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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.49 n.2 Bahía Blanca jul./dic. 2008
Best Local Approximations by Abstract Norms with Non-homogeneous Dilations
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 [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
![]() |
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 [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 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 [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
![∫ ∫ B f(x)d μɛ(x) → B f (x)dμ0(x ), f ∈ C1 (B),](/img/revistas/ruma/v49n2/2a09114x.png)
and for any ball
The assumption on the measure implies that
![( ∫ )1 ∕p p ∥F ∥ɛ = ∥F ∥Lp(μɛ) = ∥F ∥ dμɛ , B](/img/revistas/ruma/v49n2/2a09118x.png)
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 [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
![α α1 αn δɛx = δɛx = (ɛ x1,...,ɛ xn) ɛ > 0,](/img/revistas/ruma/v49n2/2a09139x.png)
![α = (α1,...,αn )](/img/revistas/ruma/v49n2/2a09140x.png)
![r(x - y)](/img/revistas/ruma/v49n2/2a09141x.png)
![ℝn,](/img/revistas/ruma/v49n2/2a09142x.png)
![r (x )](/img/revistas/ruma/v49n2/2a09143x.png)
![x ⁄= 0](/img/revistas/ruma/v49n2/2a09144x.png)
![r](/img/revistas/ruma/v49n2/2a09145x.png)
![-1 |δr (x)| = 1,](/img/revistas/ruma/v49n2/2a09146x.png)
![∑n x2 --j- = 1 . j=1 r2αj](/img/revistas/ruma/v49n2/2a09147x.png)
![r( . )](/img/revistas/ruma/v49n2/2a09148x.png)
![r(δɛx) = ɛr(x)](/img/revistas/ruma/v49n2/2a09149x.png)
![n ℝ](/img/revistas/ruma/v49n2/2a09150x.png)
![r,](/img/revistas/ruma/v49n2/2a09151x.png)
![x ⁄= 0](/img/revistas/ruma/v49n2/2a09152x.png)
![(r(x),x′)](/img/revistas/ruma/v49n2/2a09153x.png)
![δrx′ = x,](/img/revistas/ruma/v49n2/2a09154x.png)
![x′ = δr-1(x)](/img/revistas/ruma/v49n2/2a09155x.png)
![Sn- 1.](/img/revistas/ruma/v49n2/2a09156x.png)
![]() |
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 n-tuple of real numbers such that
, the measures
are given by
![]() |
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 n-tuple 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
![∑ β ∑ β1 βn p(x) = a βx = aβx1 ...xn , α.β≤m α.β≤m](/img/revistas/ruma/v49n2/2a09202x.png)
![β ∈ ℕn](/img/revistas/ruma/v49n2/2a09203x.png)
![α.β = m](/img/revistas/ruma/v49n2/2a09204x.png)
![p](/img/revistas/ruma/v49n2/2a09205x.png)
![α](/img/revistas/ruma/v49n2/2a09206x.png)
![m](/img/revistas/ruma/v49n2/2a09207x.png)
![p ∈ πm,α](/img/revistas/ruma/v49n2/2a09208x.png)
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 that
![∥F ɛ - Tɛ ∥ = o(ɛm ), ɛ → 0. m,α ɛ](/img/revistas/ruma/v49n2/2a09229x.png)
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 [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
![C- 1∥P∥0 ≤ ∥P∥ ɛ ≤ C ∥P ∥0,](/img/revistas/ruma/v49n2/2a09237x.png)
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
![|∂αp (0)| ≤ C ɛ- α.γ∥P ɛ∥ , i ɛ](/img/revistas/ruma/v49n2/2a09245x.png)
for any and
.
Proof. We consider the seminorm . Using the above proposition we have
![|∂γpɛi(x)| ≤ C ∥P ɛ∥ ɛ,](/img/revistas/ruma/v49n2/2a09249x.png)
![C > 0](/img/revistas/ruma/v49n2/2a09250x.png)
![∂ γpɛ(0 ) = ɛα.γ∂γp (0) i i](/img/revistas/ruma/v49n2/2a09251x.png)
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
![∥F ɛ - Tɛ ∥ ≤ o(ɛm ) + ɛs∥P ∥ = o(ɛl), l,α ɛ ɛ](/img/revistas/ruma/v49n2/2a09264x.png)
![s = min {α.β : l < α.β ≤ m }](/img/revistas/ruma/v49n2/2a09265x.png)
![m,α P ∈ Πk](/img/revistas/ruma/v49n2/2a09266x.png)
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
![-- -- (a) ∥E ɛ(F )∥ ɛ → ∥Φ m,α - PA,0(Φm,α)∥0.](/img/revistas/ruma/v49n2/2a09326x.png)
![-- ɛ- -- ɛ- E ɛ(F ) = Φ m,α + R m,α - PA ɛ,ɛ,α(Φ m,α + R m,α),](/img/revistas/ruma/v49n2/2a09327x.png)
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
![ɛ ɛ ɛ ɛ ɛ Q - Tm,α(Q ) = Q - (Tm,α(Q )) ∈ A 1.](/img/revistas/ruma/v49n2/2a09352x.png)
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 any
as
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
![]() |
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
![β1 βs a1x + ...+ asx ,](/img/revistas/ruma/v49n2/2a09424x.png)
![i β](/img/revistas/ruma/v49n2/2a09425x.png)
![i = 1,...,s](/img/revistas/ruma/v49n2/2a09426x.png)
![n ℕ](/img/revistas/ruma/v49n2/2a09427x.png)
![1 -- β .α = m](/img/revistas/ruma/v49n2/2a09428x.png)
![j -- β .α > m](/img/revistas/ruma/v49n2/2a09429x.png)
![j = 2,...,s](/img/revistas/ruma/v49n2/2a09430x.png)
![βj.α ⁄= βk.α](/img/revistas/ruma/v49n2/2a09431x.png)
![j ⁄= k](/img/revistas/ruma/v49n2/2a09432x.png)
![(a1,...,as) = c(v1,...,vs)](/img/revistas/ruma/v49n2/2a09433x.png)
![v1 ⁄= 0](/img/revistas/ruma/v49n2/2a09434x.png)
![c ∈ ℝ](/img/revistas/ruma/v49n2/2a09435x.png)
![V = (v1,...,vs)](/img/revistas/ruma/v49n2/2a09436x.png)
![ℝs](/img/revistas/ruma/v49n2/2a09437x.png)
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
![]() |
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
![]() |
as and
Proof. We will prove the following inequality
![]() |
Let and
be two arbitrary polynomials and set
with
and
Then
![-m- ɛ ɛ ∥E ɛ(F )∥ ɛ = ∥E ɛ(G )∥ɛ = ɛ ∥G - ¯PA ɛ,ɛ(G )∥ɛ](/img/revistas/ruma/v49n2/2a09519x.png)
![-- -- ≤ ɛ-m ∥G ɛ - (U + Zɛ)ɛ∥ɛ = ɛ-m ∥Tɛ- (G - U) - Z ɛ∥ ɛ + o(1). m,α ɛ](/img/revistas/ruma/v49n2/2a09520x.png)
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
![∥E (F) - (T-- (G - U ) - V )∥ - → 0. ɛ m,α 0 0 ɛ](/img/revistas/ruma/v49n2/2a09551x.png)
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
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
![min m,α ∥Tm,α(G - U) - V ∥0. U∈A0, V∈ Πk](/img/revistas/ruma/v49n2/2a09571x.png)
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
![]() |
where .
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