as are studied.]]>
Norma Yanzón and Felipe Zó
In memorian of Mischa Cotlar.
Abstract. We introduce a concept of best local approximation using abstract norms and nonhomogeneous dilations. The asymptotic behavior of the normalized error function as well as the limit of some net of best approximation polynomials as are studied.
2000 Mathematics Subject Classification. 41A65.
Key words and phrases. Best local approximations, function norms, nonhomogeneous dilations.
The authors where supported by UNSL, CONICET and FONCYT grants.
The notion of a best local approximation of a function has been introduced by Chui, Shisha and Smith [6] in the seventies although its origin goes as far as the paper of J. Walsh [23]. A rather general view of the problem is as follows. Let be a function in a normed space with norm . Let denote a subset of , consider points in and small neighborhoods around each point such that shrinks down to the point as , for . We wish to approximate near the points using an element of . For each we select a which minimizes

Thus we have to be the set of cluster points of the net as , which may be the empty set, a singleton or a set with more than one element, see [7] for one dimensional examples with non smooth functions and [14] for dimensional examples where or card even for functions , the algebraic polynomials of degree at most as the approximant class and to be the norm. In many situations has one element and it is called the best local approximation of [23], [6], [7],[5], [25], [24] and more recently [9], [10] for one point. The case of more than one point, sometimes called the best multipoint local approximation is fully treated in [4] where the norm is used, see [20] and [1] for other approaches to best multipoint local approximations with norms, for Orlicz norms see [13],[9],[15] and for a general family of norms [10] and [11].
The minimizing problem in (1) for the particular case , and , and the algebraic polynomials of degree at most , is related to the following problem. For simplicity let us take and let be the unique polynomial in , which minimizes

where . It is readily seen that where , and is the minimum problem described in (1) for this particular choice of norm. Not always the relationship between the best approximations and is so easily described as above, and some normalization in the norm used in problem (1) it is necessary to obtain a relationship between them see [18], [26] and [9]. Of course the problem (1) and the normalized one may have different solutions, see the last section of this paper. In this paper as it was done in [11] we study best approximation problems related to (2) and in term of these best approximations we define best approximations which play the role as the solutions of the problem (1) although in general they will give origin to different notions of best local approximations.
In [11] we studied the best local approximation problem, where the notion of closeness was given by a very general family of function seminorms acting on vector valued Lebesgue measurable functions. These seminorms embraced by far the norms used in these sort of problems, for example , Orlicz or Lorentz norms. The fact to consider best local approximation problems on vector valued functions of several variables it was due to understand better the solution to the so called multipoint best local approximation problems given in [4], [13] and [20], also this general set up gives origin to best local approximation problems not considered before, even using the standard norms.
The main goal of this paper is to consider, within the general frame given by [11], best local approximation on regions induced by dilations of the form as treated in [26], [14].
However, we should point out that our presentation doves not cover all the problems of the paper [4], for example the case when we approach a function on small neighborhood with polynomials of degree at most , the number does not divide , and the neighborhood shrink dawn to the point with different velocity at each , for . This last problem it was solved rather exhaustively for norms in [4] but it remains open in other general norms, for example in Orlicz norms, for a recent contribution in this direction see [15].
It is known, [14], that when nonhomogeneous dilations are used in best local approximation problems, the class of algebraic polynomials of degree at most is not suitable as an approximation class and should be replaced by a class which depends de tuple , see Definition 3.1. This paper extends results of [9] and [26] among others.
]]> We will work with a family of function seminorms , , acting on Lebesgue measurable functions where , and denotes the euclidean norm on .We assume the following properties for the family of function seminorms ,
(1). For and we have for every provided and
(2). If is the function we have for all
(3). For every we have as where is the set of continuous functions Moreover is a norm on
From now on, if we do not specify the contrary, the statements will be valid for an abstract family of seminorms , fulfilling conditions (1)(3).
In order to give examples of norms , with the properties (1)(3) we recall a definition of convergence of measures early given in [16]. See also [2] for the notion of weak convergence of measures in general.
Definition 2.1. Let , be a family of probability measures on . We say that the measures converge weakly in the proper sense to the measure if we have
and for any ball
]]> The assumption on the measure implies thatis actually a norm on for and where stands for any monotone norm on A seminorm fulfilling a property like (1) is called a monotone norm. We use a monotone norm on to assure property (1) for the family of seminorms , It is worthy to note we will not need this property on in proving some convergence results, see [10]
Let be in ; it is readily seen, by using the definition of weak convergence of measures, that there exists such that if for some then . Moreover we have that converges as to the norm if
In the next example, and for the most of the paper, we will consider a fixed notion of dilation on namely

where denotes the Lebesgues measure over the unit sphere and is the diagonal matrix which generates the semigroup of dilations We refer to the article [8] for an early use of this dilations in harmonic analysis or for more general dilations see [21]. In the following example we introduce measure adapted to the dilations which plays an analogous role to those introduced in [18].
Example 2.2. Let be a fix ntuple of real numbers such that , the measures are given by
]]> (4) 
where , , and . The following condition on the weight function will be assumed

We say that the weight function is a radial function with respect to if when
Remark 2.3. If the weight function is a radial function, then the measures introduced in Example 2.2 converges weakly to the measure

where is a constant depending upon the weight function
The proof of this last remark follows the same pattern of Lemma 1 in [18] and now we will point out the necessary modifications of the proof.
Let us consider the function and set for .
To deal with the weights we consider a change of variables of polar type induced by the function given by (3) .
]]> Using the above formula and following the steps of the proof of Lemma 1 in [18], we can see that 
where and Now it is easy to obtain the weak convergence of the measures to and that the constant in equation (6) it turns to be .
3. The Taylor polynomial and the limit of best approximation polynomials.
Throughout this paper will denote a fix ntuple of real numbers such that and .
Definition 3.1. The class Given a positive number we say that a real polynomial is in the class if it is of the form
Note that in the case we obtain the classical definition of polynomial of degree . We denote by the set .
It is worthy to note that when we make reference to a polynomial of degree we do not mean the classic degree of polynomial. For example, for there exist polynomials of degree and they are of the form , for in .
Given a function and a family of seminorms as in section 2, we introduce a general version of the "Peano's definition" of the Taylor polynomial, see [3], [8], [17], [18], [22], [26], [27]. We will use the notation .
]]> Definition 3.2. A function has a Taylor Polynomial of degree , if there exists such thatWe write if the function has a Taylor Polynomial of degree .
To prove the uniqueness of the Taylor polynomial we need the next result which is a consequence of an usual compactness argument. The proof is essentially given in [11], and it depends basically of the properties of the family of seminorms. See also [18].
Proposition 3.3. There exist and such that for every
for every
We make use of the standard notation, for .
Proposition 3.4. There exists a constant depending only on and the family of seminorms such that for any there holds
Proof. We consider the seminorm . Using the above proposition we have
The next proposition is a consequence of Proposition 3.4.
Proposition 3.5. The polynomial in Definition 3.2 is unique.
Proposition 3.6. If the function has the Taylor polynomial of degree , then the Taylor polynomial of degree is given by (if there exists such that ) .We set for the vector .
Proof. We have
Let , be a family of seminorms and be a fixed measurable function such that and are finite for all For any such has a meaning the following definition.
Definition 3.7. Set and for any polynomial in which minimizes
]]> Although the best approximation polynomial is not unique in general, through this paper the notation does not mean a set of best approximation polynomials but any arbitrarily chosen polynomial in this set. We have the existence of , at least for all small by Proposition 3.3.The next statement has its origin in [23] using the norm, and since then similar versions in in one and several real variables appeared. Results dealing with weighted Luxemburg norms appeared recently in [9] and [10].
Proof. In fact and by Proposition 3.4 it follows and □
As in [6] we call the limit of as the best local approximation to .
4. The asymptotic behavior of the error.
Let be a subspace of polynomials and let be a Lebesgue measurable function. Set for a polynomial which is a best approximation of the function with the seminorm Observe that is a polynomial in which is a best approximation of the function with the seminorm from the class and we will also denote it by We insist that means, in our notation, a fixed best approximation polynomial and not a set of them.
Let be the error function where . Next, we will obtain an expression for the function which has its origin in [19] see also [17, 18].
Let be in and set for the Taylor polynomial of of degree ; then by definition we have with and Moreover, observe that and where we have used that Using the equality we obtain the following result
Proposition 4.1. Let be a function in , and Then
]]>as
Proposition 4.1 is useful when for every The case with was considered in [18] and [17] for weighted norms and in [9] for the Luxemburg norm. It is easy to find and for every The following result is relevant to this matter.
Theorem 4.2. Let be a subspace of polynomials such that Then for all
Proof. Let . We shall see if for all , then for all . Let be in i.e., with As we have
Thus and there exists such that Therefore so We have proved that for all Since we get Therefore, for all .
Clearly, there is a linear space such that and , then . □
Theorem 4.3. Let be in and for every Then as as if is a strictly convex norm. We have denoted by a polynomial in which is a best approximation of with respect to the norm
]]> Proof. Let us begin with . By Proposition 4.1 we have, for anyas Therefore

Let be a sequence tending to zero such that

Set then By Proposition 3.3 we can select a convergent subsequence of which is again denoted by and then we have , as . Then Thus we have

To prove , consider any sequence and select , then We will prove which implies (b). In fact we may assume, by taking subsequences if it is necessary, that as Thus by Since is a strictly convex norm we have □
Consider the set where , for . Then and for every .
We now introduce an useful example of a subspace such that . Consider the set
]]> (8) 
where, and Now , see [11] in Proposition 4.2. Thus it is not possible to use Theorem 4.3 to study the function error with . The next condition on will be significant in the future and it was used in [11] to consider cases such as
A subspace of polynomials which does not satisfies is given in the following example.
Example 4.4. We denote by the set of all algebraic polynomials of the form
Let , where . It is clear that .
Condition 4.5. For we assume that if and then . Where and with .
Let be as in Example 4.4 , then the Condition 4.5 holds. We consider again the error function where and Set and recall that Then If we have

as and The next theorem give us a useful expression for the error function as well as we know the polynomials and used to describe it. With the notation observe that is the direct sum
]]> Theorem 4.6. Let be a function in and assume Condition 4.5 for . Set with and Then and and 
as Moreover the two families of polynomials and are uniformly bounded in for a fixed norm .
Proof. By (9) we have

Or else, since

as which is (10).
Now we will prove and are uniformly bounded in . For a norm in the expression

Since with we have .
By Proposition 3.4 we have for Thus and hence are uniformly bounded in To estimate the polynomials we note that for Then recall that , and is a norm there. □
Theorem 4.7. Let be in and assume Condition 4.5 for Then tends to

as and
Proof. We will prove the following inequality

Let and be two arbitrary polynomials and set with and Then
As using the definition of the polynomial we have

as and the right inequality of (11) holds.
To prove left inequality in (11) let be sequence tending to zero such that

By Theorem 4.6 we select a subsequence in such a way that the following limits exist:


Thus by (10) we have
]]> 

□
Proposition 4.8. Let be a strictly convex norm and assume that the subspace fulfills Condition 4.5. Then there exists a unique solution to the minimum problem in Theorem 4.7 .
Proof.
If and are solutions to the minimum problem in Theorem 4.7, we have with . Since is a strictly convex norm then by Condition 4.5, but . □
Theorem 4.9. Let be in assume Condition 4.5 for and that the minimum problem in 4.5 has a unique solution Then and as Moreover we have
Proof. By Theorem 4.6, Theorem 4.7 and (10) any convergent subsequence of the net will converge to a solution of the minimum problem in Theorem 4.7. Thus if this solution is unique, the whole net converges to the solution. □
5. The limit of best approximation polynomials.
]]> The main goal of this section will be to study the limit of as If and it will be enough to consider as sinceWe set as before Let be in then for ; see the proof of Theorem 4.6. Then Thus From Theorem 4.7 and Theorem 4.8 , this polynomial exists whenever is a strictly convex norm. Then where together with are the unique solution to the minimizing problem
Thus if we set for then in will be the unique solution to the problem

Thus, we have proved the following theorem.
Theorem 5.1. Let be in and assume Condition 4.5 for and that the minimum problem in (12) has a unique solution and denote by a polynomial in which minimizes with Then as
6. On the best local approximation using Luxemburg norm.
We denote by the measures given by (4), and let be a convex function such that , if . For any measurable set

By Proposition (2.3) in [11] we have converges to for any . Moreover the family has the properties (1),(2) y (3) of the section 2.
Recall that is the Luxemburg norm defined by (13) with the particular measure defined by (6) and denote by the Orlicz Space equipped with the norm The following result is known .
Remark 6.1. Let be a strictly convex function, then is a strictly convex Banach space with the Luxemburg norm .
By Remak 6.1 we can use Proposition 4.8 and Theorem 5.1 for the Luxemburg norm when is a strictly convex function. Also we are free to apply Theorem 5.1 in [11]. We apply these results in the particular situation described below.
Given and set and the norm as in (13) and the measure is the Lebesgue measure
Theorem 6.2. Let be a strictly convex function and let be the unique solution of the minimum problem

where . Then for a smooth function , converges to a polynomial , which is uniquely determined by the solution of the minimum problem in (4) of [11].
Now we will assume more restrictive conditions on the strictly convex function , namely and exists and it is a finite number for every . Clearly is convex function, and it is easy to see that for and if we have see [13]. From now on assume all the above conditions on the function .
]]> Theorem 6.3. For any let be the unique polynomial in which minimizes 
and Then the limit exist for smooth functions .
Theorem 6.3 may be obtained using results of [13] and [20]. For the case the polynomial it is very easy to characterize as the unique element which minimizes the problem

see [13]. For the case also can be obtained as a discrete minimum problem as in [20].
The best local approximation polynomials described in Theorem 6.2 and in Theorem 6.3 are different polynomials. Indeed, it is rather straightforward to obtain the next result when is a continuous function at each point and just a strictly convex function .
Theorem 6.4. For and let the unique polynomial which minimizes

Then the limit exist and it is characterized as the unique which minimizes
]]> 
We point out that to prove the existence of the polynomial in Theorem 6.3 still remains an open problem when is just a strictly convex function and the existence of the function is not required.
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Norma Yanzón
Instituto de Matemática Aplicada San Luis. CONICET and
Departamento de Matemática,
Universidad Nacional de San Luis,
(5700) San Luis, Argentina
nbyanzon@unsl.edu.ar
Felipe Zó
Instituto de Matemática Aplicada San Luis. CONICET and ]]>
Departamento de Matemática,
Universidad Nacional de San Luis,
(5700) San Luis, Argentina
fzo@unsl.edu.ar
Recibido: 1 de septiembrede 2008
Aceptado: 25 de noviembre de 2008