Segio Favierand Claudia Ridolfi

Instituto de Matem′atica Aplicada San Luis,Conicet and Universidad Nacional de San Luis,Ej′ercito de los Andes 950,San Luis,Argentina

?

Weighted Best Local Approximation

Segio Favier?and Claudia Ridolfi

Instituto de Matem′atica Aplicada San Luis,Conicet and Universidad Nacional de San Luis,Ej′ercito de los Andes 950,San Luis,Argentina

Abstract.In this survey,the notion of a balanced best multipoint local approximation is fully exposed since they were treated in the Lpspaces and recent results in Orlicz spaces.The notion of balanced point,which was introduced by Chui et al.in 1984 are extensively used.

Best local approximation,multipoint approximation,balanced neighborhood.

AMS Subject Classifications:41A10,41A65

Analysis in Theory and Applications

Anal.Theory Appl.,Vol.32,No.1(2016),pp.1-19

1　Introduction

The notion of a best multipoint local approximations of a function is fully treated in[2] where the Lpnorm is used.Later,other approaches to best multipoint local approximations with Lpnorms appeared in[7]and[8].And finally,for Orlicz norms,we mention[3,5,9,12,13]and for a general family of norms[6,10]and[14].However,in[2],Chui et al.introduced the concept of balanced points in Lpwhich includes different importance in each point.

More precisely,a rather general view of the problem is as follows.Let f:R→R be a function in a normed space X with norm‖·‖.Let Πmdenote the set of polynomial in R of degree less or equal than m and suppose Πm?X.Consider n points x1,···,xnin R and a net of small Lebesgue measurable neighborhoodsaround each point xisuch that the Lebesgue measuregoes to 0 as δ→0 for i=1,···,n.We select the best approximation to f near the points x1,···,xnby polynomial in Πm.Formally,for each ntuple of neighborhoods V1,···,Vn,we consider the polynomial gV∈Πm,which minimizes

forall h∈Πm,whereViandItis well knownthat a best‖·‖-approximation gValways exists since Πmhas finite dimension.If any net gVconverges to a uniqueelement g∈Πm,as δ→0,then g is said to be a best local approximation to f at the points x1,···,xn.We will mention in this survey all the works which consider that the velocity of convergence|Vi|→0,as|V|→0,can be different at each point xi.According to[2],this problem has been treated considering the concept of balanced neighborhoods in local approximation and it reflects the different importance of the points x1,···,xn.We need to deal with the necessary definition of balanced neighborhoods in each context.

As we pointed out above,Chui et al.study in[2]this problem when the space X is the usual Lpspace,with the norm

where B is a measurable set.They get results for balanced and non balanced neighborhoods.At last they generalize the results to the case of Rkinstead of R.On the other hand,in[4],the authors get balanced results in Lpusing other technique.We will discuss the Lpproblem with more details in Section 3.

In[11]and[12]the authors study the problem in Orlicz spaces,it means,

where φ is a convex function,non negative,defined on R+0,and B is a Lebesgue measurable set.In these two works,the authors studied the best local approximation problem with the Luxemburg norm

and get results for balanced and non balanced neighborhoods.Furthermore,we can consider a different Luxemburg norms in Orlicz Spaces,that is

and it can generate different best approximation functions gVthan those obtained with the standard Luxemburg norm given in(1.2).In[13]the authors study the balanced neighborhoods problem with this norm using a different technique than that used in[11] and[12].

Moreover,in[9]the authors study the balanced problem in Orlicz spaces Lφwhen the error(1.1)does not come from a norm,but considering

The last three problems in Lpare equivalent,but in Orlicz spaces they are different problems and have different conceptsof balanced neighborhoods.InSection 4 we will present the three problems in Orlicz spaces in detail.

2　Notations

We now introduce some notation.Let B?R be a bounded open set,and set|B|for its Lebesgue measure.Denote by M the system of all equivalence classes of Lebesgue measurable real valued functions defined on B.

Let x1,···,xnbe n distinct points in B.Considera net of measurable setssuch thatwhere Viis a neighborhood of the point xiand

as|V|→0.It is easy to see that Vi=xi+|Vi|Ai,i≤i≤n,where Aiis a measurable set with measure 1.Henceforward,we assume the sets Aiare uniformly bounded.

For each p,1≤p≤∞,we consider the space Lp=Lp(B)and the following norms

and

for h∈M and p＜∞.Sometimes we writeinsteadwhere χBdenotes the characteristic function of the set W?B.

Let Φ be the set of convex functions φ:[0,∞)?→[0,∞),with φ(x)＞0 for x＞0,and φ(0)=0.For φ∈Φ define the Orlicz space

This space can be endowed with the Luxemburg norm‖f‖φdefined in(1.2)as well as with the norm‖f‖φ,Bdefined in(1.3).If φ(x)=xp,the last norm coincides with the norm‖h‖p,B.Sometimes we write‖f‖Lφ(W)instead of‖fχW‖φ.The space Lφwith both norms is a Banach space,and we refer to[1]for a detailed study of Orlicz spaces.

We recall that a function φ∈Φ satisfies the?2-condition if there exists a constant k＞0 such that φ(2x)≤kφ(x),for x≥0.We also say that φ∈Φ satisfies the?′condition if there exists a constant C＞0 such that φ(xy)≤Cφ(x)φ(y)for x,y≥0.Note that it is easy to see that?′condition implies?2condition.

Let f∈PCm(B),where PCm(B)is the class of functions in Lφ(B)with m?1 continuous derivatives and with bounded piecewise continuous mthderivative on B.

3　Balanced and non balanced problem in space Lp

In this section we present the results given in[2]and[4],both in Lpspaces,p≥1.In[2] Chui et al.introduce the balanced concept as follows.For each α∈R and k,1≤k≤n,wedenote

and assume the following condition which allows us to compare Vk(α)with each other as functions of α.

For any nonnegative real numbers α and β,and any pair j,k,1≤j,k≤n,

Given a collection of neighborhoodsand a set of n non negative real numbers α1,···,αnin R,we say that Vj(αj)is maximal if for all k,Vk(αk)=O(Vj(αj)).When it happens we write Vj(αj)=max{Vk(αk)}.

In the balanced case,the neighborhoods can have different measure,but it is not at random,there is a relationship between the measure of the sets|Vk|and the amount of information of f over the points ik.

Definition 3.1.A n-tuple of non negative integers(ik)is balanced if for each j such that ij＞0,max{Vk(ik+1/p)}=o(Vk(ij?1+1/p)).In this case,we say thatis a balanced integer and the neighborhoods Vkare balanced.

It is easy to see that to each balanced integer m+1 there corresponds exactly one balanced n-tuple(ik)such that

Example 3.1.If Lp=L2and the neighborhoods areand V2=then(0,0),(0,1),(1,1),(1,2)and(1,3)are balanced n-tuples,while (2,2),(1,0),(2,0)and(2,1)are non balanced n-tuples.

There exists a simple way to find all the balanced n-tuples.

Algorithm 3.1.It begins with the balanced n-tuplecorresponding to the balanced integer 0.Let(ik(l))be a balanced n-tuple.LetmaxTo build the next n-tuple,,putforandfor k∈/C.

In[2],the authors prove that this algorithm generates exactly all the balanced ntuples.

The following Lemma gives an order of the error produced in the approximation(1.1) with the norm‖·‖p.Also,this Lemma exposes how to define a maximal element.

Lemma 3.1.Let(ik)be an ordered n-tuple of nonnegative integers.Suppose h∈PC(l)(B),where l=max{ik}and h(j)(xk)=0,0≤j≤ik?1,1≤k≤n.Then

We now present the Lemma 3 stated in[2],which will be used in the sequel.This lemma have importance in the proof of the main Theorems.

Lemma 3.2.Let 1≤p≤∞and let Λ be a family of uniformly bounded measurable subsets of the real line with measure 1.Then there exists a constant M(depending on m and p)such that for all the polynomials P∈Πm,and all A∈Λ,

Now we present the first main result given in[2],which solved the problem of best local approximation,for balanced neighborhoods.In the proof they used the above lemmas.

Theorem 3.1.If m+1 is a balanced integer with balanced n-tuple(ik)and f∈PCl(B),l= max{ik},then the best local approximation to f from Πmis the unique g∈Πmdefined by the m+1 interpolation conditions f(j)(xk)=g(j)(xk),0≤j≤ik?1,1≤k≤n.

Given a balanced integer m+1,as a consequence of the Algorithm,there exists a following and previous balanced integerand for example,the following balanced integer will be m+Car(C),where Car(C)is the cardinality of the set C.So we have the following definition.

Definition 3.2.Given neighborhoods V1,···,Vn,p,and an integer m+1 we define:

?m+1 the smallest balanced integer greater than or equal to m+1.

We present the following auxiliary lemma from[2]which is used to prove one of the main results.

From Lemma 3.3,if there exists a best local approximation g,then it satisfy the equations f(j)(xk)=g(j)(xk),since

The n-tuple of neighborhoods(V1,···,Vn)satisfy

where ekis a fixed constant.

Remark 3.2.Given m+1,set C={k:1≤k≤n and=maxFrom the algorithm ek=0 for k/∈C and ek6=0 for k∈C.

Now we present the main Theorem from[2].

Theorem 3.2.Suppose m+1 is not balanced.Assume that each Akis either an interval for each Vkor is independent of the net{Vk}.Assume that the measure(|V1|,···,|Vn|)satisfies for each k,

with eka constant independent of the net{Vk}.Let JA(i,p)denote the minimum Lpnorm over the measurable set A of an ithdegree polynomial with unit leading coefficient.If f∈PCl(B), where l=and 1＜p≤∞,then the best local approximation to f from πmis the unique solution of the constrained lpminimization problem

where,if Akis an interval,we can replace

If p=1 the lpminimization may not have a unique solution;if it does,however,it is the best local approximation.

By the other hand,in[4]the balanced result(Theorem 3.1)is proved with other technique.The authors prove a Polya-type inequality for polynomials in Lpspaces and it has an application to best local approximation.The Polya-type inequality is the following.

Theorem 3.3.Let 0＜p≤∞,and m,n∈N.Let ik,1≤k≤n,be n positive integers such that i1+···+in=m+1.Then there exists a constant K depending on p,ik,for 1≤k≤n,such that

In[2]the authors prove that if(i1,···,in)is a balanced n-tuple and f is a function sufficiently differentiable in a neighborhood of the n-points x1,···,xn,the best local approximation is the classical Hermite polynomial on the points x1,···,xn,fixed from the interpolation conditions of the function f in xkup to order ik?1,i≤k≤n.In[4],the authors get a similar result for more general functions f.They introduce the following class of Lebesgue measurable functions.

Definition 3.3.Given p＞0 and m+1=i1+···+in,a function f belongs to the class Hm,p(i1,···,in)if f∈Lp(B)and there exists a polynomial H∈πmsatisfying

These classes are similar to those introduced in[4]and[14],for n=1.As a consequence of Theorem 3.3 it follows that the polynomial H is unique if f∈Hm,p(i1,···,in). It is called the generalized Hermite polynomial of f on x1,···,xnwith respect to the n-tuple (i1,···,in).Moreover,

Theorem 3.4.Let f∈Hm,p(i1,···,in).Then the best local approximation to f from πm,say H, is the generalized Hermite polynomial of f on x1,···,xnwith respect to the n-tuple(i1,···,in).

In particular,under certain differentiability conditions of the function f,from Lemma 3.1 the polynomial H,which interpolates the data f(j)(xk),0≤j≤ik?1,1≤k≤n,satisfies

If in addition,(i1,···,in)is a balanced n-tuple,then f∈Hm,p(i1,···,in)and H fulfills(3.2). Therefore,it is obtained as a consequence of Theorem 3.3 the analogous result for balanced k-tuples,proved in Theorem 1 of[2].However in[4]the authors do not assume any condition of classic differentiability over the function f.

Here we have exposed two techniques to get the existence of multipoint local approximation with balanced neighborhoods.They can be generalizated to Orlicz spaces as we show in the next section.

4　Balanced and non balanced problems in Orlicz spaces Lφ

Inthissectionwepresentthebestlocal approximationpolynomialsusingbalanced neighborhoods in Orlicz spaces Lφ.According to the norm that we consider to minimize the error(1.1),we obtain three different problems which we include in the following subsection.

4.1Best Local Approximation in Lφwith norm‖·‖φ

Inthis subsectionwe will exposethe existenceof best multipoint local‖·‖-approximation to a function f from Πnfor a suitable integer n,it means,for the balanced and non balanced cases.This problem is considered in an arbitrary Orlicz space Lφwith the Luxemburg norm‖·‖φ.We refer to[1]for a detailed treatment of Orlicz spaces.For this purpose,we introduce the concept of‖·‖-balanced integer in this context.The following results follow the pattern given in[2]for Lpspaces and they appeared in[11]and[12]. Now we assume in this article that φ∈Φ and it satisfies the?2-condition and recall thatis a net of union of neighborhoods of the points x1,···,xnand denote by gVa best‖·‖φ?approximation to f from Πmon V,it means,

For each α＞0 and 1≤k≤n,we denote

and instead of 3.1 we assume in this context that for any nonnegative integers α and β, and any pair j,k,1≤j,k≤n,either

Let(ik)be an ordered n-tuple of nonnegative integers.We say that vj(ij)is maximal if vk(ik)=O(vj(ij))for all 1≤k≤n.We denote it by

Definition 4.1.An n-tuple(ik)of nonnegative integers is said to be‖·‖φ-balanced if for each ij＞0, If(ik)is‖·‖φ-balanced,we say thatis a‖·‖φ-balanced integer.

Next we set,from[11],an example of‖·‖φ-balanced integers.

Example 4.1.Define

It can be seen that φ satisfies the?2-condition(see[1],pp.30).Given x1,x2and let the neighborhoods satisfy|V2|=|V1|2.Thus these neighborhoods satisfy the conditions(4.1) and every integer is‖·‖φ-balanced.

In[11]an algorithm is presented and it generates all the‖·‖φ-balanced integer as in Lp.

Algorithm 4.1.Begin with the‖·‖φ-balanced n-tuple〉corresponding to the‖·‖φ-balanced integer 0.Then,givenfor s≥0,set=We build the next‖·‖φ-balanced n-tupletakingfor k∈C,andfor k∈/C.

Remark 4.1.It is proved in[11]that to each‖·‖φ-balanced integer there corresponds exactly one‖·‖φ-balanced n-tuple.Also an integer m+1 is‖·‖φ-balanced if only if m+1=for some(ik)generated by this algorithm.

Now,we cite from[11]the following auxiliary lemmas and the first main result.Instead of Lemma 3.1,in[11],the authors prove the following auxiliary result.

Lemma 4.1.Let(ik)be an increasing ordered n-tuple of nonnegative integers.Suppose h∈PCl(X),where l=max{ik},and h(j)(xk)=0,0≤j≤ik?1,1≤k≤n.Then

Instead of Lemma 3.2 we have

Lemma 4.2.Let Λ be a family of uniformly bounded measurable subsets of the real line with measure 1 and let 0＜r＜1,then there exists a constant s＞0 such that

for all A∈Λ,and for all P∈Πm.

Proposition 4.1.Given an integer m+1,consider the Definition 3.2 for the Luxemburg norm,then

a)If ij+1=then max

We now present the first important result from[11]concerning the behavior of a net {gV}|V|＞0of best‖·‖φ-approximations from Πm,as|V|→0.

Theorem 4.1.Let m+1 be a positive integer and l=If f∈PCl(X)and{gV}|V|＞0is a net of best‖·‖φ-approximations of f from πmon V,then{gV}|V|＞0is uniformly bounded on X.

Using the same technique it is obtained

Lemma 4.3.Given an integer m+1,set l=max{ik}.If f∈PCl(X),and{gV}|V|＞0is a net of best‖·‖φ-approximations of f from πmon V,then

0≤j≤ik?1,1≤k≤n.

Thus,using the‖·‖φ-balanced definition,it follows the main result of[11].

Theorem 4.2.Let(ik)be a‖·‖φ-balanced n-tuple and let 0＜m+1=∑ik.If l=max{ik}, f∈PCl(X),then the best local‖·‖φ-approximation to f from Πmis the unique g∈Πmdefined by the m+1 interpolation conditions

where 0≤j≤ik?1,1≤k≤n.

Now,we cite the following results from[12],which are a continuity of the above analysis.

for k=1,2,···,n and j=0,1,···,ik?1.As a consequence of Lemma 4.3 we have

Consider the following basis for Πm,say{uj,k}∪{wr},with k=1,2,···,n,j=0,1,···,and r=1,2,···,which satisfies

where g(j)(xk)=aj,k,k=1,2,···,n and j=0,1,···

Thus,if thereexistsabestlocal approximation g,since(4.6)it will satisfytheequations f(j)(xk)=g(j)(xk),The remainingdegrees of freedom must then be chosen so as to minimize the local Lperror around thepoints. It required a delicate analysis and it appears in[12].There are many auxiliary lemmas here to prove the main result,which solve the best local approximation problem when (m+1)is not a balanced integer.As an example of the auxiliary lemmas we expose the following(see[12]).

Given a non balanced integer m+1,set

Lemma 4.4.There holds

Lemma 4.5.Let φ∈Φ satisfying the?2-condition.If for each x≥0 there exists

then ψ(x)=xpfor some p≥1.

Lemma 4.6.For every k∈C,set

such that

If

exists for x≥0 and

for each k∈C,then

where

We now can give,under certain conditions,the existence of the best local‖·‖φapproximation.

Theorem 4.3.Let φ∈Φ satisfying the?2-condition.Assume that there exists

for all x≥0,and therefore this limit is xpfor some p≥1.Let m+1 be a non‖·‖φ-balanced integer and

For each k∈C suppose

where,for k∈C,JAk(ik,p)is the minimum Lpnorm over Akof an ikthdegree polynomial with unit leading coefficient.In particular,if(4.8)has a unique solution g,then

and therefore this is a best local‖·‖φ-approximation to f from Πmon{x1,···,xn}.

The following example shows that lim|V|→0gVmay not exist if φ does not satisfy the assumption that

exists for all x≥0.The proof is in[12].

Example 4.2.Let x1=0,x2=1,A1=A2=|V1|=2δ,|V2|=δ,forand let Πm=Π0be the subspace formed by the constant functions in Lφ.Define

4.2Best local approximation with the norm‖·‖φ,B

In this section we expose the analysis given in[13]to prove the existence of the best local approximation to a function f,with balanced neighborhoods,when the error(1.1)is the following.Denote gV∈Πmsuch that

These best approximation can be different to that given with the Luxemburg norm‖·‖Lφ(V).

The analysis in[13]follows the pattern used in[4]for Lpspaces.We begin with the following auxiliary lemmas and properties.

If φ satisfies the?′-condition,it is easy to see that there exists a constant K＞0 such that

We assume in this section that φ∈Φ and it satisfies the?′-condition.

Proposition 4.2.The family of all seminorms‖·‖φ.Vwith|V|＞0,has the following properties:

(b)If f,g∈Lφ(X)satisfy|f|≤|g|on V,then‖f‖φ,V≤‖g‖φ,V.The inequality is strict if |f|＜|g|on some subset of V with positive measure.

(c)There exists a constant M＞0 such that

for all pair of measurable sets G,D,with G?D and|G|＞0.

Lemma 4.7.There exists a constant M＞0 such that

for all P∈Πm,[a??,a+?]?B,and 0≤j≤m.

Lemma 4.8.Let C?B be an interval,E?C,|E|＞0.For all P∈Πm,there exists an interval F:=F(E,P)?C such that

Now,we present the main result concerning Po′lya inequality in Lφ.

Theorem 4.4.Let φ∈Φ,and n,m∈N.Let ik,1≤k≤n,be n positive integers such that=m+1.Let Ek,1≤k≤n,be disjoint pairwise compact intervals in R,with 0＜|Ek|≤1. Then there exists a positive constant M depending on φ,ik,and Ek,1≤k≤n,such that

Now,we will introduce the concept of balanced integer in that context.For each α∈R and k,1≤k≤n,we denote

The following condition allows us that Ak(α)can be compared with each other as functions of α when|V|→0.

For any nonnegative integers α and β,and any pair j,k,1≤j,k≤n,

Let(ik)be an ordered n-tuple of nonnegative integers.We say that Aj(ij)is a maximal element of(Ak(ik))if Ak(ik)=O(Aj(ij))for all 1≤k≤n.We denote it by

Observe that

Definition 4.2.An n-tuple〈ik〉of nonnegative integers is balanced if

To each balanced integer there corresponds exactly one balanced n-tuple.Moreover, there are an algorithm which gives all balanced n-tuples which it is proved in[13].

Given(ik),set

Algorithm 4.2.Let vqbe a balanced integer and letbe the corresponding balanced n-tuple.To build the next n-tuple,,put+1 for k∈Candfor k∈/C

The algorithm generates n-tuples candidates to be balanced.We can observe it with the following example.

Example 4.3.Define φ(x)=x3(1+|lnx|),x＞0,and φ(0)=0.Consider two points x1,x2

with|V1|=δ4/3,|V2|=δ1/3,and A1=A2=[0,1].The 2-tuple(0,1)is balanced.Here,the set C((0,1))={0},however(1,1)is not a balanced 2-tuple.

Lemma 4.9.Let(ik)be an ordered n-tuple of nonnegative integers.Suppose h∈PCl(X),where l=max{ik}and h(j)(xk)=0,0≤j≤ik?1,1≤k≤n.Then

If a polynomial P∈Πm,m+1=satisfies P(j)(xk)=f(j)(xk),1≤j≤ik?1,1≤k≤n, we call it the Hermite interpolating polynomial of the function f on{x1,···,xn}.

Now,we are in condition to prove the main result in this Section.

Theorem 4.5.Let(ik)be a balanced n-tuple and m+1=If l=max{ik}and f∈PCl(X), then the best local approximation to f from Πmon{x1,···,xk}is the Hermite interpolating polynomial of f on{x1,···,xn}.

Proof.Let H∈Πmbe the Hermite interpolating polynomial and let{gV}be a net of best approximations of f from Πmrespect to‖·‖φ,V.From Lemma 4.9,

Using Theorem 4.4 and the equivalence of the norms in Πm,we get

So,the definition of balanced n-tuple implies gV→H,as|V|→0.

4.3Best Local φ-approximation

In this section we present the analysis of the problem given in[9].Here the authors study the existence of the best local approximation,with balanced neighborhoods,when the error(1.1)is the following

The technique used in[9]follows the pattern used in[2].Assume that φ∈Φ satisfies the?′-condition.

Given a net of neighborhoods{V},denote for each 1≤k≤n and β∈R

Assume for any α,β≥0 and any j,k such that 1≤j,k≤n,that either

orboth.Then,cj(αj)is themaximal ofthe n-tuple(ck(αk)),with αk∈R,if forall k,1≤k≤n, ck(αk)=O(cj(αj)).We denote it by

Definition 4.3.An n-tuple(ik)of nonnegative integers is said to be φ-balanced if for each j such that ij＞0 and εj＞0,

If(ik)is φ-balanced,thenis said to be a φ-balanced integer.

The n-tuple(Vk)is said to be φ-balanced neighborhoods if the dimension m+1 of the space Πmis a φ-balanced integer.

To each φ-balanced integer there corresponds exactly one φ-balanced(ik).

Remark 4.2.If φ(x)=xp,1≤p＜∞the last definition of φ-balanced is equivalent to those considered by Chui et al.in[2].

Example 4.4.Let φ(x)=x3(1+|lnx|)with φ(0)=0 a convex function that satisfies the?′condition and(|V1|,|V2|)=(δ,e?1/δ),for δ＞0;then each integer m is a φ-balanced integer.

Now we state an algorithm that generates all the φ-balanced n-tuples.

Algorithm 4.3.Begin with the φ-balanced n-tuplecorresponding to the φbalanced integer 0.Givendetermine a maximal element ofsaymaxand definefor k6=k?and+1 for k=k?.

In[9],the authors proved the following lemma.

Lemma 4.10.For

a)The above algorithm generates all φ-balanced(ik).

As we see in the following example,the lemma gives a way to find candidates of φ-balanced n-tuples.

Example 4.5.If φ(x)=x3(|lnx|+1)with φ(0)=0 and(|V1|,|V2|)=(δ,δ4),for δ＞0,then in the first step the algorithm generates the 2-tuple(1,0)and the corresponding maximal max{ck(ik)}=c1(1)is unique.However the second 2-tuple generated by the algorithm is (2,0)and it is not φ-balanced.

Now we expose the auxiliary lemmas given in[9].

Lemma 4.11.Let i1,···,inbe nonnegative integers.Suppose h∈PCl(X),where l=max{ik}, and that h(j)(xk)=0,1≤j≤ik?1,1≤k≤n.Then

As a Corollary of Lemma 3.1,we mention the following result.

Proposition 4.3.Let Λ be a family of uniformly bounded measurable subsets of the real line with measure 1.Let P(x)=b0+b1x+···+bmxmbe an arbitrary polynomial of degree m.Then there exists a constant M(depending on m)such that for all P(x)and all A∈Λ,

where 0≤k≤m.

Instead of lemma 3.3,in[9]the authors prove the following two lemmas.

Lemma 4.12.Given C,set a φ-balanced n-tuple(ik)such that m+1=and define l= max{ik}.If f∈PCl(X)and{gV}is a net of best φ-approximations,then there exists M＞0 such that for all|V|＞0,

Lemma 4.13.Given V and a φ-balanced n-tuple〈ik〉such that m+1=define l= max{ik}.If f∈PCl(X)and{gV}is a net of best φ-approximations,then for each k

where 0≤j≤ik?1.

Using lemma 4.13 and the definition of φ-balanced n-tuple it is obtained the main result in that context,which solve the best local approximation problem when the neighborhoods are φ-balanced.

Theorem 4.6.If m+1 is a φ-balanced integer with φ-balanced(ik)and f∈PCl(X),(l= max{ik}),then the best local φ-approximation to f from Πmis the unique g∈Πmdefined by the m+1 interpolation conditions

where 0≤j≤ik?1,1≤k≤n.

Acknowledgments

The author would like to thank Professor M.Marano for his helpful comments about this survey.

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10.4208/ata.2016.v32.n1.1

4 November 2014;Accepted(in revised version)11 April 2015

?Corresponding author.Email addresses:sfavier@unsl.edu.ar(S.Favier),ridolfi@unsl.edu.ar (C.Ridolfi)