A Moment Problem for Discrete Nonpositive Measures on a Finite Interval

Abstract

We will estimate the upper and the lower bounds of the integral ∫¹₀Ω(𝑡)𝑑𝜇(𝑡), where 𝜇 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 {𝑢₀,…,𝑢_𝑘} be a Chebyshev system on [0, 1]. A function 𝑓, defined on [0, 1], is said to be convex relative to the system {𝑢₀,…,𝑢_𝑘} (we will write 𝑓∈𝐶(𝑢₀,…,𝑢_𝑘)) if 𝑢₀𝑡₀𝑢₀𝑡₁⋯𝑢₀𝑡_𝑘+1𝑢⋯⋯⋯⋯_𝑘𝑡₀𝑢_𝑘𝑡₁⋯𝑢_𝑘𝑡_𝑘+1𝑓𝑡₀𝑓𝑡₁⋯𝑓𝑡_𝑘+1≥0 (1.1) for all choices of 0<𝑡₀<𝑡₁<⋯<𝑡_𝑘+1<1.

In particular, if 𝑢₀(𝑥)≡1, then 𝐶(𝑢₀) is a cone of all increasing functions on (0, 1). If 𝑢₀(𝑥)≡1, 𝑢₁(𝑥)≡𝑥, then 𝐶(𝑢₀,𝑢₁) 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 𝑘≥0, 𝜎=(𝜎₀,…,𝜎_𝑘)∈ℝ^𝑘+1 with 𝜎_𝑖∈{−1,0,1}, 𝜎₀,𝜎_𝑘≠0. As usual, ℝ denotes the set of real numbers, and ℝ^𝑛 denotes the vector space of all real 𝑛-tuples (columns).

Denote by 𝑊_𝑙+1 the set of all continuous functions defined on [0, 1] and convex relative to the system {𝑢₀,…,𝑢_𝑙}, that is,𝑊_𝑙+1∶=𝑓∈ℂ[0,1]∶𝑓∈𝐶𝑢₀,…,𝑢_𝑙,𝑙=0,…,𝑘−1. (1.2) Denote 𝑊₀∶={𝑓∈ℂ[0,1]∶𝑓≥0}. Following ideas of [2] we consider the cone𝑊_0,𝑘(𝜎)=^𝑘_𝑙=0𝜎_𝑙𝑊_𝑙. (1.3) For example, if 𝑘=2, 𝜎=(1,0,1), 𝑢₀(𝑥)≡1, 𝑢₁(𝑥)≡𝑥, then 𝑊_0,2(𝜎) is the cone of all positive and convex continuous functions defined on [0, 1].

Let 0≤𝑥₁<𝑥₂<⋯<𝑥_𝑛≤1, and denote 𝐼𝑔=(𝑔(𝑥₁),…,𝑔(𝑥_𝑛))^𝑇∈ℝ^𝑛, 𝑔∈ℂ[0,1]. Let𝑉_0,𝑘(𝜎)∶=𝐼𝑓∈ℝ^𝑛∶𝑓∈𝑊_0,𝑘(𝜎). (1.4) Denote by𝑉^∗_0,𝑘(𝜎)∶=𝜇∈ℝ^𝑛∶𝐼𝑓^𝑇𝜇≥0∀𝐼𝑓∈𝑉_0,𝑘(𝜎) (1.5) the dual cone.

Let {𝑓₀,…,𝑓_𝑝} be a Chebyshev system on [0, 1]. Let us consider the moment space with respect to the system {𝑓₀,…,𝑓_𝑝} defined by𝑀_𝑝+1,𝑘(𝜎)∶=𝑐=𝑐₀,…,𝑐_𝑝∈ℝ^𝑝+1∶𝐼𝑓_𝑖^𝑇𝜇=𝑐_𝑖,𝑖=0,…,𝑝, (1.6) where 𝜇 runs over 𝑉^∗_0,𝑘(𝜎).

Given 𝑐⁰=(𝑐⁰₀,𝑐⁰₁,…,𝑐⁰_𝑝)∈𝑀_𝑝+1,𝑘(𝜎), denote𝐾_0,𝑘𝑐⁰=𝜇∈𝑉^∗_0,𝑘(𝜎)∶𝐼𝑓_𝑖^𝑇𝜇=𝑐⁰_𝑖,𝑖=0,1,…,𝑝. (1.7) In this paper we find the lower and upper bound of the value (𝐼𝑓)^𝑇𝜇, where 𝜇∈𝐾_0,𝑘(𝑐⁰). 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 𝑐⁰ be an internal point of 𝑀_𝑝+1,𝑘(𝜎), and let 𝑓∈ℂ[0,1] be such that 𝑃₊ and 𝑃₋ are nonempty sets, then sup_{𝜇∈𝐾0,𝑘(𝑐0)}𝐼𝑓^𝑇𝜇=inf_{𝑔∈𝑃−}𝑔𝑐⁰,inf_{𝜇∈𝐾0,𝑘(𝑐0)}𝐼𝑓^𝑇𝜇=sup_𝑔∈𝑃+𝑔𝑐⁰, (1.8)(1.9) where 𝑃₊=𝑔∈span𝑓₀,…,𝑓_𝑝∶𝐼𝑔−𝑓∈𝑉_0,𝑘(𝜎,𝑃₋)=𝑔∈span𝑓₀,…,𝑓_𝑝∶𝐼𝑓−𝑔∈𝑉_0,𝑘(𝜎). (1.10)

Note that the motivation of consideration of the problemssup_{𝜇∈𝐾0,𝑘(𝑐0)}𝐼𝑓^𝑇𝜇,inf_{𝜇∈𝐾0,𝑘𝑐0}𝐼𝑓^𝑇𝜇 (1.11) 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 𝐴∈ℝ^𝑚×𝑛, 𝑏∈ℝ^𝑚, 𝑐∈ℝ^𝑛, 𝑚<𝑛, 𝑚,𝑛∈ℕ.

Consider the problemmin_𝑥∈𝑀𝑐^𝑇𝑥,where𝑀∶=𝑥∈ℝ^𝑛∶𝐴𝑥=𝑏,𝑥∈𝑉^∗_0,𝑘(𝜎). (2.1) It follows from [4], that the dual problem can be written in the following way:max_{𝑦∈𝑀∗}𝑏^𝑇𝑦,where𝑀^∗∶=𝑦∈ℝ^𝑚∶𝐴^𝑇𝑦+𝑠=𝑐,𝑠∈𝑉_0,𝑘(𝜎). (2.2)

Lemma 2.1. The set 𝑄∶={𝐴𝑥∶𝑥∈𝑉^∗_0,𝑘(𝜎)} is a nonempty, convex, closed set.

Proof. It is clear that 𝑄 is a convex set. Moreover, since the origin of ℝ^𝑛 belongs to 𝑄, the set 𝑄 is nonempty. To show that 𝑄 is closed, suppose that 𝑞^𝑘 is a sequence in 𝑄, such that 𝑞^𝑘→𝑞. Our goal is to show that 𝑞∈𝑄.
Consider the optimization problem min_{𝑥∈𝑉∗0,𝑘(𝜎)}𝑞−𝐴𝑥_∞, (2.3) where ‖⋅‖_∞ defined by ‖𝑎‖_∞=max_𝑖|𝑎_𝑖|, 𝑎=(𝑎_𝑖)^𝑛_𝑖=1∈ℝ^𝑛.
It can be rewritten as follows: min𝑡, (2.4) where minimum is taken over all 𝑥∈𝑉^∗_0,𝑘(𝜎) such that 𝑡≥𝑞_𝑖−𝐴𝑥^𝑇_𝑖,𝑖=1,…,𝑛,−𝑡≥−𝑞_𝑖+𝐴𝑥^𝑇_𝑖,𝑖=1,…,𝑛. (2.5)(2.6) Note that 𝑥∈𝑉^∗_0,𝑘(𝜎) if and only if 𝑥^𝑇𝑣^∗≥0, where 𝑣^∗ runs over all extreme rays of the cone 𝑉_0,𝑘(𝜎). Thus, the set of all 𝑥∈𝑉^∗_0,𝑘(𝜎) satisfying inequalities (2.5), (2.6) is a nonlinear polyhedron. It is obvious that there is an optimal solution (𝑥^∗,𝑡^∗) such that 𝑡^∗≥0. Assume that 𝑡^∗>0. Since 𝑞^𝑘→𝑞, there is an index 𝑘′ such that ‖𝑞−𝑞^𝑘′‖_∞=𝑡′<𝑡^∗, where 𝑞^𝑘′∈𝑄. Let 𝑥′∈𝑉^∗_0,𝑘(𝜎) be such that 𝑞^𝑘′=𝐴𝑥′. It implies that (𝑥′,𝑡′) is a feasible solution of the system (2.5), (2.6). It contradicts that (𝑥^∗,𝑡^∗) is optimal. Thus, we have 𝑡^∗=0 which implies 𝑞=𝐴𝑥^∗, and therefore 𝑞∈𝑄.

Lemma 2.2. Let 𝐴∈ℝ^𝑚×𝑛, 𝑏∈ℝ^𝑚. Only one of the following sets is not empty: 𝑥∈ℝ^𝑛∶𝐴𝑥=𝑏,𝑥∈𝑉^∗_0,𝑘(𝜎),𝑦∈ℝ^𝑚∶−𝐴^𝑇𝑦∈𝑉_0,𝑘(𝜎),𝑏^𝑇𝑦>0. (2.7)(2.8)

Proof. Assume the opposite, that is, there exist 𝑥_∗∈ℝ^𝑛 and 𝑦_∗∈ℝ^𝑚 which belong to the sets (2.7) and (2.8), respectively. It follows from −𝐴^𝑇𝑦_∗∈𝑉_0,𝑘(𝜎) and 𝑥_∗∈𝑉^∗_0,𝑘(𝜎) that 0≥(𝐴^𝑇𝑦_∗)^𝑇𝑥_∗=(𝑦^𝑇_∗𝐴)𝑥_∗=𝑦^𝑇_∗(𝐴𝑥_∗)=𝑦^𝑇_∗𝑏=𝑏^𝑇𝑦_∗. This contradicts to 𝑏^𝑇𝑦_∗>0. 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 𝑄={𝐴𝑥,𝑥∈𝑉^∗_0,𝑘(𝜎)}. Since (2.7) is empty, we have 𝑏∉𝑄. It follows from the separating hyperplane theorem that there exists 𝑦∈ℝ^𝑚 such that 𝑦^𝑇(𝐴𝑥)<𝑦^𝑇𝑏 for all 𝑥∈𝑉^∗_0,𝑘(𝜎). As 0∈𝑄, 𝑏^𝑇𝑦>0. Since 𝑏^𝑇𝑦>0, the definition of set (2.7) implies −𝐴^𝑇𝑦∈𝑉_0,𝑘(𝜎).

Lemma 2.3. Suppose the feasible sets 𝑀 and 𝑀^∗ of problems (2.1) and (2.2) are both not empty. Let 𝑥_∗∈ℝ^𝑛 be the optimal solution of (2.1) and 𝑦_∗∈ℝ^𝑚 the optimal solution of (2.2). Then 𝑏^𝑇𝑦_∗≤𝑐^𝑇𝑥_∗.

Proof. The proposition follows from 𝑏^𝑇𝑦_∗=𝐴𝑥_∗^𝑇𝑦_∗=𝑥^𝑇_∗𝐴^𝑇𝑦_∗≤𝑥^𝑇_∗𝐴^𝑇𝑦_∗+𝑠=𝑥^𝑇_∗𝑐, (2.9) where 𝑠∈𝑉_0,𝑘(𝜎), 𝑥_∗∈𝑉^∗_0,𝑘(𝜎), and 𝐴^𝑇𝑦_∗+𝑠=𝑐 by definition.

Theorem 2.4 (strong duality theorem). If the problem (2.1) has an optimal solution 𝑥_∗∈ℝ^𝑛, then the problem (2.2) also has an optimal solution 𝑦_∗∈ℝ^𝑚 and 𝑏^𝑇𝑦_∗=𝑐^𝑇𝑥_∗. (2.10)

Proof. Assume that the feasible set 𝑀 of the problem (2.1) is not empty, and denote by 𝑥_∗∈ℝ^𝑛 the optimal solution of the problem (2.1). Let us show that the set of all (𝑥,𝑡), 𝑥∈ℝ^𝑛, 𝑡∈ℝ, satisfying 𝐴𝑥−𝑏𝑡=0,𝑐^𝑇𝑥−𝑐^𝑇𝑥_∗𝑡=−1<0,𝑥∈𝑉^∗_0,𝑘(𝜎),𝑡≥0, (2.11) is empty.
Assume that (𝑥′,𝑡′) satisfies the system (2.11) and 𝑡′>0. Then (𝑥′/𝑡′) is a solution of (2.1) and 𝑐^𝑇(𝑥′/𝑡′)<𝑐^𝑇𝑥_∗, which contradicts to the optimality of 𝑥_∗. On the other hand, if 𝑡′=0, then 𝑥_∗+𝑥′ is a solution of (2.1), and 𝑐^𝑇(𝑥_∗+𝑥′)=𝑐^𝑇𝑥_∗−1<𝑐^𝑇𝑥_∗, which contradicts to the optimally of 𝑥_∗.
Now, it follows from [4]that there is 𝑦_∗∈ℝ^𝑚 such that 𝑐−𝐴^𝑇𝑦_∗≥0 and −𝑐^𝑇𝑥_∗+𝑏^𝑇𝑦_∗≥0. It implies that 𝑦_∗ is a solution of (2.2). Moreover, it follows from Lemma 2.2 that 𝑏^𝑇𝑦_∗≤𝑐^𝑇𝑥_∗.

Now we are ready to prove Theorem 1.1. Note that the set 𝑀_𝑝+1,𝑘(𝜎) defined in Section 1 is a closed convex cone. Let 0≤𝑡₀<𝑡₁<⋯<𝑡_𝑝≤1 be arbitrary points in [0, 1]. Since {𝑓₀,…,𝑓_𝑝} is a Chebyshev system, we may conclude that points𝑐_𝑖=𝑓₀𝑡_𝑖,…,𝑓_𝑝𝑡_𝑖∈𝑀_𝑝+1,𝑘(𝜎),𝑖=0,…,𝑝, (2.12) are linearly independent. Thus, the cone 𝑀_𝑝+1,𝑘(𝜎) is not contained in any 𝑝-dimensional subspaces.

Proof of Theorem 1.1. We will prove (1.9). Consider the conic programming problem min_{𝜇∈𝐾0,𝑘(𝑐0)}𝐼𝑓^𝑇𝜇. (2.13) Denote 𝑀^∗_0,𝑘(𝜎)∶=𝑦∈ℝ^𝑝+1∶𝐼𝑓−^𝑝_𝑖=0𝑦_𝑖𝑓_𝑖∈𝑉_0,𝑘(𝜎). (2.14) The dual problem of the problem (2.13) is the problem max_{𝑦∈𝑀∗0,𝑘(𝜎)}𝑐^𝑇𝑦. (2.15) It follows from Lemma 2.3 that inf_{𝜇∈𝐾0,𝑘(𝑐0)}𝐼𝑓^𝑇𝜇=max_{𝑦∈𝑀∗0,𝑘(𝜎)}𝑦^𝑇𝑐⁰. (2.16) Equality (1.9) follows from the equality max_{𝑦∈𝑀∗0,𝑘(𝜎)}𝑦^𝑇𝑐⁰=sup_𝑔∈𝑃+𝑔𝑐⁰. (2.17) Equality (1.8) can be proved similarly.

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

Let 0≤𝑥₁<𝑥₂<⋯<𝑥_𝑛≤1, 𝐼𝑓=(𝑓(𝑥₁),…,𝑓(𝑥_𝑛))^𝑇∈ℝ^𝑛, 𝑓∈ℂ[0,1]. Let Φ denote the class of all linear algorithms 𝐴∶ℝ^𝑛→ℝ based on information 𝐼. The error of the problem of optimal linear interpolation of 𝑓∈ℂ[0,1] at point 𝜁∈[0,1] on the basis of information 𝐼𝑓, 𝑓∈𝑊, is defined by𝑒_𝜁𝑓,𝐼∶=inf_𝐴∈Φ𝑓(𝜁)−𝐴𝐼𝑓. (3.1) Note that for every 𝐴∈Φ there exists 𝜇∈ℝ^𝑛 such that 𝐴(𝐼𝑓)=(𝐼𝑓)^𝑇𝜇 for all 𝑓∈ℂ[0,1]. Then𝑒_𝜁𝑓,𝐼=inf_{𝜇∈ℝ𝑛}𝑓(𝜁)−𝐼𝑓^𝑇𝜇. (3.2) 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 𝑓 has some shape properties, then it usually means that the element 𝑓 belongs to a cone in ℂ[0,1].

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 𝐾 be a cone in ℂ[0,1]. Let Φ(𝐾) denote the class of all linear algorithms 𝐴∶ℝ^𝑛→ℝ, based on information 𝐼 and such that 𝐴(𝑣)≥0 for all 𝑣∈𝑉, 𝑉∶={𝐼𝑓∶𝑓∈𝐾}⊂ℝ^𝑛.

Define by𝑒_𝜁𝑓,𝐼,𝐾∶=inf_{𝐴∈Φ(𝐾)}𝑓(𝜁)−𝐴𝐼𝑓 (3.3) the error of optimal linear interpolation of 𝑓∈ℂ[0,1] at point 𝜁∈[0,1] on the basis of information 𝐼𝑓, 𝑓∈𝑊, with respect to the cone 𝐾.

Denote by 𝑉^∗∶={𝑢∈ℝ^𝑛∶𝑢^𝑇𝑣≥0forall𝑣∈𝑉} the cone dual to 𝑉. Note that for every 𝐴∈Φ(𝐾) there exists 𝜇∈𝑉^∗ such that 𝐴(𝐼𝑓)=(𝐼𝑓)^𝑇𝜇 for all 𝑓∈ℂ[0,1]. Then𝑒_𝜁𝑓,𝐼,𝐾=inf_{𝜇∈𝑉∗}𝑓(𝜁)−𝐼𝑓^𝑇𝜇. (3.4) The next proposition demonstrates how we can use Theorem 1.1 to obtain the error of optimal linear interpolation.

We will consider the case 𝑘=2, 𝜎=(1,0,1), 𝑢₀(𝑥)≡1, 𝑢₁(𝑥)≡𝑥. Then 𝑊_0,2(𝜎) is the cone of all positive and convex functions on [0,1], 𝑉_0,2(𝜎)={𝐼𝑓∶𝑓∈𝑊_0,2(𝜎)} and, 𝑉^∗_0,2(𝜎)={𝜇∈ℝ^𝑛∶(𝐼𝑓)^𝑇𝜇≥0forall𝐼𝑓∈𝑉_0,2(𝜎)}.

In the next proposition we consider the problem of interpolation by means of shape-preserving algorithms 𝐴, which have some properties of shape-preserving projections (i.e., 𝐴(𝑓)=𝑓 for every 𝑓 from a certain finite-dimensional subspace). Note that a deep study of linear shape preserving projections was undertaken in papers [10–12].

Corollary 3.1. Let 𝑓∈ℂ[0,1] be a strictly convex function on [0,1], 𝜁∈[0,1], and let 1≤𝑘≤𝑛−1 be such that 𝑥_𝑘<𝜁<𝑥_𝑘+1. Denote 𝐷∶=𝜇∈𝑉^∗_0,2(𝜎)∶𝐼𝑢_𝑖^𝑇𝜇=𝑢_𝑖(𝜁),𝑖=0,1. (3.5) Then inf_𝜇∈𝐷𝑓(𝜁)−𝐼𝑓^𝑇𝜇=𝑥_𝑘+1−𝜁𝜁−𝑥_𝑘𝑥_𝑘,𝜁,𝑥_𝑘+1𝑓, (3.6) where [𝑥_𝑘,𝜁,𝑥_𝑘+1]𝑓 denotes the divided difference of 𝑓 at 𝑥_𝑘<𝜁<𝑥_𝑘+1.

Proof. Consider the problem min_𝜇∈𝐷𝑓(𝜁)−𝐼𝑓^𝑇𝜇. (3.7) It follows from Theorem 1.1 that min_𝜇∈𝐷𝑓(𝜁)−𝐼𝑓^𝑇𝜇=max𝑓(𝜁)+𝑦₀𝑢₀(𝜁)+𝑦₁𝑢₁(𝜁), (3.8) where maximum is taken over all 𝑦₀,𝑦₁∈ℝ such that 𝐼(𝑓+𝑦₀𝑢₀+𝑦₁𝑢₁)∈𝑉_0,2(𝜎).
It follows from 𝑥_𝑘<𝜁<𝑥_𝑘+1 and strictl convexity of 𝑓 that max𝑓(𝜁)+𝑦₀𝑢₀(𝜁)+𝑦₁𝑢₁(𝜁)=𝑥_𝑘+1−𝜁𝜁−𝑥_𝑘𝑥_𝑘,𝜁,𝑥_𝑘+1𝑓. (3.9) Consider the problem min_𝜇∈𝐷−𝑓(𝜁)+𝐼𝑓^𝑇𝜇. (3.10) It follows from Theorem 1.1 that min_𝜇∈𝐷−𝑓(𝜁)+𝐼𝑓^𝑇𝜇=max−𝑓(𝜁)+𝑦₀𝑢₀(𝜁)+𝑦₁𝑢₁(𝜁), (3.11) where maximum is taken over all 𝑦₀,𝑦₁∈ℝ such that 𝐼(−𝑓+𝑦₀𝑢₀+𝑦₁𝑢₁)∈𝑉_0,2(𝜎).
It follows from 𝑥_𝑘<𝜁<𝑥_𝑘+1 and strictl convexity of 𝑓 that max𝑓(𝜁)+𝑦₀𝑢₀(𝜁)+𝑦₁𝑢₁(𝜁)=𝑥_𝑛−𝜁𝜁−𝑥₁𝑥₁,𝜁,𝑥_𝑛𝑓. (3.12) 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.