A Moment Problem for Discrete Nonpositive Measures on a Finite Interval

We will estimate the upper and the lower bounds of the integral , where runs over all discrete measures, positive on some cones of generalized convex functions, and satisfying certain moment conditions with respect to a given Chebyshev system. Then we apply these estimations to find the error of optimal shape-preserving interpolation. 1. Introduction Let be a Chebyshev system on [0,？1]. A function , defined on [0,？1], is said to be convex relative to the system (we will write ) if for all choices of . In particular, if , then is a cone of all increasing functions on (0,？1). If , , then is a cone of all convex functions on (0,？1). The review of some results of the theory of generalized convex functions can be found in the book in [1]. Let , with , . As usual, denotes the set of real numbers, and denotes the vector space of all real -tuples (columns). Denote by the set of all continuous functions defined on [0,？1] and convex relative to the system , that is, Denote . Following ideas of [2] we consider the cone For example, if , , , , then is the cone of all positive and convex continuous functions defined on [0,？1]. Let , and denote , . Let Denote by the dual cone. Let be a Chebyshev system on [0,？1]. Let us consider the moment space with respect to the system defined by where runs over . Given , denote In this paper we find the lower and upper bound of the value , where . This problem is similar to the classical moment problem (see, e.g., [1, Chapter？？2] and [3, Chapter？？4]), but the measure we are interested in is discrete and positive on some cones of generalized convex functions. The main result of this paper can be stated as follows. Theorem 1.1. Let be an internal point of , and let be such that and are nonempty sets, then where Note that the motivation of consideration of the problems has arisen from the theory of shape-preserving approximation. As we will show in Section 3, the estimation of the error of optimal recovery by means of shape-preserving algorithms can be reduced to the problems of type (1.11). 2. Duality Theorems and the Proof of Theorem 1.1 First we consider a conic programming problem, and we prove weak and strong duality theorems relative to this problem. Let , , , , . Consider the problem It follows from [4], that the dual problem can be written in the following way: Lemma 2.1. The set is a nonempty, convex, closed set. Proof. It is clear that is a convex set. Moreover, since the origin of belongs to , the set is nonempty. To show that is closed, suppose that is a sequence in , such that . Our goal is to show that . Consider the