Shape Preserving Approximation by Polynomials

We are going to survey recent developments and achievements in shape preserving approximation by polynomials. We wish to approximate a function f deened on a nite interval, say ?1; 1], while preserving certain intrinsic \shape" properties. To be speciic we demand that the approximation process preserve properties of f , like its sign in all or part of the interval, its monotonicity, convexity, etc. We will refer to these properties as the shape of the function. x1. Introduction We are going to discuss the degree of constrained approximation of a function f in either the uniform norm or in the L p ?1; 1], norm 0 < p < 1, and we will use the notation L 1 ?1; 1] for C ?1; 1], whenever we state a result which is valid both for C ?1; 1] as well as for L p ?1; 1], for a proper range of p's. The degree of approximation will be measured by the appropriate (quasi-)norm which we denote by k k p. The approximation will be carried out by polynomials p n 2 n , the space of polynomials of degree not exceeding n, which have the same shape in which we are interested, as f, namely, have the same sign as f does in various parts of ?1; 1], or change their monotonicity or convexity exactly where f does in ?1; 1]. Most of the proofs of the statements in this survey and especially those of the aarmative results, are technically involved and will be omitted. All we are going to say about the technique of proof is that we usually rst approximate f well by splines or just continuous piecewise polynomials with the same shape as f, and then we replace the polynomial pieces by polynomials of the same shape. Thus, while this survey deals only with polynomial approximation, there are similar aarmative results for continuous