On Monotone and Convex Spline Interpolation

Abstract

This paper is concerned with the problem of existence of monotone and/or convex splines, having degree n and order of continuity k, which interpolate to a set of data at the knots.The interpolating splines are obtained by using Bernstein polynomials of suitable continuous piecewise linear functions; they satisfy the inequality k < n -k.The theorems presented here are useful in developing algorithms for the construction of shape-preserving splines interpolating arbitrary sets of data points.Earlier results of McAllister, Passow and Roulier can be deduced from those given in this paper.1. Introduction.Let x0 < xx < < xN be real numbers, let and k, 0 ^ k < -1, be integers, and let Snk[x0,xN) = [s e Ck [x0,xN]: s(x) e P", x e [x,,x, + 1], i = 0,1,..., N -l) be the set of splines of degree and deficiency -k at knots x, i = 1,..., N -1.In a previous paper [2], a necessary and sufficient condition for the existence of monotone and convex s e 53[x0, x^] interpolating a set of data at the knots was given, and in [3] an algorithm for the construction of shape-preserving interpolating splines was described.Further investigations led to an extension of the above results to se Sk [x0, xN], where 1 < k < -k and s(j)(xi) = 0, j' -2,..., k, i = 0,1,..., N, whenever k^-2.Such splines are obtained from Bernstein polynomials of monotone and/or convex interpolating linear splines with two knots in (x,, x, + 1), i = 0,..., N -1.The well-known theorems of [9] and [7] (see also [8] for related results), where linear splines with one knot in (x,, x, + 1), i = 0,...,N -1, are used, can be deduced from those presented here.This paper is divided into three parts.In Section 2, and k are prescribed and a data-dependent condition is given.In Section 3 the degree is computed to ensure the existence of a monotone and/or convex interpolating spline, and a degree-dependent condition is provided.Section 4 is then devoted to final conclusions and remarks.The results of Sections 2 and 3 enable us to derive an algorithm for the construction of shape-preserving interpolating s e Sk [x0,xN].The spline can be selected by the user to perform Lagrange or Hermite interpolation, to be piecewise monotone and/or convex (concave), and k can be assigned or automatically computed by the code, according to Theorem 3. A detailed description of this algorithm, together with numerical and graphical examples, will be reported elsewhere