Shape Preserving C Spline Interpolation

In this paper we summarize the main results of where an algo rithm of shape preserving C spline interpolation for arbitrary D discrete data is developed We consider a classi cation of such data to separate the sec tions of linearity the angles and the breaks For remaining data we give a local algorithm of C interpolation by generalized splines with automatic choice of the parameters to retain the monotonicity and convexity properties of the data x Introduction It is well known that polynomial splines generally do not retain the geometric prop erties of the given data To obtain the necessary solution many authors introduce some parameters in the structure of the spline Then they choose these parameters in such a way to satisfy the geometric constraints The key idea here is to develop algorithms for automatic selection of these parameters This paper de nes a class of functions I V having shape properties determined by a given set of points V fPi xi fi IR x x xNg Based on the de nition necessary and su cient inequality conditions on V are given in order that I V be non empty A local algorithm for covex and monotone interpolation by C generalized splines with automatic choice of the parameters is obtained Its application enables us to give a complete solution to the shape preserving inter polation problem for D data of arbitrary form and to isolate the sections of linearity the angles and the breaks x The Class of Shape Preserving Interpolants Let the sequence of points V fPi j i Ng Pi xi fi on the plane IR be xed where a x x xN b forms a partition of the interval a b We introduce the notation for the rst two devided di erences if fi fi hi hi xi xi i N if if i f i N As usual we shall say that the initial data increases monotonically On leave from Institute of Computational Technologies Russian Academy of Sciences Novosibirsk Russia