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International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 545780, 8 pages
http://dx.doi.org/10.1155/2011/545780
Research Article

A Moment Problem for Discrete Nonpositive Measures on a Finite Interval

1Department of Mechanics and Mathematics, Saratov State University, Astrakhanskaya 83, Saratov 410060, Russia
2School of Information Systems, Computing and Mathematics, Brunel University, Uxbridge UB8 3PH, UK

Received 29 December 2010; Revised 16 March 2011; Accepted 22 March 2011

Academic Editor: Palle E. Jorgensen

Copyright © 2011 M. U. Kalmykov and S. P. Sidorov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 optimization problem where defined by , .
It can be rewritten as follows: where minimum is taken over all such that Note that if and only if , where runs over all extreme rays of the cone . Thus, the set of all satisfying inequalities (2.5), (2.6) is a nonlinear polyhedron. It is obvious that there is an optimal solution such that . Assume that . Since , there is an index such that , where . Let be such that . It implies that is a feasible solution of the system (2.5), (2.6). It contradicts that is optimal. Thus, we have which implies , and therefore .

Lemma 2.2. Let , . Only one of the following sets is not empty:

Proof. Assume the opposite, that is, there exist and which belong to the sets (2.7) and (2.8), respectively. It follows from and that . This contradicts to . Hence, we conclude that at most one of (2.7) or (2.8) is not empty.
Now, it remains to show that if (2.7) is empty, then (2.8) is not. Consider the nonempty closed and convex set . Since (2.7) is empty, we have . It follows from the separating hyperplane theorem that there exists such that for all . As , . Since , the definition of set (2.7) implies .

Lemma 2.3. Suppose the feasible sets and of problems (2.1) and (2.2) are both not empty. Let be the optimal solution of (2.1) and the optimal solution of (2.2). Then .

Proof. The proposition follows from where , , and by definition.

Theorem 2.4 (strong duality theorem). If the problem (2.1) has an optimal solution , then the problem (2.2) also has an optimal solution and

Proof. Assume that the feasible set of the problem (2.1) is not empty, and denote by the optimal solution of the problem (2.1). Let us show that the set of all , , , satisfying is empty.
Assume that satisfies the system (2.11) and . Then is a solution of (2.1) and , which contradicts to the optimality of . On the other hand, if , then is a solution of (2.1), and , which contradicts to the optimally of .
Now, it follows from [4]that there is such that and . It implies that is a solution of (2.2). Moreover, it follows from Lemma 2.2 that .

Now we are ready to prove Theorem 1.1. Note that the set defined in Section 1 is a closed convex cone. Let be arbitrary points in [0, 1]. Since is a Chebyshev system, we may conclude that points are linearly independent. Thus, the cone is not contained in any -dimensional subspaces.

Proof of Theorem 1.1. We will prove (1.9). Consider the conic programming problem Denote The dual problem of the problem (2.13) is the problem It follows from Lemma 2.3 that Equality (1.9) follows from the equality Equality (1.8) can be proved similarly.

3. The Error of Optimal Interpolation by Means of Shape Preserving Algorithms

Let , , . Let denote the class of all linear algorithms based on information . The error of the problem of optimal linear interpolation of at point on the basis of information , , is defined by Note that for every there exists such that for all . Then Optimal recovery problems arise in many applications of the approximation theory and have received much attention. In-depth study can be found in papers [5, 6], and in book in [7].

In various applications it is necessary to approximate a function preserving properties such as monotonicity, convexity, and concavity. In the theory of shape-preserving approximation by means of polynomials and splines the last 25 years have seen extensive research. The most significant results were summarized in [8, 9].

If a function has some shape properties, then it usually means that the element belongs to a cone in .

One of the tasks of the theory of shape-preserving approximation is to estimate value (3.1), where infimum is taken over all linear algorithms, which are satisfied additional (shape-preserving) properties.

Let be a cone in . Let denote the class of all linear algorithms , based on information and such that for all , .

Define by the error of optimal linear interpolation of at point on the basis of information , , with respect to the cone .

Denote by the cone dual to . Note that for every there exists such that for all . Then The next proposition demonstrates how we can use Theorem 1.1 to obtain the error of optimal linear interpolation.

We will consider the case , , , . Then is the cone of all positive and convex functions on , and, .

In the next proposition we consider the problem of interpolation by means of shape-preserving algorithms , which have some properties of shape-preserving projections (i.e., for every from a certain finite-dimensional subspace). Note that a deep study of linear shape preserving projections was undertaken in papers [1012].

Corollary 3.1. Let be a strictly convex function on , , and let be such that . Denote Then where denotes the divided difference of at .

Proof. Consider the problem It follows from Theorem 1.1 that where maximum is taken over all such that .
It follows from and strictl convexity of that Consider the problem It follows from Theorem 1.1 that where maximum is taken over all such that .
It follows from and strictl convexity of that Now (3.6) follows from (3.8), (3.9), (3.11), and (3.12).

Acknowledgments

This work is supported by RFBR (Grant 10-01-00270) and the President of the Russian Federation (NS-4383.2010.1). The authors thank the referees for many kind suggestions and comments.

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