational quintic C² splines. Assume that the lower and upper obstacles are compatible with the data set. Then the tension parameters occurring in the mentioned spline classes can be always determined in such a way that range restricted interpolation is successful.     

Cubic spline interpolation with quasiminimal B-spline coefficients

Summary The end conditions for cubic spline interpolation with equidistant knots will be defined so as to make the (slightly modified) B-spline coefficients minimal. This produces good approximation results as compared e.g. with the not-a-knot spline.     

Chebyshev rational interpolation

The Lanczos method and its variants can be used to solve efficiently the rational interpolation problem. In this paper we present a suitable fast modification of a general look-ahead version of the Lanczos process in order to deal with polynomials expressed in the Chebyshev orthogonal basis. The proposed approach is particularly suited for approximating analytic functions by means of rational interpolation at certain nodes located on the boundary of an elliptical region of the complex plane. In fact, in this case it overcomes some of the numerical difficulties which limited the applicability of the look-ahead Lanczos process for determining the coefficients both of the numerators and of the denominators with respect to the standard power basis. This revised version was published online in August 2006 with corrections to the Cover Date.     

Spline interpolation at knot averages

It is well known that when interpolation points coincide with knots, the knot sequence must obey some restriction in order to guarantee the existence and boundedness of the interpolation projector. But, when the interpolation points are chosen to be the knot averages, the corresponding quadratic or cubic spline interpolation projectors are bounded independently of the knot sequence. Based on this fact, de Boor in 1975 made a conjecture that interpolation by splines of orderk at knot averages is bounded for anyk. In this paper we disprove de Boor's conjecture fork 20.Communicated by Wolfgang Dahmen.     

保单调的有理三次样条插值

构造了一种C^1连续的保单调的有理三次插值函数。由于函数表达式中含有调节参数，使得插值曲线更具灵活性。     

Constrained interpolation and smoothing

Numerical and theoretical questions related to constrained interpolation and smoothing are treated. The prototype problem is that of finding the smoothest convex interpolant to given univariate data. Recent results have shown that this convex programming problem with infinite constraints can be recast as a finite parametric nonlinear system whose solution is closely related to the second derivative of the desired interpolating function. This paper focuses on the analysis of numerical techniques for solving the nonlinear system and on the theoretical issues that arise when certain extensions of the problem are considered. In particular, we show that two standard iteration techniques, the Jacobi and Gauss-Seidel methods, are globally convergent when applied to this problem. In addition we use the problem structure to develop an efficient implementation of Newton's method and observe consistent quadratic convergence. We also develop a theory for the existence, uniqueness, and representation of solutions to the convex interpolation problem with nonzero lower bounds on the second derivative (strict convexity). Finally, a smoothing spline analogue to the convex interpolation problem is studied with reference to the computation of convex approximations to noisy data.     

Bayesian Free-Knot Monotone Cubic Spline Regression

This article proposes a new Bayesian approach for monotone curve fitting based on the isotonic regression model. The unknown monotone regression function is approximated by a cubic spline and the constraints are represented by the intersection of quadratic cones. We treat the number and locations of knots as free parameters and use reversible jump Markov chain Monte Carlo to obtain posterior samples of knot configurations. Given the number and locations of the knots, second-order cone programming is used to estimate the remaining parameters. Simulation results suggest the method performs well and we illustrate the approach using the ASA car data.     

On the denominators of rational points on elliptic curves

Let x(P) = A_P/B²_P denote the x-coordinate of the rational pointP on an elliptic curve in Weierstrass form. We consider whenB_P can be a perfect power or a prime. Using Faltings' theorem,we show that for a fixed f > 1, there are only finitely manyrational points P with B_P equal to an fth power. Where descentvia an isogeny is possible, we show that there are only finitelymany rational points P with B_P equal to a prime, that thesepoints are bounded in number in an explicit fashion, and thatthey are effectively computable. Finally, we prove a strongerversion of this result for curves in homogeneous form.     

The Newton differential correction algorithm for rational Chebyshev approximation with constrained denominators

An algorithm for constrained rational Chebyshev approximation is introduced that combines the idea of an algorithm due to Hettich and Zencke, for which superlinear convergence is guaranteed, with the auxiliary problem used in the well-known original differential correction method. Superlinear convergence of the algorithm is proved. Numerical examples illustrate the fast convergence of the method and its advantages compared with the algorithm of Hettich and Zencke.     

Approximation by rational functions with polynomials of positive coefficients as the denominators

The present paper proves that if f(x) ∈ C[0,1], changes its sign exactly l times at 0 ＜ y1 ＜ y2 ＜y1＜1 in(0,1),then there exists a pn (x)пn( ),such that |f(x)-p(x)/pn(x)|≤ Cωφ(f,n-1/2),where ρ(x) is defined by ρ(x)=l∏i=1(x - yi), if f(x) ≥ 0 for x ∈ (yl, 1),-1∏i=1(x-yi), iff(x) ＜ 0 for x ∈ (y1,1).which improves and generalizes the result of [7].