On-line version ISSN 1669-9637
Rev. Unión Mat. Argent. vol.49 no.2 Bahía Blanca July/Dec. 2008
Norma Yanzón and Felipe Zó
In memorian of Mischa Cotlar.
Abstract. We introduce a concept of best local approximation using abstract norms and non-homogeneous 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, non-homogeneous 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  in the seventies although its origin goes as far as the paper of J. Walsh . 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
where and If converges as to an element then is said to be a best local approximant of at the points .
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  for one dimensional examples with non smooth functions and  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 , , ,, ,  and more recently ,  for one point. The case of more than one point, sometimes called the best multipoint local approximation is fully treated in  where the norm is used, see  and  for other approaches to best multipoint local approximations with norms, for Orlicz norms see ,, and for a general family of norms  and .
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 ,  and . 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  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  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 ,  and , also this general set up gives origin to best local approximation problems not considered before, even using the standard norms.
However, we should point out that our presentation doves not cover all the problems of the paper , 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  but it remains open in other general norms, for example in Orlicz norms, for a recent contribution in this direction see .
It is known, , that when non-homogeneous 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  and  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 . See also  for the notion of weak convergence of measures in general.
and for any ball
The assumption on the measure implies that
is 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 
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  for an early use of this dilations in harmonic analysis or for more general dilations see . In the following example we introduce measure adapted to the dilations which plays an analogous role to those introduced in .
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  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 , 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 .
Throughout this paper will denote a fix n-tuple of real numbers such that and .
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 , , , , , , . We will use the notation .
We 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 , and it depends basically of the properties of the family of seminorms. See also .
We make use of the standard notation, for .
for any and .
Proof. We consider the seminorm . Using the above proposition we have
The next proposition is a consequence of Proposition 3.4.
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.
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  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  and .
Proof. In fact and by Proposition 3.4 it follows and □
As in  we call the limit of as the best local approximation to .
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 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 is useful when for every The case with was considered in  and  for weighted norms and in  for the Luxemburg norm. It is easy to find and for every The following result is relevant to this matter.
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 . □
Proof. Let us begin with . By Proposition 4.1 we have, for any
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
where, and Now , see  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  to consider cases such as
A subspace of polynomials which does not satisfies is given in the following example.
Let , where . It is clear that .
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
is a norm on Here we are using that the subspace fulfills Condition 4.5.
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
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
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. □
The main goal of this section will be to study the limit of as If and it will be enough to consider as since
We 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.
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  we have converges to for any . Moreover the family has the properties (1),(2) y (3) of the section 2.
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 . 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
where . Then for a smooth function , converges to a polynomial , which is uniquely determined by the solution of the minimum problem in (4) of .
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 . From now on assume all the above conditions on the function .
and Then the limit exist for smooth functions .
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 .
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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Instituto de Matemática Aplicada San Luis. CONICET and
Departamento de Matemática,
Universidad Nacional de San Luis,
(5700) San Luis, Argentina
Instituto de Matemática Aplicada San Luis. CONICET and
Departamento de Matemática,
Universidad Nacional de San Luis,
(5700) San Luis, Argentina
Recibido: 1 de septiembrede 2008
Aceptado: 25 de noviembre de 2008