Monotone cubic interpolation

About: Monotone cubic interpolation is a research topic. Over the lifetime, 1740 publications have been published within this topic receiving 38111 citations.

A reliable modification of the cross rule for rational Hermite interpolation

Abstract: Claessens' cross rule [8] enables simple computation of the values of the rational interpolation table if the table is normal, i.e. if the denominators in the cross rule are non-zero. In the exceptional case of a vanishing denominator a singular block is detected having certain structural properties so that some values are known without further computations. Nevertheless there remain entries which cannot be determined using only the cross rule.

Cardinal hermite interpolation with box splines

Abstract: The study of cardinal interpolation (CIP) by the span of the lattice translates of a box spline has met with limited success. Only the case of interpolation with the box spline determined by the three directionsd 1=(1, 0),d 2=(0, 1), andd 3=(1, 1) inR 2 has been treated in full generality [2]. In the case ofR d,d ≥ 3, the directions that define the box spline must satisfy a certain determinant condition [6], [9]. If the directions occur with even multiplicities, then this condition is also sufficient. For Hermite interpolation (CHIP) both even multiplicities and the determinant condition for the directions does not prevent the linear dependence of the basis functions. This leads to singularities in the characteristic multiplier when using the standard Fourier transform method. In the case of derivatives in one direction, these singularities can be removed and a set of fundamental splines can be given. This gives the existence of a solution to CHIP inL p (R d) for data inl p (Z d), 1≤p≤2.

An unusual cubic representation problem

Abstract: For a non-zero integer N , we consider the problem of finding 3 integers (a, b, c) such that N = a b+ c + b c+ a + c a+ b . We show that the existence of solutions is related to points of infinite order on a family of elliptic curves. We discuss strictly positive solutions and prove the surprising fact that such solutions do not exist for N odd, even though there may exist solutions with one of a, b, c negative. We also show that, where a strictly positive solution does exist, it can be of enormous size (trillions of digits, even in the range we consider).

Improved Approximation Order of Surface Spline Interpolation

Abstract: ABSTRACT The purpose of this paper is to study the Lp-approximation order of interpolation using a Surface Splinesφ for 1 ≤ p ≤ ∞. The current theories of this approach provide optimal erro r bounds when the approximand f is in a certain reproducing kernel Hilbert space [1–3]. Howe ver, we are particularly interested in approximating to functions f whose derivatives satisfy certain Lipschitz continuities. It turns out to provide more accura te theoretical results than the currently known estimate. Some numerical results are presented.

The Fractional Cubic Spline Interpolation without Using the Derivative Values

Abstract: The paper introduces a function value based fraction of cubic spline interpolation, which is used for studying the curves and surfaces. The interpolation function has a simple and explicit mathematical representation, convenient both in practical application and in theoretical studies. It should be mentioned that the interpolating surfaces are C 1 in the interpolating region under the condition that the interpolation is only based on the function values. Moreover, properties and views are shown in matrix notation, and then the error is calculated.