Applied Mathematics and Computation 247 (2014) 417–430

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

On approximation of linear functionals over convexfunctions: Construction techniques and new directions

http://dx.doi.org/10.1016/j.amc.2014.09.0300096-3003/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author.E-mail addresses: achchab@estb.ac.ma (B. Achchab), allal.guessab@univ-pau.fr (A. Guessab).

Boujemâa Achchab a, Allal Guessab b,⇑a LM2CE, LAMSAD, Univ. Hassan 1 FSJES and EST, B.P. 218, Berrechid, Moroccob Laboratoire de Mathématiques et de leurs Applications, UMR CNRS 4152, Université de Pau et des Pays de l’Adour, 64000 Pau, France

a r t i c l e i n f o a b s t r a c t

Keywords:Under/over-estimationsNon-negative linear functionalsConvex functionsJensen’s inequality

Recently there has been renewed interest in the problem of finding under and overestimations on the set of convex functions to a given non-negative linear functional; thatis, approximations which estimate always below (or above) the functional over a family ofconvex functions. The most important example of such an approximation problem is givenby the multidimensional versions of the midpoint (rectangle) rule and the trapezoidal rule,which provide under and over estimations to the true value of the integral on the set ofconvex functions (also known as the Hermite–Hadamard inequality). In this paper, weintroduce a general method of constructing new families of under/over-estimators onthe set of convex functions for a general class of linear functionals. In particular, underthe regularity condition, namely the functions belonging to C2ðXÞ (not necessarily convex),we will show that the error estimations based on such estimators are always controlled bythe error associated with using the quadratic function. The result is also extended to theclass of Lipschitz functions. We also propose a modified approximation technique to derivea general class of under/over estimators with better error estimates.

� 2014 Elsevier Inc. All rights reserved.

1. Some background and motivation

The problem of under and over-estimations of functionals in its most general form can be described in the following way:Let X be a real vector sub-space of real-valued continuous functions on X, a nonempty fixed compact convex set in Rd, and T alinear functional on X. If a function f is given in X, it should be noted that in many situations, of course, the true value of T½f � is,in general, impossible or not easy to compute, or often not know but only we can evaluate T on a given subset of X containingsome simple functions. One popular numerical approach is to replace T½f � by another simple approximating functional A½f �,which can relatively easy to evaluate numerically. We also wish to construct A½f � with a possibly small error. The key idea toquantify the quality of the numerical approximation A½f � is to use two different functionals, say T and T , to estimate the abso-lute value of the error T½f � � A½f �j j by T½f � � T½f �

�� ��. If no other information is available, we are forced to accept this (or somescaling of it) as the error estimate of T½f � � A½f �j j. However, to get a good estimate of T½f �, we need some a priori informationsabout T in a given subset G of X (not necessarily containing f). The common practice in such a case is to construct lower andupper bounding functionals T and T in such a way that

T½g� 6 T½g� 6 T½g� ð1Þ

for any g 2 G.

418 B. Achchab, A. Guessab / Applied Mathematics and Computation 247 (2014) 417–430

A natural question to ask is therefore: Given a linear functional T on X and f 2 CðXÞ. Can we select appropriate functionalsA½f �; T½f �; T½f �, and also decide how the deviation of A½f � from T½f � should measured?

Obviously, if we know that f belongs to the some set G, we can sometimes better evaluate and estimate the exact value ofT½f �. Indeed, if that is the case, we may take the approximation functional A :¼ ð1� kÞT þ kT , any convex combination of Tand T then, obviously, for any k 2 ½0;1�, the error estimate can always be controlled as follows:

T½f � � A½f �j j 6 T½f � � T½f �: ð2Þ

Eq. (2), clearly, shows that when T½f � � T½f � is small, we are confident that T½f � � A½f �j j is also small. Hence, we are interestedin solving the following under and over approximation problem.

Problem 1.1. For a given subset G of X, a linear functional T and a function f 2 X (no necessary in G,) how we can determinethe functionals T; T; A with T 6 T; A 6 T in G and, in such a way that, we are able to control the error T½f � � A½f �j j?

Clearly, a basic issue of the success of this approach is in the ability to construct valid and rigorous under and over esti-mators. In order to make this idea more precise, a specific example may help to clarify most of the main points of thisapproach. We adopt the following notation: for a measurable set D � Rd, we shall always denote by Dj j the measure of D.The set of all continuous convex functions defined on X will be denoted by KðXÞ.

Example 1.2. As a concrete example, we take X a non-degenerate simplex in Rd with vertices v0; . . . ;vd. Denote by v� itscenter of gravity, which is defined by

v� ¼R

X xdxXj j :

For simplicity, let us consider the particularly important case when T is the normalized functional integral

T½f � ¼R

X f ðxÞdxXj j ;

which we may under and over estimate respectively by:

T½g� ¼ gðv�Þ and T½g� ¼ 1dþ 1

Xd

i¼0

gðv iÞ;

when we take G to be the set of all convex functions KðXÞ. The verification of constraints hypothesis (1) of under and overestimations of T and T is based on the use of the multidimensional versions of the midpoint rule and the trapezoidal rule, see[15]. Indeed, T can be under and over estimated as follows:

T½g� 6 T½g� 6 T½g�; for all g 2 KðXÞ; ð3Þ

see [15, Theorems 2.1 and 2.2]. For a given f 2 CðXÞ, we now choose as an approximation to the true value of T½f � any convexcombination of T½f � and T½f �. Then clearly A½f � guarantees a better approximation than T½f � and T½f � in KðXÞ, and for any k, suchthat 0 6 k 6 1=ðdþ 1Þ, the functional A½f � ¼ ð1� kÞT½f � þ kT½f � systematically over estimates T½f � in KðXÞ, see [15, Theorem4.1]. Moreover, as observed previously, under the assumption f 2 KðXÞ, we have the error estimate (2). However, if moreoverf belongs to C2ðXÞ, then for any k, such that 0 6 k 6 1=ðdþ 1Þ, we have the estimate, see [15, Corollary 6.5],

T½f � � A½f �j j 6 jD2f jX2

1dþ 2

� kdþ 1

� �Xd

i¼0

kv i � v�k2 ð4Þ

with k � k denoting the usual Euclidean norm for vectors in Rd and

jD2f jX :¼ supx2X

sup D2y f ðxÞ

��� ��� : y 2 Rd; kyk ¼ 1n o

:

The above Problem 1.1 is obviously too general to be dealt with under a unifying aspect. From now on, we restrict our-selves to the case where the set G ¼ KðXÞ and we are faced now with two issues: Given a linear functional T on X andf 2 CðXÞ.

(P1) Do there exist two linear functionals T; T , that under and over estimate T as follows:

T½g� 6 T½g� 6 T½g�; for all g 2 KðXÞ? ð5Þ

(P2) Once we found an under and over estimator, how can we determine a ‘good’ approximation for T½f �?

In the case of functional integrals, the answer to the first issue has attracted the interest of people working in generalinequalities, who refer to (5) as the Hermite–Hadamard inequalities (see [15]). We shall join them in doing so. In particular,we shall call the first and the second inequality in (5) the lower and the upper Hermite–Hadamard inequality, respectively.There is a big qualitative jump going from one to more dimensions when dealing with the Hermite–Hadamard inequalities

B. Achchab, A. Guessab / Applied Mathematics and Computation 247 (2014) 417–430 419

for multivariate convex functions. This can be appreciated readily by looking at the existing literature, where a largelyone-dimensional approach has been considered both theoretically and numerically, essentially for integral functionals(the monograph of Dragomir and Pearce [2] gives good insights into a variety of the Hermite–Hadamard inequality). How-ever, to date, relatively very little work has been done in the multivariate setting. In this regard, it is worth mentioning thefollowing contributions [5–15], which played a relevant role in determining the best (or optimal) cubature formulas. Thepresent paper may be viewed as a continuation of this study obtained for a class of integral functionals.

Besides its own importance in approximation theory, and numerical analysis, the motivation to consider the above prob-lems is that it provides a simple and unified approach to estimate, with error control, any non-negative linear functional, bydeveloping and analyzing new appropriate improved under/over estimators in the form of a weighted sums at suitablepoints. Indeed, under the regularity condition the functions belonging to C2ðXÞ or only to the Lipschitz class, we will showthat the error estimation based on such estimators has a remarkable property. It is always controlled by the error associatedwith the quadratic function k � k2, see Lemma 2.4, Theorems 4.6 and 5.2.

The rest of this paper is divided into the following sections: In Section 2, notations and definitions for the non-negativelinear functional linear together with a study of the general problem (P2) in an abstract setting are given. In subsequent sec-tions these results are applied to more specific problems. In particular, we give an affirmative answer to Problem (P2) byintroducing a general technique for constructing a class of improved estimators. Indeed, assuming that we are given underand over estimators in KðXÞ, we show that for every function in C2ðXÞ (not necessarily convex) it is possible to determine aclass of modified under/over estimators, which produce small error bounds than the original estimators. In Section 3, weintroduce the notion of the center of gravity with respect to a given functional. The first subsection deals mainly with someextremal properties of the center of gravity with respect to a linear functional. The second subsection is devoted to give adetailed analysis of its location. In this general context, we also obtain some characterization results of the center of gravity.The rest of the paper considers estimators in the form of averages or weighted sums at suitable points. To do so, severalgeneral classes of under estimators are proposed which are based on the notion of the center of gravity with respect a linearfunctional. Finally, Section 5 shows that analogous results, stated for under approximation of non-negative linear function-als, are also valid for over approximation on arbitrary convex polytopes. We also use our general technique developed inSection 2 to derive new modified and refined under estimators.

2. Improved under/over estimating functionals

After establishing some notational conventions which will be used throughout the rest of the paper, this section focuseson the theoretical framework requisite for generating a new class of modified under/over-estimators on the set of convexfunctions KðXÞ. We also give an affirmative answer to problem (P2) by introducing a general strategy for constructing mod-ified and improved estimators. Indeed, we show that for every function in C2ðXÞ (not necessarily convex) it is possible todetermine under/over estimators with small error bounds.

We first specify some notational conventions. We use boldface letters to denote vectors. Throughout the paper it isassumed that T is a given nonzero linear functional defined on X ¼ CðXÞ. By k � kX, we denote the supremum norm onCðXÞ, that is,

kfkX ¼ supx2X

f ðxÞj j:

We will assume that CðXÞ is equipped with the uniform (convergence) topology induced by k � kX.The following definition is important for the rest of the paper. Let X be a subset of CðXÞ. A functional T will be called non-

negative on X if T½f �P 0 for all f 2 X. That is a non-negative functional on X takes non-negative values when applied to eachfunction in the set X. Throughout the paper, CþðXÞ will denote the set of all non-negative functions in CðXÞ. Thus, a non-neg-ative linear functional on CþðXÞ is a linear functional T such that T½f �P 0 whenever f ðxÞP 0 on X.

We will use the following definition.

Definition 2.1. Let A; T be two linear functionals defined on CðXÞ. Let R ¼ T � A, we say that A under (resp., over) estimates Ton KðXÞ if the functional R (resp. �R) is non-negative on KðXÞ.

Clearly, if A under (resp., A over) estimates T on KðXÞ, then this means exactly that the lower and the upper Hermite–Had-amard inequality are satisfied. For any linear functional, we shall say that it admits a lower or an upper Hermite–Hadamardinequality if there exists an under or over estimator, which approximates the value of the functional at any convex functionfrom below or above, respectively. We shall see that such estimators exist, and we can always establish for them sharp errorbounds.

We now ready to present the following simple observations.

Proposition 2.2. Let A; T be two linear functionals defined on CðXÞ. If A under(resp. over) estimates T on KðXÞ, then the errorfunctional R ¼ T � A satisfies the following properties:

(i) R½l� ¼ 0, for all affine function;(ii) R½k � k2� ¼ R½k � �y�k2�, for any given y� 2 X.

Proof. We prove only the case where A under estimates T on KðXÞ, since the other case is similar. Indeed, let l be a fixed

420 B. Achchab, A. Guessab / Applied Mathematics and Computation 247 (2014) 417–430

affine function. Since �l are both convex, we then have R½�l�P 0. Consequently, the linearity of R implies the inequalities�R½l�P 0. These inequalities taken together show the desired result (i).

To prove (ii), we first observe that, for any fixed y� 2 X, we have k � �y�k2 ¼ k � k2 þ l, where l is an affine function.Therefore, applying R and employing the above property, then both sides of this equation reduce to (ii). h

Let us now remark that it is not possible to ask for more than second order accuracy for error functionals R, no matter howsmooth the functions to be approximated are. Indeed, we have the following result.

Theorem 2.3. Let A; T be two linear functionals defined on CðXÞ. Assume that A under (resp., over) estimates T on KðXÞ. If the

error functional R :¼ T � A satisfies R½k � k2� ¼ 0, then R½f � ¼ 0 for any continuous function f 2 CðXÞ.Before proceeding to the proof of Theorem 2.3, we need the following result, which presents an upper bound on the

approximation error in the case that the function f 2 C2ðXÞ.

Lemma 2.4. Let A; T be two linear functionals defined on CðXÞ. Assume that A under (resp., over) estimates T on KðXÞ. Iff 2 C2ðXÞ, then the functional R ¼ T � A satisfies the following error bound:

jR½f �j 6 jD2f jX2jR½ k � k2h i

j:

Proof. Let f be any twice continuously differentiable function. Then, it is easily seen that the Hessian matrix of the twofunctions

g� :¼ jD2f jX2k � k2 � f ð6Þ

is positive semi-definite. This means that gþ and g� are both convex functions. Applying R and using its linearity we imme-diately get the desired result. h

We now have all the ingredients needed to prove Theorem 2.3.

Proof of Theorem 2.3. We only prove the case where A under estimates T. In this case, by definition the functional R satisfies

R½f �P 0 ð7Þ

for all f 2 KðXÞ.This means that R ¼ T � A is non-negative linear functional on KðXÞ, then it follows that R is automatically continuous on

ðCðXÞ; k � kXÞ, see Aliprantis and Tourky [1, Theorem 2.32]. Taking into account the classical result that C2ðXÞ is dense in CðXÞby the Stone–Weierstrass theorem, see, e.g., [18, p. 162, Theorem 7.32], we conclude by employing the continuity of R that itsuffices to prove the conclusion of the theorem for f 2 C2ðXÞ only. The result now comes directly from Lemma 2.4. h

The rest of this section is devoted to a useful elementary technique, which solves problem (P2) and improves the qualityof under or over approximation of linear functionals, when we know a priori that the given function belongs to C2ðXÞ.

In order to proceed in developing our argument, we need some additional necessary background and notations.By dd the set of all d� d symmetric matrices in R. Let A 2 dd, and bi½A�; i ¼ 1; . . . ; d, the (real) eigenvalues of A, we define

bmin½A� :¼ min16i6d

bi½A� ¼ minkyk¼1

Ay; yh i:

We say A 2 dd is positive semi-definite if Ay; yh iP 0, for every y 2 Rd. The set of positive semi-definite matrices (all eigen-values P0) is denoted by dþd .

By D2f ðxÞ, we mean the d� d matrix whose entries are the second-order partial derivatives of f at x. It is well known thatwhen f is a C2ðXÞ-function, its convexity is characterized by the fact that for all x 2 X;D2f ðxÞ 2 dþn (see e.g. [19]). For every

x 2 X, the Hessian matrix D2f ðxÞ, as real-valued and symmetric matrix, has real-valued eigenvalues. For every function f

in C2ðXÞ, we define

kmin½f � :¼ infx2X

bmin½D2f ðxÞ�:

We shall call kmin½f � the ‘globally’ smallest eigenvalue of the Hessian D2f ðxÞ on X. Now, let f be any C2ðXÞ-function and set

g :¼ f � kmin½f �2k � k2

: ð8Þ

The Hessian matrix of g is

B. Achchab, A. Guessab / Applied Mathematics and Computation 247 (2014) 417–430 421

D2gðxÞðxÞ ¼ D2f ðxÞðxÞ � kmin½f �Id;

where Id denotes the d� d identity matrix. Hence, for y 2 Rd such that kyk ¼ 1, we have

y;D2gðxÞðxÞyD E

¼ y;D2f ðxÞðxÞyD E

� kmin½f �;

where h�; �i denotes the standard inner product on Rd.It is clear from the definition of kmin½f � that, for every, x 2 X, the last term in the above equation is non-negative. This

means that the Hessian matrix of g is positive semi-definite for all x of the set X, and consequently g is convex.Hence, an arbitrary non-convex twice continuously differentiable function is made convex after adding to it the quadratic

� kmin ½f �2 k � k2. It should be noted that for every a 6 kmin½f �, the function

ga :¼ f � a2k � k2

is convex. Indeed, we have

ga ¼ g þ kmin½f � � a2

k � k2;

where g is described by the formula (8). Then ga is convex as sum of two convex functions.Note that kmin½f � is not necessarily zero if f is convex over X. On the other hand, if kmin½f �P 0 then f is convex over X.We shall assume now that A� (resp. AþÞ under (resp. over) estimates T. With this notation, the next Theorem shows that if

f is a twice continuously differentiable function f (not necessarily convex), then there exist better estimators than A�½f � andAþ½f �. In fact, in such a case we can get the following more precise estimates.

Theorem 2.5. Let f be an arbitrary twice continuously differentiable function defined on X. Assume that the functionals A�; T; Aþsatisfy the estimates

A�½g� 6 T½g� 6 Aþ½g� ð9Þ

in KðXÞ. Set

R� :¼ T � A�Rþ :¼ Aþ � T:

Then the following estimates hold

A�½f � þkmin½f �

2R�½k � k2� 6 T½f � 6 Aþ½f � �

kmin½f �2

Rþ½k � k2�: ð10Þ

Let us emphasize before proving Theorem 2.5 that by comparing the estimates (9) and (10), we can see that if f is strictly

convex (then kmin½f � > 0), and therefore (10) always provides better bounds than those of (9). Note also that, since R�½k � k2�and Rþ½k � k2� are two non-negative real numbers, then the new bounding expressions are obtained, by adding and subtract-ing, respectively, non-negative correction terms to the left-hand side and from the right-hand side of the estimates (9).

We may now prove Theorem 2.5.

Proof. Under the present assumption about the function f, it is immediately obvious that the auxiliary function

g :¼ f � kmin½f �2k � k2

is convex. Hence applying the inequality (9) to g and rearranging terms give to the required inequality. h

3. The center of gravity: its extremal properties and its computation

Another key tool in the construction of suitable under estimators on KðXÞ is the notion of the center of gravity withrespect to a given functional T. In the first subsection, we give a list of the more important properties of the center of gravitywith respect to a linear functional. The second subsection is devoted to give a detailed analysis of its location; the resultspresented here might be of some independent interest outside the context of under and over-estimations.

3.1. Extremal properties of the center of gravity

For simplicity, in this subsection, we assume that T is a given linear functional on CðXÞ such that T½1� > 0.Here ei denotes the projection ei : x ¼ ðx1; . . . ; xdÞ ! xi. We write e :¼ ðe1; . . . ; edÞ for the identity on Rd, that is, eðx1; . . . ;

xdÞ ¼ ðx1; . . . ; xdÞ. When f ¼ ðf 1; . . . ; f dÞ 2 ðCðXÞÞd, where ðCðXÞÞd is the Cartesian product of d copies of CðXÞ, we define

422 B. Achchab, A. Guessab / Applied Mathematics and Computation 247 (2014) 417–430

T½f � :¼ ðT½f 1�; . . . ; T½f d�Þ:

In this way, T is extended to a functional T : ðCðXÞÞd ! R.Let now introduce the vector

xcg ½T� ¼T½e�T½1� : ð11Þ

We call the point xcg ½T� the center of gravity with respect to the functional T. This definition coincides with the one com-monly used for the integral functionals. In fact, in physics, this concept is associated with material bodies. The center of grav-ity is given by the formula

xcg ¼Z

Xxdx

�ZX

dx

and it is usual to say that xcg is the barycenter of X.Let us start with a simple localization of the center of gravity. This result is reminiscent of the work [13, Lemma 3.3], but

for the sake of completeness we give here a more direct proof. For notational convenience, for any linear functional T, we willrefer to the functional

Tcg ½f � :¼ T½1� f ðxcg ½T�Þ; ð12Þ

the center of gravity linear functional and we will denote the corresponding error functional by

Rcg ½f � :¼ T½f � � Tcg ½f �: ð13Þ

Proposition 3.1. The center of gravity, with respect to a non-negative linear functional T on CþðXÞ is an interior point of thedomain X. Moreover, the error functional Rcg defined by (13) is non-negative on KðXÞ.

Proof. To prove that the center of gravity is an interior point of the domain X, assume, on contrary, that xcg ½T� R intðXÞ.Then, due to the separation Theorem for closed convex sets (see, e.g., [20, p. 65, Theorem 2.4.1]), there exists a pointx� 2 X such that the affine function

lðxÞ :¼ xcg ½T� � x�; x� x�� �

; ð14Þ

satisfies lðxÞ 6 0, for all x 2 X. Since l 6 0 and T is non-negative and linear, it follows that T½l� 6 0. Now, a simple calculation

shows T½l� ¼ T½1�lðxcgÞ. Consequently, lðxcg ½T�Þ ¼ kxcg ½T� � x�k26 0. This clearly implies xcg ½T� ¼ x� and therefore establishes

that xcg ½T� must belong to intðXÞ.We will now prove the non negativity of the functional T � Tcg . Assume the contrary to derive a contradiction. Then, there

exists a convex function f 2 CðXÞ such that

f ðxcg ½T�Þ >T½f �T½1� ð15Þ

Define epiðf Þ the epigraph of the function f, that is the subset of Rdþ1

epiðf Þ :¼ ðx; tÞ 2 Rdþ1 : x 2 X; t 2 R; t P f ðxÞ�

:

It is well known that f is a convex function if and only if its epigraph is a convex set. Moreover, it is easy to prove that, if f iscontinuous, then its epigraph is closed. Hence, the epigraph of f is at the same time closed and convex set in Rdþ1. Observethat by (15), ðxcg ½T�; T½f �=T½1�Þ does not belong to the epigraph of f. Thus the separation theorem can be applied for epiðf Þ. Thenwe find that there exists a point ðx�; t�Þ 2 epiðf Þ such that the affine function

lðx; tÞÞ :¼ ðxcg ½T�; T½f �=T½1�Þ � ðx�; t�Þ; ðx; tÞ � ðx�; t�Þ� �

ð16Þ

satisfies lðx; tÞÞ 6 0, for all ðx; tÞ 2 epiðf Þ. Now we define the function h : X! R as follows

hðxÞ :¼ lðx; f ðxÞÞ:

We obviously have lðx; tÞÞ 6 0, for all ðx; tÞ 2 ðX; f ðXÞÞ, as ðX; f ðXÞÞ is always a subset of epiðf Þ. Hence h 6 0 on X and conse-quently T½h� 6 0. Proceeding as before we get:

T½h� ¼ T½1�lððxcg ½T�; T½f �=T½1�Þ :¼ kxcg ½T� � x�k2 þ ðT½f �=T½1� � t�Þ2 6 0:

This immediately shows that ðxcg ½T�; T½f �=T½1�Þ ¼ ðx�; t�Þ and contradicts the fact that ðxcg ½T�; T½f �=T½1�Þ R epiðf Þ. h

B. Achchab, A. Guessab / Applied Mathematics and Computation 247 (2014) 417–430 423

We are now able to present two characterization results. The first one shows that the center of gravity characterizes thepositivity of the functional Rcg defined by (13), in the sense that the point xcg ½T� is the only point in X which satisfiesRcg ½f �P 0, for all function in KðXÞ.

Proposition 3.2. The center of gravity, with respect to a linear functional T on CðXÞ, is the only point of X, such that the errorfunctional Rcg is non-negative on KðXÞ. Moreover, it is uniquely defined by the relation

Rcg ½f � ¼ 0; ð17Þ

for all affine function f.

Proof. Indeed, assume that there is y 2 X such that

Ry½f � :¼ T½f � � T½1� f ðyÞP 0 ð18Þ

for any function f 2 KðXÞ. Since, by Proposition 2.2 and 3.1, the equality sign will occur for any affine function, we then have

lðyÞ ¼ lðxcg ½T�Þ; for any affine function l: ð19Þ

Therefore, for the affine function lðxÞ ¼ y � xcg ½T�; x� xcg ½T�� �

, we obtain lðyÞ ¼ 0, and then we can deduce y ¼ xcg ½T�.Now, assume that condition (17) holds, then

T½f �T½1� ¼ f ðxcgÞ ð20Þ

is satisfied for any affine function. In particular, if we choose the projection functions ei; i ¼ 1; . . . ; d, we can then get thedesired result as an immediate consequence of the definition of the center of gravity. h

The center of gravity has also the following remarkable property.

Proposition 3.3. Let A; T be two linear functionals defined on CðXÞ. Assume that A under estimates T on KðXÞ, then T and A havethe same center of gravity.

Proof. By Proposition 2.2, it follows that A½l� ¼ T½l�, for any affine function. Adopting vector notation, in particular, we haveA½e� ¼ T½e� and A½1� ¼ T½1� then we arrive to the announced equality. h

As a corollary of the above results, we now derive an extremal result for the center of gravity functional Tcg .

Proposition 3.4. Let A; T be two linear functionals defined on CðXÞ. Assume that A under estimates T on KðXÞ, then Tcg underestimates A on KðXÞ.

Proof. We only need to prove that A� Tcg is non-negative on KðXÞ. Since T � A is non-negative on KðXÞ, by Proposition 3.3, Tand A have the same center of gravity. Hence, xcg ½T� ¼ xcg ½A�. Since, T½1� ¼ A½1�, this implies f ðxcg ½T�ÞT½1� ¼ f ðxcg ½A�ÞA½1� andthen by Proposition 3.1, f ðxcg ½T�ÞT½1� 6 A½f �, for every convex function. Which shows that A� Tcg is non-negative on KðXÞ,and gives the desired result. h

3.2. Computation of the center of gravity

For practical applications, it is natural to ask, for a given T, how does one find its center of gravity?It is a well-known fact (see, e.g., [15, Theorem 2.2]) that if X is a non-degenerate simplex of Rd and T is the integral

functional then the center of gravity xcg ½T� is the arithmetic mean of its vertices.For an arbitrary linear functional T, its center of gravity in a simplex is given by:

Lemma 3.5. Let S be a non-degenerate simplex of Rd with vertices x0; . . . ; xd, and T a non-negative linear functional on CþðXÞ.Then the center of gravity xcg ½T� is given by the formula

xcg ½T� ¼Xd

i¼0

T½ki�T½1� xi; ð21Þ

where k0; . . . ; kd are the barycentric coordinate functions with respect to the vertices x0; . . . ; xd.

Proof. Let xcg ½T� :¼ T½e�T½1�, Proposition 3.1 says that xcg ½T� 2 intðXÞ then since S is a simplex, there exist (unique) non-negative

real numbers ai; i ¼ 0; . . . ; d, such thatPd

i¼0ai ¼ 1 and xcg ½T� ¼Pd

i¼0aixi. Consequently, we get

424 B. Achchab, A. Guessab / Applied Mathematics and Computation 247 (2014) 417–430

lðxcg ½T�Þ ¼Xd

i¼0

ailðxiÞ; ð22Þ

for every affine function. In view of linear precision of barycentric coordinates

x ¼Xd

i¼0

kiðxÞxi; ðx 2 SÞ; ð23Þ

and since xi; i ¼ 0; . . . ; d, are extreme points of S, we have kiðxjÞ ¼ dij, where dij is Kronecker’s delta. Therefore, from (22) andthe fact that barycentric coordinates are affine functions, we obtain ai ¼ kiðxcg ½T�Þ; i ¼ 0; . . . ; d. Thus, the center of gravity hasalso the representation formula

xcg ½T� ¼Xd

i¼0

kiðxcgÞxi: ð24Þ

Again, as barycentric coordinates are affine functions, then by using Proposition 3.2 it follows that kiðxcg ½T�Þ ¼ T½ki �T½1� . This shows

that xcg ½T� is therefore given by identity (21) and completes the proof of the lemma. h

The remainder of this section will be used to consider the general case when X is a (convex) polytope with X ¼ convðWÞand W ¼ x0; . . . ; xnf g a set of finite points of Rd. To this end let us first recall that a triangulation of a polytope X is a set of afinite number of non degenerate simplices T ¼ S1; S2; . . . ; Smf g with the following properties.

1. [Si2T Si ¼ X;2. intðSiÞ \ intðSjÞ ¼ ; if i: – j;3. Si \W is the set of vertices of Si for each Si 2 T ;4. Si \ Sj is a simplex with vertices in W, for all i; j.

There is no theoretical difficulty to decompose a polytope into simplices. For an explicit statement on the existence oftriangulations with a proof based on an algorithmic method, see [16, Theorem 3, part a]. There also exist efficient algorithmsfor computing a triangulation with respect to a finite set W whose convex hull is X, see [4].

Given a triangulation T . Let S1; . . .Sm be the simplices of T , we denote by Ni the set of all integers j such that xi is a vertexof Sj. If x 2 Sj and j 2 Ni, then we denote by kijðxÞ the barycentric co-ordinate of x with respect to xi for the simplex Sj. It iseasily verified that if x 2 Sj \ Sk, then kijðxÞ ¼ kikðxÞ if j; k 2 Ni and kijðxÞ ¼ 0 if j 2 Ni; k R Ni. Therefore, setting

/iðxÞ :¼kijðxÞ if x 2 Sj and j 2 Ni;

0; otherwise;

ð25Þ

it is straightforward to verify that for any i; i ¼ 0; . . . ;n, the function /i is well-defined and piecewise linear, non-negativereal-valued continuous function and that the functions f/0; . . . ;/ng sum to one at all points in X and they also satisfy thelinear precision property, i.e.

x ¼Xn

i¼0

/iðxÞxi ðon XÞ: ð26Þ

When X is a polytope, the following theorem gives an explicit representation formula of the center gravity xcg ½T�, using onlythe vertices of the simplices in the triangulation of X.

Theorem 3.6. Let T be a non-negative linear functional on CþðXÞ. Then the center of gravity xcg ½T� is determined by the followingformula

xcg ½T� ¼Xn

i¼0

Xj2Ni

T½kij�T½1�

!xi: ð27Þ

Proof. Just use the key formula:

x ¼Xn

i¼0

Xj2Ni

kijðxÞ !

xi ð8x 2 XÞ ð28Þ

and adapt the argumentation of the proof of Lemma 3.5. h

From now on we will concentrate on the under and over estimators of the following form.

B. Achchab, A. Guessab / Applied Mathematics and Computation 247 (2014) 417–430 425

Problem 3.7. Let T be a linear functional defined on CðXÞ. Fix an integer n. The construction problems of under and overestimations of functionals based on utilizing some point evaluation functionals can be formatted as follows.

Find non-negative real values a0; . . . ; an and ðnþ 1Þ-points x0; . . . ; xn 2 X (independent of f) so that the error functional

R½f � :¼ T½f � �Xn

i¼ 0

aif ðxiÞ; ð29Þ

has a constant sign in KðXÞ.

4. Construction of under estimators of linear functionals

In the this sections a general method is proposed for the construction of a class of under estimators of a given linear func-tion defined on CðXÞ, such that its error functional is of the form (29). We say that a system f/0; . . . ;/ng of real-valuedcontinuous functions is a partition of unity (PU) on X with respect to T if the following properties hold:

(i) T½/i� > 0; i ¼ 1; . . . ;n;(ii) /iðxÞP 0 on X; i ¼ 0; . . . ;n;

(iii) /0ðxÞ þ � � � þ /nðxÞ ¼ 1 on X.

Given a linear functional T and a PU, U ¼ f/1; . . . ;/ng, for each function /i, we define its center of gravity with respect to Tby

xcg ½/i; T� ¼T½/ie�T½/i�

: ð30Þ

We call the vector ðxcg ½/0; T�; . . . ; xcg ½/n; T�Þ the center of gravity of T with respect to U.

Theorem 4.1. Let ðxcg ½/0; T�; . . . ; xcg ½/n; T�Þ be the vector center of gravity of T with respect the partition of unity U. Then

A½f � :¼Xn

i¼0

T½/i�f ðxcg ½/i; T�Þ ð31Þ

under estimates T on KðXÞ.

Proof. For every i; i ¼ 0; . . . ;n, we set Ai½f � ¼ f ðxcg ½/i; T�ÞT½/i�. Then a careful inspection of the proof of Proposition 3.1 showsthat for every i, the functional Ai under estimates Ti on KðXÞ, where Ti½f � :¼ T½/if �. Hence, summing over i from 0 to n, we getA½f � 6

Pni¼0T½/if � on KðXÞ. Then, using the linearity of T and the fact that the functions f/0; . . . ;/ng sum to 1, we immediately

get the desired result. This proves the theorem. h

Now, we discuss how centers of gravity can be used to define tighter under estimators.For a given partition of unity, f/0; . . . ;/ng, with respect to a non-negative linear functional T consider functionals of the

form

AY ½f � ¼Xn

i¼0

T½/i�f ðyiÞ; ð32Þ

where Y ¼ fy0; . . . ; yng is a set of fixed points in X. Theorem 4.1 says that for every PU, there is at least one linear functional ofthe form (32), which under estimates T on KðXÞ. We will be interested in the following extremal problem: Find in the class ofall under estimators on KðXÞ of the form (32) the functional AY� which minimizes the error functional T½f � � AY ½f �, when apriori know that f belongs to a certain class of functions M � CðXÞ.

In physical situation, the constraint imposed on AY is some energy that we try to minimize, see [3]. Let us examine whathappens when we fix M the class of Lipschitz continuous functions with, say, Lipschitz constant L. For L > 0, let LipLðXÞ be thespace of Lipschitz continuous functions defined in X with values in R, such that

jf ðxÞ � f ðyÞj 6 Lkx� yk ðx; y 2 XÞ:

For every f 2 LipLðXÞ, the error functional R½f � :¼ T½f � � AY ½f � can be estimated as follows:

Theorem 4.2. Assume that AY under estimates T on KðXÞ. If f 2 LipLðXÞ, then

R½f �j j 6 L T½1�Xn

i¼0

T /ik � �yik2

h i !1=2

: ð33Þ

426 B. Achchab, A. Guessab / Applied Mathematics and Computation 247 (2014) 417–430

Before the proof of Theorem 4.2, we need the following lemma which presents a version of Cauchy–Schwarz’s inequalityfor non-negative linear functionals on CþðXÞ. A function q 2 CðXÞ is called a weight function if qðxÞP 0; 8x 2 X.

Lemma 4.3. Let q be a weight function and U a non-negative linear functional on CþðXÞ. Then U satisfies

U½qf �j j 6 U½q fj j�; ð34Þ

together with the Cauchy–Schwarz inequality

U2½qfg� 6 U½qf 2�T½qg2�: ð35Þ

Proof. By reasoning analogous to the proof of the usual Cauchy–Schwarz inequality, the inequality (34) results from apply-ing U to jf j � f P 0, and (35) is a consequence of the following non-negative quadratic expression in k 2 R:

0 6 U½qðf þ kgÞ2� ¼ U½qf 2� þ 2kU½qfg� þ k2U½qg2�: �

Using the above lemma, it is now possible to prove Theorem 4.2.

Proof of Theorem 4.2. Let first observe that

Xn

i¼0

T½/i� ¼ TXn

i¼0

/i

" #¼ T½1�

Xn

i¼0

T½/if � ¼ T fXn

i¼0

/i

" #¼ T½f �:

A simple calculation also shows that

R½f � ¼Xn

i¼0

T½/iðf � f ðyiÞÞ�:

On the right-hand side, for each i, we apply the Cauchy–Schwarz inequality (35), with q ¼ /i, to the functional T to obtain

R½f �j j 6Xn

i¼0

T1=2½/i�T1=2 /iðf � f ðyiÞÞ2

h i:

If we now apply the Cauchy–Schwarz inequality but this time for sums, and taking into the account thatPn

i¼1T½/i� ¼ T½1�, weget

R½f �j j 6 T½1�Xn

i¼0

T /iðf � f ðyiÞÞ2

h i !1=2

:

But since f 2 LipLðXÞ, we conclude that

R½f �j j 6 L T½1�Xn

i¼0

T /ik � �yik2

h i !1=2

:

This is exactly the required estimation what we are looking for. h

Motivated by the above estimation, using a l2-norm as a distance metric, the problem of finding the tightest possibleunder estimator AY of T on KðXÞ can be formulated as follows: find the set of points Y� ¼ fy�0; . . . ; y�ng such that

Xn

i¼0

T /ik � �y�i k2

h i¼min

Y

Xn

i¼0

T /ik � �yik2

h i: ð36Þ

Note that the existence of the optimal under estimator may be guaranteed by:

infy2X

GðyÞ; where GðyÞ ¼ T /ik � �yk2h i

: ð37Þ

Clearly, G is convex and continuous on the compact set X, and consequently we obtain the existence of solutions of (37);however, the solution may not be unique.

The next Lemma shows that the minimizer is the center of gravity with respect to /i and T. Indeed, the following extremalproperty of the center of gravity holds.

B. Achchab, A. Guessab / Applied Mathematics and Computation 247 (2014) 417–430 427

Lemma 4.4. The center of gravity xcg ½q; T� of a non-negative linear functional on CþðXÞ with respect to a weight function qsatisfies

T qk � �xcg ½q; T�k2h i

¼ infy2X

T qk � �yk2h i

: ð38Þ

Proof. Indeed, we have, for any point y 2 X,

T qk � �xcg ½q; T�k2h i

¼ T½q � � xcg ½q; T�; � � y� �

�:

Using the Cauchy–Schwarz inequality for non-negative functionals, see Lemma 4.3, we then have

T qk � �xcg ½q; T�k2h i

6 T qk � �xcg ½q; T�k2h i� �1=2

T qk � �yk2h i� �1=2

:

But this inequality just asserts that, for every y 2 X,

T qk � �xcg ½q; T�k2h i

6 T qk � �yk2h i

:

This, together with the fact that xcg ½q; T� belongs to X shows that the center of gravity has the optimal desired property(38). h

We are now interested in the general optimization problem (36).

Theorem 4.5. A necessary condition for

Fðy0; . . . ; ynÞ :¼Xn

i¼0

T /ik � �yik2

h i

to be minimized is that for each i; yi is the center of gravity of T with respect to /i.

Proof. Indeed, in view of the above Lemma 4.4, applied for each i, with the weight function q ¼ /i, we have

T /ik � �xcg ½/i; T�k2

h i6 T /ik � �yik

2h i

: ð39Þ

Therefore, summing over i from 0 to n, we obtain

Fðxcg ½/0; T�; . . . ; xcg ½/n; T�Þ 6 Fðy0; . . . ; ynÞ: ð40Þ

This shows that ðxcg ½/0; T�; . . . ; xcg ½/n; T�Þ is a global minimum of F on Xnþ1. Let us now prove that the point

ðxcg ½/0; T�; . . . ; xcg ½/n; T�Þ is the unique minimum. From the definition of xcg ½/i; T�, it is easily seen that the individualcontribution of each function /i to the function F can be simplified to:

T /ik � �yik2

h i¼ kxcg ½/i; T� � yik

2T½/i� þ Ei; ð41Þ

where we have set Ei ¼ T /ik � �xcg ½/i; T�k2

h i, which depends only on /i. By summing over i from 0 to n, we can obviously

deduce

Fðy0; . . . ; ynÞ ¼Xn

i¼0

kxcg ½/i; T� � yik2T½/i� þ F xcg ½/0; T�; . . . ; xcg ½/n; T�

�: ð42Þ

Observe that Fðxcg ½/0; T�; . . . ; xcg ½/n; T�Þ does not depend on the data y0; . . . ; yn, we can then safely omit their computation tominimize

Gðy0; . . . ; ynÞ ¼Xn

i¼0

kxcg ½/i; T� � yik2T½/i�:

Since, the coefficients T½/i� are assumed to be positive, we see that, due to representation (42), the only minimizer isðxcg ½/0; T�; . . . ; xcg ½/n; T�Þ. h

According to Theorem 4.1 the functional A½f � ¼Pn

i¼0T½/i�f ðxcg ½/1; T�Þ under estimates T on KðXÞ. From Lemma 2.4 we knowthat if f 2 C2ðXÞ then the functional R ¼ T � A satisfies the following error bound:

jR½f �j 6 jD2f jX2

R½ k � k2h i��� ���:

428 B. Achchab, A. Guessab / Applied Mathematics and Computation 247 (2014) 417–430

If f were only Lipschitz with constant L, then we would have another explicit error estimate of the error functional R.

Theorem 4.6. The functional A½f � ¼Pn

i¼0T½/i�f ðxcg ½/1; T�Þ under estimates T on KðXÞ. Moreover, for every f 2 LipLðXÞ, the errorestimate

R½f �j j 6 L T½1� R½k � k2���� ���� �1=2

ð43Þ

holds.

Proof. This is an immediate consequence of Theorem 4.2 together with the fact that

R½k � k2� ¼Xn

i¼0

T /ik � �xcg ½/i; T�k2

h i: �

5. Construction of over estimators of linear functionals

In this section we describe a method for computing over estimators used in (29) to approximate any given linear func-tional. There are two fundamental differences between under and over approximation on KðXÞ of linear functionals. Indeed,while the domain X for under estimators can be taken any compact convex set, we will see that the admissible domains forover estimators cannot be other than polytopes. The second remarkable difference is that the vertices of the polytope con-stitute a subset of the nodes for any over estimator of the form (29). Thus, in this section X will always denote a (convex)polytope with a non-empty interior (that is, the convex hull of ðmþ 1Þ vertices v0;v1; . . . ;vmf g in Rd). This section showsthat analogous results, stated for under approximation of non-negative linear functionals, are also valid for over approxima-tion on arbitrary convex polytopes.

We define the notion of (generalized) barycentric coordinates in the remainder of this paper as follows: let x be an arbi-trary point of X. We call barycentric coordinates of x with respect to the set x0; . . . ; xnf g any set of real coefficients kiðxÞf gn

i¼0

depending on the vertices of X and on x such that all the three following properties hold true:

kiðxÞ P 0; i ¼ 0; . . . ; n; ð44ÞXn

i¼0

kiðxÞ ¼ 1; ð45Þ

x ¼Xn

i¼0

kiðxÞv i: ð46Þ

Recall that these coordinates exist for more general types of polytopes. The first result on their existence was due to Kalman[17, Theorem 2] (1961). Eq. (46) is said to be a barycentric coordinate representation of any x 2 X. Here and in what followsthe set of points x0; . . . ; xnf g satisfies X ¼ convð x0; . . . ; xnf gÞ. For any given non-negative functional U on CðXÞ the functional

Uvert½f � ¼Xm

i¼ 0

U kverti

� �f ðv iÞ ð47Þ

will play an essential role in all the following discussion. Note that this functional uses only the vertices of the polytope.Throughout this section kvert

i ðxÞ; i ¼ 0; . . . ;m, will denote the barycentric coordinates of x with respect to all verticesv0; . . . ;vm of the polytope X.

Throughout this section, we assume that T is a given strictly non-negative function on CðXÞ, i.e. T½f � > 0 wheneverf 2 CðXÞ, f ðxÞP 0, and f is non-identically null on X.

We now collect some useful properties of the strict non-negativity of the linear functional T. Result given in Property 1shows one of the fundamental difference between the theories of under and over approximations of non-negative linearfunctionals.

Theorem 5.1. Let T be a strictly non-negative linear functional on CðXÞ. Assume that there are n points y0; . . . ; yn 2 X, and realnumbers a0; . . . ; an, such that, for any convex function f, it holds

T½f � 6Xn

i¼ 0

aif ðyiÞ: ð48Þ

Then the following statements are satisfied:

1. X is the convex hull of y0; . . . ; yn.2. n P m.3. The set fy0; . . . ; yng contains all the vertices of X.

B. Achchab, A. Guessab / Applied Mathematics and Computation 247 (2014) 417–430 429

4. The linear functional Tvert over estimates T on KðXÞ. Moreover, if A is any linear functional, which over estimates T on KðXÞ andX is a simplex then Tvert ¼ Avert .

Proof. We only give a sketch of the proof of (4), since the other facts are proved, in [15, Theorem 2.3] in the setting of inte-gral functionals, but similar arguments apply to the general functionals. To this end, we first observe the fact that Tvert overestimates T on KðXÞ is an immediate consequence of the classical Jensen’s inequality and the non-negativity of T. We onlyneed to prove that Tvert ¼ Avert on CðXÞwhen X is a simplex. By Proposition 2.2, T and A coincide for affine functions and sincebarycentric coordinates for a simplex are affine functions, it follows that for all f 2 CðXÞ the following hold:

Tvert½f � :¼Xm

i¼ 0

T½kverti �f ðv iÞ ¼ T

Xm

i¼ 0

kverti f ðv iÞ

" #¼ A

Xm

i¼ 0

kverti f ðv iÞ

" #¼Xm

i¼ 0

A kverti

� �f ðv iÞ :¼ Avert ½f �: �

The classical Jensen’s inequality is a useful result in several theoretical and applied fields. It provides a fundamental toolto over estimate convex functions at every point. Indeed, if f is convex then for any distribution of x0; . . . ; xn we have thefollowing:

f ðxÞ 6Xn

i¼0

kiðxÞf ðxiÞ; ðfor every x 2 X ¼ convð x0; . . . ; xnf gÞ: ð49Þ

Note that the definition of convexity is simply the statement that Jensen’s inequality holds.By (49), the functional

A½f � :¼Xn

i¼ 0

T½ki�f ðxiÞ ð50Þ

systematically over-estimates T½f � for any convex function f 2 KðXÞ. Hence, using Jensen’s inequality, we can always gener-ated a linear functional, which over estimates T on KðXÞ.

Lemma 2.4 shows that if f 2 C2ðXÞ then the error functional R ¼ T � A can be estimate as follows:

jR½f �j 6 jD2f jX2

R½ k � k2h i��� ���: ð51Þ

When the function is only Lipschitz, it is quite remarkable that also in the case of over estimation, it is possible to replace theerror bound (51) by:

Theorem 5.2. If f 2 LipLðXÞ then there holds the error estimate

R½f �j j 6 L T½1� R½k � k2���� ���� �1=2

: ð52Þ

Proof. We follow closely what was done in Theorem 4.2. Indeed, if f 2 LipLðXÞ, then arguing as in the proof of Theorem 4.2,we immediately get

R½f �j j 6 L T½1�Xn

i¼0

T kik � �xik2h i !1=2

:

As the reader may verify, the desired result is now an immediate consequence of the fact that

R k � k2h ih ��� ¼Xn

i¼0

T½kik � �xi�k2

�����#;

which can be easily obtained by using the properties of barycentric coordinates given in equations (44)–(46). h

In order to construct more refined over estimators, we now apply our technique developed in Section 2. Under thecondition f 2 C2ðXÞ, the next result, see [5], provides a refined version of the classical Jensen’s inequality.

Lemma 5.3. For every f 2 C2ðXÞ, the following estimate holds, for every x 2 X,

f ðxÞ 6Xn

i¼0

kiðxÞf ðxiÞ �kmin½f �

2

Xn

i;j¼0;i<j

kiðxÞkjðxÞkxi � xjk2: ð53Þ

On the basis of this observation, we are now ready to prove:

430 B. Achchab, A. Guessab / Applied Mathematics and Computation 247 (2014) 417–430

Theorem 5.4. Let X ¼ conv x0; . . . ; xnf gð Þ. Then, for every f 2 C2ðXÞ, we have

T½f � 6Xn

i¼ 0

T½ki�f ðxiÞ �kmin½f �

2

Xn

i;j¼0;i<j

T½kikj�kxi � xjk2: ð54Þ

Proof. Simply apply T on both sides of (53). h

Acknowledgments

The authors would like to thank financial support from the Volubilis Hubert Curien Program (Grant No. MA/13/286), andHassan I University, Settat, Morocco for hosting and supporting us during the research stay that led to this collaboration. Wewould also like to thank the referees for their careful reading of the paper.

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