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Revista de la Unión Matemática Argentina

On-line version ISSN 1669-9637

Rev. Unión Mat. Argent. vol.49 no.2 Bahía Blanca July/Dec. 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 P ɛ as ɛ → 0 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.

1. Introduction.

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 f : ℝn → ℝ be a function in a normed space X with norm ∥.∥ . Let V denote a subset of X , consider k points x1,...,xk in ℝn and small neighborhoods V (x) ɛ i around each point x i such that V ɛ(xi) shrinks down to the point xi as ɛ → 0 , for i = 1,...,k . We wish to approximate f near the points x1, ...,xk using an element of V . For each ɛ > 0 we select a Pɛ(f) = Pɛ ∈ V which minimizes

∥(f - P )XVɛ∥, (1)

where P ∈ V and  ⋃ Vɛ = ki=1 Vɛ(xi). If Pɛ converges as ɛ → 0 to an element P0 (f) ∈ V then P0(f) is said to be a best local approximant of f at the points x ,...,x 1 k .

Thus we have P (f) 0 to be the set of cluster points of the net {P (f)} ɛ as ɛ → 0 , 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 n -dimensional examples where P0(f ) = ∅ or cardP0 (f) > 1 even for functions f ∈ C ∞ , the algebraic polynomials of degree at most m as the approximant class and ∥.∥ to be the L2 norm. In many situations P (f) 0 has one element and it is called the best local approximation of f [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 Lp norm is used, see [20] and [1] for other approaches to best multipoint local approximations with Lp 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 - 1 ≤ xi ≤ 1 , V ɛ(xi) = (xi - ɛ,xi + ɛ) and  ∫ 1 1 ∥f∥ = ( - 1|f(x)|pdx)p , 1 < p < ∞ and V = πm the algebraic polynomials of degree at most m , is related to the following problem. For simplicity let us take k = 1,x = 0 1 and let P-(f) be the unique polynomial in  m π , which minimizes

∥(f - P )∥, (2)

where  m P ∈ π . It is readily seen that -- ɛ x P (f )(ɛ) = Pɛ(f)(x) where  ɛ f (x ) = f (ɛx) , and Pɛ(f ) is the minimum problem described in (1) for this particular choice of norm. Not always the relationship between the best approximations Pɛ(f) and -- P (f) 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 P (f ) ɛ 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 Lp , 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 Lp 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  α α δɛx = (ɛ 1x1,...,ɛ nxn ) 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 V (x ),...,V (x ) ɛ 1 ɛ k with polynomials of degree at most n , the number k does not divide n + 1 , and the neighborhood V ɛ(xi) shrink dawn to the point xi with different velocity at each xi , for i = 1,...,k . This last problem it was solved rather exhaustively for Lp 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  α1 αn (ɛ x1,...,ɛ xn ) are used in best local approximation problems, the class  m Π of algebraic polynomials of degree at most m is not suitable as an approximation class and should be replaced by a class Πm, α which depends m de n -tuple α = (α1,...,αn) , see Definition 3.1. This paper extends results of [9] and [26] among others.

2. The norm set up.

We will work with a family of function seminorms ∥.∥ɛ , 0 ≤ ɛ ≤ 1 , acting on Lebesgue measurable functions F : B ⊂ ℝn → ℝk, where  n B = {x ∈ ℝ : |x| ≤ 1 } , and | . |  denotes the euclidean norm on  n ℝ .

We assume the following properties for the family of function seminorms ∥.∥ɛ , 0 ≤ ɛ ≤ 1.

(1). For F = (f1,...fk), and G = (g1,...,gk), we have ∥F ∥ɛ ≤ ∥G ∥ɛ for every ɛ > 0, provided |fi(x)| ≤ |gi(x)|, i = 1, ...,k, and x ∈ B.

(2). If 1 is the function F (x) = (1,...,1), we have ∥1∥ɛ < ∞, for all ɛ ≥ 0.

(3). For every F ∈ Ck (B), we have ∥F∥ ɛ → ∥F ∥0, as ɛ → 0, where C (B ) k is the set of continuous functions F : B ⊂ ℝn → ℝk. Moreover ∥F ∥0 is a norm on Ck(B ).

From now on, if we do not specify the contrary, the statements will be valid for an abstract family of seminorms ∥.∥ɛ , 0 ≤ ɛ ≤ 1, fulfilling conditions (1)-(3).

In order to give examples of norms ∥.∥ɛ , 0 ≤ ɛ ≤ 1 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 μ ɛ , 0 ≤ ɛ ≤ 1, be a family of probability measures on B . We say that the measures μɛ converge weakly in the proper sense to the measure μ0 if we have

∫ ∫ B f(x)d μɛ(x) → B f (x)dμ0(x ), f ∈ C1 (B),

and  ′ μ0 (B ) > 0 for any ball  ′ B ⊆ B.

The assumption on the measure μ0 implies that

 ( ∫ )1 ∕p p ∥F ∥ɛ = ∥F ∥Lp(μɛ) = ∥F ∥ dμɛ , B

is actually a norm on Ck (B ) for ɛ = 0 and 1 ≤ p < ∞, where ∥ .∥ stands for any monotone norm on ℝk. A seminorm fulfilling a property like (1) is called a monotone norm. We use a monotone norm on ℝk to assure property (1) for the family of seminorms ∥.∥ ɛ , 0 ≤ ɛ ≤ 1. It is worthy to note we will not need this property on ∥ .∥ in proving some convergence results, see [10]

Let F be in Ck (B) ; it is readily seen, by using the definition of weak convergence of measures, that there exists ɛ0 = ɛ0(F ) > 0 such that if ∥F ∥ɛ = ∥F ∥Lp(μɛ) = 0, for some 0 < ɛ ≤ ɛ0 then F = 0 . Moreover we have that ∥F ∥ɛ = ∥F∥Lp (μɛ) converges as ɛ → 0 to the norm ∥F ∥ = ∥F ∥ p 0 L (μ0) if F ∈ C (B). k

In the next example, and for the most of the paper, we will consider a fixed notion of dilation on  n ℝ , namely

 α α1 αn δɛx = δɛx = (ɛ x1,...,ɛ xn) ɛ > 0,
where α = (α1,...,αn ) are given positive numbers. Associated with the above (nonisotropic) dilation is the metric r(x - y) on ℝn, where r (x ) is defined for any x ⁄= 0 as the unique positive number r which is the solution of  -1 |δr (x)| = 1, that is,
∑n x2 --j- = 1 . j=1 r2αj
The function r( . ) has the homogeneity property r(δɛx) = ɛr(x) and there is polar like decomposition of  n ℝ relative to r, i.e. to each vector x ⁄= 0 we assign (r(x),x′) where δrx′ = x, or x′ = δr-1(x) is in the unit sphere Sn- 1. Finally we can integrate in polar like coordinates, according to the next formula
∫ ∫ ∞ ∫ |α|-1 ′ ′ ′ ′ ℝn f(x)dx = 0 r ( Sn-1 f(δrx )(P x ,x )dx ) dr, (3)

where  ′ dx denotes the Lebesgues measure over the unit sphere  n-1 S , |α| = Σni=1αi and P is the diagonal matrix which generates the semigroup of dilations δɛ = exp P ln ɛ. 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 μ ɛ adapted to the dilations δɛx which plays an analogous role to those introduced in [18].

Example 2.2. Let α = (α1,...,αn) be a fix n-tuple of real numbers such that αi ≥ 1 , the measures μ ɛ are given by

 ∫ μ ɛ(E ) = ɛ|α|w (δɛt)W -1(ɛ)dt, E (4)

where ,  α1 αn δɛt = (ɛ t1,...,ɛ tn) , B ɛ = {δɛt : t ∈ B} and  ∫ W (ɛ) = w (t)dt Bɛ . The following condition on the weight function w will be assumed

W (ɛ) = A ɛβ+|α|(1 + o (1)),as ɛ → 0, A > 0, β + |α| > 0. (5)

We say that the weight function w is a radial function with respect to r if w(x) = w (y) when r(x) = r(y).

Remark 2.3. If the weight function w is a radial function, then the measures μɛ(E ) introduced in Example 2.2 converges weakly to the measure

 ∫ β μ0(E ) = C (|α | + β) r (x)dx, E (6)

where C is a constant depending upon the weight function w.

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 w ɛ(t) = ɛ|α |W - 1(ɛ)w (δɛt) and set  ∫ ∥Q ∥p(p,wɛ) = ɛ|α|W - 1(ɛ) B |Q (t)|pw (δɛt)dt for Q ∈ πm .

To deal with the weights w ɛ we consider a change of variables of polar type induced by the function r given by (3) .

Using the above formula and following the steps of the proof of Lemma 1 in [18], we can see that

|∥Q ∥p - ∥Q ∥p --| ≤ o(1 )∥Q ∥p --, (p,wɛ) (p,w) (p,w ) (7)

where -- -1 β w(t) = w n (β + |α |)r(t) and  -1 ∫ ′ ′ ′ wn = Sn- 1(P x ,x )dx . Now it is easy to obtain the weak convergence of the measures μ ɛ to μ0 and that the constant C in equation (6) it turns to be w -n1 .

3. The Taylor polynomial and the limit of best approximation polynomials.

Throughout this paper α = (α1,...,αn) will denote a fix n-tuple of real numbers such that αi ≥ 1 and min αi = 1 .

Definition 3.1. The class πm,α. Given a positive number m we say that a real polynomial p is in the class  m,α π if it is of the form

 ∑ β ∑ β1 βn p(x) = a βx = aβx1 ...xn , α.β≤m α.β≤m
where each β ∈ ℕn and at least one of them satisfies α.β = m . A polynomial p is of α -degree m if p ∈ πm,α

Note that in the case α = (1,...,1) we obtain the classical definition of polynomial of degree m . We denote by Πm,k α the set {P = (p ,...,p ) : p ∈ πm, α} 1 k i .

It is worthy to note that when we make reference to a polynomial of α -degree m we do not mean the classic degree of polynomial. For example, for α = (1, 3∕2) there exist polynomials of α -degree 3∕2 and they are of the form ax (0,0) + bx(1,0) + cx (0,1) , for a, b,c in ℝ .

Given a function F : B ⊂ ℝn → ℝk, and a family of seminorms ∥.∥ɛ, 0 ≤ ɛ ≤ 1, 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  ɛ α1 αn F (x ) = F(δɛx) = F (ɛ x1,...,ɛ xn) .

Definition 3.2. A function  n k F : B ⊂ ℝ → ℝ , has a Taylor Polynomial of α -degree m , if there exists  m,α Tm,α = Tm,α(F ) ∈ Πk such that

∥F ɛ - Tɛ ∥ = o(ɛm ), ɛ → 0. m,α ɛ

We write  m,α F ∈ t if the function F has a Taylor Polynomial of α -degree m .

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 C = C(m, k,α ) and 0 < ɛ(m, k,α) such that for every 0 < ɛ ≤ ɛ(m,k, α),

C- 1∥P∥0 ≤ ∥P∥ ɛ ≤ C ∥P ∥0,

for every  m,α P ∈ Π k .

We make use of the standard notation, ∂γf = ∂γγ11...-∂γnγn(f) ∂x1 ∂xn for  n γ ∈ ℕ .

Proposition 3.4. There exists a constant C > 0, depending only on m,k, n,α and the family of seminorms ∥ .∥ ɛ such that for any  m,α P = (p1,...,pk) ∈ Πk there holds

|∂αp (0)| ≤ C ɛ- α.γ∥P ɛ∥ , i ɛ

for any 0 ≤ α.γ ≤ m and i = 1,...,k .

Proof. We consider the seminorm ∥P ∥ = ∥∂γP (x)∥ = max ∥ ∂γp (x )∥ i i . Using the above proposition we have

|∂γpɛi(x)| ≤ C ∥P ɛ∥ ɛ,
for a constant C > 0 . Since ∂ γpɛ(0 ) = ɛα.γ∂γp (0) i i , the proposition follows. □

The next proposition is a consequence of Proposition 3.4.

Proposition 3.5. The polynomial Tm,α = Tm,α(F ) ∈ Πm,k α in Definition 3.2 is unique.

Proposition 3.6. If the function F has the Taylor polynomial of α -degree m ,  ∑ Tm,α (x ) = 0≤α.β≤m Aβx β, then the Taylor polynomial of α -degree l ≤ m is given by T (x ) = ∑ A x β l,α 0≤α.β≤l β (if there exists β such that α.β = l ) .We set  β ∂ F(0) for the vector β!Aβ .

Proof. We have

∥F ɛ - Tɛ ∥ ≤ o(ɛm ) + ɛs∥P ∥ = o(ɛl), l,α ɛ ɛ
where s = min {α.β : l < α.β ≤ m } and  m,α P ∈ Πk . □

Let ∥F ∥ ɛ , 0 ≤ ɛ ≤ 1, be a family of seminorms and  n k F : B ⊂ ℝ → ℝ , be a fixed measurable function such that ∥F ∥ɛ and  ɛ ∥F ∥ɛ are finite for all ɛ. For any such F has a meaning the following definition.

Definition 3.7. Set ∥F ∥*ɛ = ∥F ɛ∥ɛ, and P ɛ,α = P ɛ,α(F ) for any polynomial in Πm,k α which minimizes ∥F - P ∥*ɛ, P ∈ Πm,k α.

Although the best approximation polynomial P ɛ,α(F ) is not unique in general, through this paper the notation P ɛ,α(F ) does not mean a set of best approximation polynomials but any arbitrarily chosen polynomial in this set. We have the existence of Pɛ,α(F ) , at least for all small ɛ, by Proposition 3.3.

The next statement has its origin in [23] using the  ∞ L norm, and since then similar versions in  p L in one and several real variables appeared. Results dealing with weighted Luxemburg norms appeared recently in [9] and [10].

Theorem 3.8. If F ∈ tm,α, then Pɛ,α → Tm,α(F ) as ɛ → 0.

Proof. In fact  ɛ ɛ ɛ ɛ m ∥P ɛ,α - Tm,α(F )∥ɛ ≤ 2∥F - T m,α(F)∥ɛ = o(ɛ ), and by Proposition 3.4 it follows |[P (ɛ,βα)]i(0 ) - [Tm(β,)α(F )]i(0)| ≤ ɛ-α.βC ∥P ɛɛ,α - Tmɛ,α(F )∥ɛ, and 0 ≤ α.β ≤ m.

As in [6] we call the limit of Pɛ,α(F) as ɛ → 0, the best local approximation to F .

4.  The asymptotic behavior of the error.

Let A be a subspace of polynomials Πm, α⊆ A ⊆ Πl,α k k and let  n k F : B ⊂ R → ℝ , be a Lebesgue measurable function. Set P ɛ,α ∈ A for a polynomial which is a best approximation of the function F with the seminorm ∥F∥ *ɛ = ∥F ɛ∥ɛ. Observe that Pɛɛ,α is a polynomial in A ɛ = {P ɛ : P ∈ A } which is a best approximation of the function Fɛ with the seminorm ∥ .∥ ɛ from the class A ɛ, and we will also denote it by  ɛ ɛ PA ,ɛ,α(F ). We insist that  ɛ ɛ PA ,ɛ,α(F ) means, in our notation, a fixed best approximation polynomial and not a set of them.

Let Eɛ(F ) be the error function  -m- ɛ ɛ ɛ (F - Pɛ,α) where -- m = min {h ∈ (m, ∞ ) : ∃ β with α. β = h} . Next, we will obtain an expression for the function E ɛ(F ) which has its origin in [19] see also [1718].

Let F be in t(m,α) and set T -- m,α for the Taylor polynomial of F of α -degree -- m ; then by definition we have  ɛ ɛ- m- ɛ- F = T m,α + ɛ R m,α, with  ɛ ∥R m,α∥ɛ = o(1), and  - m- Rm,α(x) = ɛ (F (x) - Tm,α(x)). Moreover, observe that λPA ɛ,ɛ,α(F ɛ) = PA ɛ,ɛ,α(λF ɛ) and T ɛm,α + PA ɛ,ɛ,α (F ɛ) = PA ɛ,ɛ,α((Tm,α + F )ɛ), where we have used that T ∈ A. m,α Using the equality Φ-- + R ɛ- = ɛmF ɛ - ɛmT m,α m,α m,α we obtain the following result

Proposition 4.1. Let F be a function in m,α t , and Φm,-α = Tm,α - Tm,α. Then

 -- -- (a) ∥E ɛ(F )∥ ɛ → ∥Φ m,α - PA,0(Φm,α)∥0.
 -- ɛ- -- ɛ- E ɛ(F ) = Φ m,α + R m,α - PA ɛ,ɛ,α(Φ m,α + R m,α),

∥R ɛ- ∥ɛ = o(1), m,α as ɛ → 0.

Proposition 4.1 is useful when  ɛ A = A for every ɛ > 0. The case  m,α A = Π k with α = (1,...,1) was considered in [18] and [17] for weighted  p L norms and in [9] for the Luxemburg norm. It is easy to find Πm,k α ⊊ A ⊆ Πl,αk ,m < l, and A ɛ = A for every ɛ > 0. The following result is relevant to this matter.

Theorem 4.2. Let A be a subspace of polynomials such that  m,α m,α Π k ⊆ A ⊆ Π k . Then  ɛ A = A for all ɛ > 0.

Proof. Let A1 = {P - Tm,α(P ) : P ∈ A } . We shall see if A ɛ1 = A1 for all ɛ > 0 , then A ɛ = A for all ɛ > 0 . Let H be in A ɛ, i.e., H = Q ɛ with Q ∈ A. As Q - T (Q) ∈ A m,α 1 we have

 ɛ ɛ ɛ ɛ ɛ Q - Tm,α(Q ) = Q - (Tm,α(Q )) ∈ A 1.

Thus Q ɛ - T (Q ɛ) ∈ A m,α 1 and there exists V ∈ A such that  ɛ ɛ Q - Tm,α(Q ) = V - Tm,α(V ). Therefore  ɛ Q - V ∈ A, so H ∈ A. We have proved that Aɛ ⊂ A for all ɛ > 0. Since  1 A ɛ ⊂ A we get A = (A 1ɛ)ɛ ⊂ A ɛ. Therefore, A ɛ = A for all ɛ > 0 .

Clearly, there is a linear space W ⊂ ℝk such that A = {Ax β : α.β = m} 1 and {A ∈ W } , then A ɛ = A 1 1 . □

Theorem 4.3. Let F be in tm,α, and A ɛ = A for every ɛ > 0. Then  -- -- (a) ∥E ɛ(F )∥ɛ → ∥ Φm,α - PA,0(Φ m,α )∥0, as ɛ → 0. (b) ∥E ɛ(F ) - (Φm,α - PA,0(Φm,α ))∥ɛ → 0, as ɛ → 0 if ∥ ∥0 is a strictly convex norm. We have denoted by PA,0(Φm,α) a polynomial in A which is a best approximation of Φ-- m,α with respect to the norm ∥ .∥ . 0

Proof. Let us begin with (a) . By Proposition 4.1 we have, for any P ∈ A,

∥Eɛ(F )∥ɛ ≤ ∥Φm,α + R ɛm,α - P ∥ɛ = ∥Φm,α - P ∥ɛ + o(1) = ∥Φm,α - P ∥0 + o(1), as ɛ → 0. Therefore

lim-∥E (F )∥ ≤ ∥ Φ-- - P ∥ . ɛ→0 ɛ ɛ m,α A,0 0

Let (ɛk) be a sequence tending to zero such that

lim-∥E ɛ(F)∥ɛ = lim ∥Eɛk(F )∥ɛk. ɛ→0 ɛk→0

Set Pk = PA ɛk,ɛ (Φm,α + R ɛk ); k m,α then ∥Pk∥ɛ ≤ 2∥Φm,-α + Rɛk ∥ ɛ = 2∥Φm,-α∥ɛ + o(1). k m,α k k By Proposition 3.3 we can select a convergent subsequence of Pk which is again denoted by Pk and then we have ∥Φm,-α - Pk∥0 = ∥Φm,-α - Pk∥ɛk + o(1) , as ɛk → 0 . Then  ɛ ∥Φm,-α - PA,0(Φm,α)∥0 ≤ ∥ Φm,α - Pk∥0 = ∥ Φm,α - Pk∥ ɛk + o(1) = ∥Φm,-α + Rmk,α - Pk∥ ɛk + o(1). Thus we have

∥Φm,-α - PA,0(Φm,α)∥0 ≤ lim--∥Eɛ (F )∥ɛ . ɛk→0 k k

To prove (b) , consider any sequence ɛk → 0 and select Pk = PAɛk,ɛk(Φm,-α + R ɛmk,α) , then ∥E ɛk(F )∥ɛk = ∥Φm,-α + R ɛmk,α - Pk∥ɛk. We will prove Pk → PA,0(Φm,-α), which implies (b). In fact we may assume, by taking subsequences if it is necessary, that P → P ∈ A, k 0 as ɛk → 0. Thus by (a)  -- -- -- ∥Φ m,α - P0∥0 = ∥Φ m,α - PA,0(Φ m,α)∥0. Since ∥.∥0 is a strictly convex norm we have  -- P0 = PA,0(Φ m,α).

Consider the set A = ×k πmi, m ≤ m i = 1,...,k. i=1 i where α.β = m i , for  n β ∈ ℕ . Then  m,α l,α Π k ⊆ A ⊆ Π k , and  ɛ A = A, for every ɛ > 0 .

We now introduce an useful example of a subspace A such that  ɛ A ⁄= A . Consider the set

A (l,k) = A (l;x ,...,x ) = {Lp : p ∈ πl,α}, 1 k (8)

where, Lp (s) = (p(x1 + s),...,p(xk + s)), and - 1 < x1 < ...< xk < 1. Now  ⋂ ɛ A A = { (c,...,c) : c ∈ ℝ} , see [11] in Proposition 4.2. Thus it is not possible to use Theorem 4.3 to study the function error with A = A (l, k) . The next condition on A will be significant in the future and it was used in [11] to consider cases such as A (l,k ).

A subspace of polynomials which does not satisfies A ɛ = A is given in the following example.

Example 4.4. We denote by QV,β1,...,βs the set of all algebraic polynomials of the form

 β1 βs a1x + ...+ asx ,
where  i β , i = 1,...,s are fixed vectors in  n ℕ ,  1 -- β .α = m ,  j -- β .α > m for j = 2,...,s and βj.α ⁄= βk.α for j ⁄= k . Moreover (a1,...,as) = c(v1,...,vs) whit v1 ⁄= 0 , c ∈ ℝ and V = (v1,...,vs) is a fixed vector in ℝs .

Let  ⊕ A = Πm,k α B , where B = {(p1,...,pk) : pi ∈ QVi,β1,i,...,βs,i} . It is clear that Aɛ ⁄= A .

Condition 4.5. For  m,α l,α Πk ⊆ A ⊆ Π k , we assume that if P ∈ A and Tm,α(P ) = 0, then P = 0 . Where -- m = min {h ∈ ℝ : h > m } and ∃β with {α.β = h} .

Let  m,α⊕ A = Π k B be as in Example 4.4 , then the Condition 4.5 holds. We consider again the error function  -m- ɛ ɛ E ɛ(F ) = ɛ (F - Pɛ ), where P ɛ ∈ A and  ɛ ɛ Pɛ = PA ɛ,ɛ(F ). Set G = F - Tm,α(F) and recall that Tm,α (F ) ∈ A.  Then Eɛ(F ) = Eɛ(G ). If  -- F ∈ tm,α we have

 -- -m- ɛ E ɛ(F ) = Φ m,α - ɛ PA ɛ,ɛ(G ) + o(1), (9)

as ɛ → 0, and Φm,α = Tm,α(F) - Tm,α(F ). The next theorem give us a useful expression for the error function E ɛ(F ) as well as we know the polynomials {Uɛ}ɛ>0 and {Pɛ}ɛ>0 used to describe it. With the notation A = {P ∈ A : T (P) = 0}, 0 m,α observe that A is the direct sum  m,α⊕ Π k A0.

Theorem 4.6. Let F be a function in tm,α, and assume Condition 4.5 for A . Set Pɛ = P ɛ ((F - T (F))ɛ), ɛ A ,ɛ m,α with ¯P ∈ A, ɛ  ¯ ¯ U ɛ = Pɛ - Tm,α(Pɛ), and  -m- ɛ -- V ɛ = ɛ T m,α(Pɛ). Then U ɛ ∈ A0 and  m,α V ɛ ∈ Π k and

 -- -- E ɛ(F ) = Φ m,α - T m,α(Uɛ) - Vɛ + o(1), (10)

as ɛ → 0. Moreover the two families of polynomials {U ɛ} ɛ>0 and {V ɛ}ɛ>0 are uniformly bounded in ɛ for a fixed norm ∥ . ∥ .

Proof. By (9) we have

 -- -- Eɛ(F ) = Φm,α - ɛ- m(Uɛɛ + Tm,αɛ(P ɛ)) + o(1).

Or else, since Tm,α(U ɛ) = 0,

E (F ) = Φ-- - T -- (U ) - ɛ -mT ɛ(P- ) + o(1), ɛ m,α m,α ɛ m,α ɛ

as ɛ → 0, which is (10).

Now we will prove {U ɛ}ɛ>0 and {V ɛ} ɛ>0 are uniformly bounded in ɛ . For a norm ∥ ∥ in ℝk, the expression

 γ αm.aγx≤m-∥∂ P (0)∥,

is a norm on A. Here we are using that the subspace A fulfills Condition 4.5.

Since --ɛ P ɛ = PA ɛ,ɛ(G ɛ), with G = F - Tm,α(F ), we have  -- -- ∥P¯ɛɛ∥ɛ ≤ 2∥G ɛ∥ɛ ≤ 2∥Tmɛ,α (G )∥ɛ + o(ɛm) = O (ɛm) .
By Proposition 3.4 we have ∥∂γP¯(0)∥ = O (ɛ-α.γ∥P¯ɛ∥ ) = O(ɛm--α.γ), ɛ ɛ ɛ for  -- α.γ ≤ m. Thus  ¯ Pɛ and hence  ¯ ¯ U ɛ = P ɛ - Tm, α(Pɛ) are uniformly bounded in ɛ > 0. To estimate the polynomials  - (m-) ɛ V ɛ = ɛ Tm,α( ¯Pɛ), we note that  -- ∂ γVɛ(0) = ɛ-mɛα.γ∂γP¯ɛ(0 ), for α.γ ≤ m. Then ∥∂γV ɛ(0)∥ = O (1); recall that Vɛ ∈ Πm, α k , and max α.γ≤m ∥∂ γVɛ(0)∥ is a norm there. □

Theorem 4.7. Let F be in  m,α t and assume Condition 4.5 for A. Then ∥E ɛ(F)∥ɛ tends to

 -- m,α min{∥T m,α(G - U ) - V ∥0 : U ∈ A0, V ∈ Π k },

as ɛ → 0, and G = F - T (F). m,α

Proof. We will prove the following inequality

---- lim ∥Eɛ(F )∥ɛ ≤ inf m,α ∥Tm,α(G - U) - V ∥0 ≤ lim-∥E ɛ(F )∥ɛ. ɛ→0 U∈A0, V∈ Πk ɛ→0 (11)

Let U ∈ A0 and V ∈ Πmk,α be two arbitrary polynomials and set Z ɛ ∈ Πm, α k with Z ɛ = ɛm,αV ɛ and U + Z ɛ ∈ A. Then

 -m- ɛ ɛ ∥E ɛ(F )∥ ɛ = ∥E ɛ(G )∥ɛ = ɛ ∥G - ¯PA ɛ,ɛ(G )∥ɛ
 -- -- ≤ ɛ-m ∥G ɛ - (U + Zɛ)ɛ∥ɛ = ɛ-m ∥Tɛ- (G - U) - Z ɛ∥ ɛ + o(1). m,α ɛ

As  -- T ɛm,α (G - U ) = ɛmTm,α (G - U ), using the definition of the polynomial Z ɛ we have

 -- -- ∥E ɛ(F )∥ɛ ≤ ∥T m,α (G - U) - V ∥ɛ+ o(1) ≤ ∥T m,α(G - U )- V ∥0+ o(1 ),

as ɛ → 0, and the right inequality of (11) holds.

To prove left inequality in (11) let (ɛk) be sequence tending to zero such that

∥Eɛj(F )∥ ɛj -→ lim-∥E ɛ(F )∥ɛ. ɛ→0

By Theorem 4.6 we select a subsequence ɛj → 0 in such a way that the following limits exist:

lim U ɛj = U˜0, ˜U0 ∈ A0. j→∞
 m,α jli→m∞ Vɛj = ˜V0, ˜V0 ∈ Πk .

Thus by (10) we have

 -- ˜ ˜ liɛ→m0-∥E ɛ(F )∥ ɛ = ∥ Φs,α - T m,α(U0) - V0∥0
≥ inf m,α∥Tm,α(G - U ) - V ∥0. U∈A0, V ∈Πk

Proposition 4.8. Let ∥ .∥0 be a strictly convex norm and assume that the subspace A fulfills Condition 4.5. Then there exists a unique solution  m (U,V ) ∈ A0 × Π k to the minimum problem in Theorem 4.7 .

Proof.

If (U1,V1 ) and (U2,V2) are solutions to the minimum problem in Theorem 4.7, we have ∥Tm,α(U1 ) + V1 ∥0 = ∥Tm,α(U2) + V2∥0 with (Ui,Vi) ∈ A0 × Πm,k α, i = 1,2 . Since ∥ . ∥0 is a strictly convex norm Tm,α(U1 + V1) = Tm,α(U2 + V2 ), then by Condition 4.5, U + V = U + V , 1 1 2 2 but A = A ⊕ Πm,α 0 k . □

Theorem 4.9. Let F be in tm,α, assume Condition 4.5 for A, and that the minimum problem in 4.5 has a unique solution  m,α (U0, V0) ∈ A0 × Π k . Then Uɛ → U0 and Vɛ → V0 as ɛ → 0. Moreover we have

∥E (F) - (T-- (G - U ) - V )∥ - → 0. ɛ m,α 0 0 ɛ

Proof. By Theorem 4.6, Theorem 4.7 and (10) any convergent subsequence of the net {(Uɛ,Vɛ)}ɛ 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 PA, ɛ,α(F) as ɛ → 0. If  -- F ∈ tm,α and G = F - Tm,α (F ) it will be enough to consider P (G) A,ɛ,α as ɛ → 0, since P (F) = P (G ) + T (F ). A,ɛ,α A,ɛ,α m,α

We set as before P (G) = P (G) = P (G ) - T (P (G )) + T (P (G )) = U + T (P (G)). ɛ,α A,ɛ,α ɛ,α m,α ɛ,α m,α ɛ,α ɛ m,α ɛ,α Let F be in  m,α t , then  γ m--α.γ ∥ ∂ Pɛ,α(0)∥ ≤ O (ɛ ), for  -- α.γ ≤ m ; see the proof of Theorem 4.6. Then Tm, α(Pɛ,α(G )) - → 0, ɛ → 0. Thus lim ɛ→0 P ɛ,α(G ) = lim ɛ→0U ɛ = U0. From Theorem 4.7 and Theorem 4.8 , this polynomial exists whenever ∥ .∥0 is a strictly convex norm. Then lim ɛ→0 PA, ɛ,α(F ) = Tm, α(F) + lim ɛ→0 Uɛ = Tm,α(F ) + U0, where U 0 together with V 0 are the unique solution to the minimizing problem

 min m,α ∥Tm,α(G - U) - V ∥0. U∈A0, V∈ Πk

Thus if we set P = T (F ) + U ∈ A 0 m,α 0 for lim P (F ) ɛ→0 ɛ,α then P0, in AF = A0 + Tm,α(F ), will be the unique solution to the problem

 inf ∥T -- (F - P ) - V∥ . P∈AF , V∈Πmk,α m,α 0 (12)

Thus, we have proved the following theorem.

Theorem 5.1. Let F be in  m,α t and assume Condition 4.5 for A, and that the minimum problem in (12) has a unique solution  m,α (P0,V0 ) ∈ AF × Π k , and denote by PA,ɛ,α(F ) a polynomial in A which minimizes ∥F - P ∥*ɛ = ∥F ɛ - P ɛ∥ɛ, with P ∈ A. Then PA,ɛ,α(F) → P0, as ɛ → 0.

6. On the best local approximation using Luxemburg norm.

We denote by μ ɛ the measures given by (4), and let φ be a convex function such that φ(0) = 0 , φ (x) > 0 if x > 0 . For any measurable F : B ⊂ ℝn → ℝk, set

 ∫ ∑ k ( |fi(t)|) ∥F ∥ɛ = inf{λ > 0 : φ ------ dμɛ(t) ≤ 1}, B i=1 λ (13)

where F(t) = (f (t),...,f (t)) 1 k .

By Proposition (2.3) in [11] we have ∥F∥ ɛ converges to ∥F ∥0, for any F ∈ Ck(B ) . Moreover the family ∥F ∥ɛ,0 ≤ ɛ ≤ 1 has the properties (1),(2) y (3) of the section 2.

Recall that ∥F ∥0 is the Luxemburg norm defined by (13) with the particular measure μ0 defined by (6) and denote by Lφ0(B ) the Orlicz Space equipped with the norm ∥F ∥0. The following result is known .

Remark 6.1. Let φ be a strictly convex function, then  φ L0(B ) is a strictly convex Banach space with the Luxemburg norm ∥.∥0 .

By Remak 6.1 we can use Proposition 4.8 and Theorem 5.1 for the Luxemburg norm ∥.∥0 when φ 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 f : [- 1,1] → ℝ and - 1 < x1 < ...< xk < 1, set F (t) = (f (x1 + t),...,f(xk + t)) and the norm ∥F ∥ɛ = ∥F ∥ as in (13) and the measure dμ ɛ is the Lebesgue measure dt.

Theorem 6.2. Let φ be a strictly convex function and let  m P ɛ ∈ π be the unique solution of the minimum problem

 ∫ ( ) * k xi+ɛ |f(x) - P (x)| dx ∥f - P ∥ɛ = inf {λ > 0 : Σi=1 φ -------------- ---≤ 1}, xi-ɛ λ ɛ

where  m P ∈ π . Then for a smooth function f , P ɛ converges to a polynomial  m P0 ∈ π , 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 φ , namely  φ(x) limx →0 x = 0,  φ(x)- limx →∞ x = ∞ and φ^(λ) = limx →∞ φ(λxx)- exists and it is a finite number for every λ ≥ 0 . Clearly ^φ is convex function, ^φ(0) = 0 and it is easy to see that φ^(x) = xp, for x ≥ 0, and if ^φ(2) > 2, we have 1 < p < ∞ see [13]. From now on assume all the above conditions on the function φ .

Theorem 6.3. For any ɛ > 0, let Qɛ(f) be the unique polynomial in πm which minimizes

∥(f - P)XV ɛ∥0,

 m P ∈ π and  ⋃k V ɛ = i=1(xi - ɛ,xi + ɛ). Then the limit Q0 (f ) = lim ɛ→0Q ɛ(f) exist for smooth functions f .

Theorem 6.3 may be obtained using results of [13] and [20]. For the case m + 1 < k, the polynomial Q0(f) it is very easy to characterize as the unique element Q0(f ) ∈ πm which minimizes the problem

 k ∑ p |(f (xi) - Q (xi))|, i=1

Q ∈ πm, see [13]. For the case m + 1 = kq + r, r > 0 also Q0 (f) can be obtained as a discrete minimum Lp problem as in [20].

The best local approximation polynomials P (f) 0 described in Theorem 6.2 and Q0 (f) in Theorem 6.3 are different polynomials. Indeed, it is rather straightforward to obtain the next result when f is a continuous function at each point x1,...,xk and φ just a strictly convex function φ (0) = 0 .

Theorem 6.4. For m + 1 < k, and ɛ > 0 let Pɛ the unique polynomial which minimizes

 ∫ xi+ɛ ( ) inf {λ > 0 : Σk φ |f(x)---P-(x)| dx-≤ 1}, i=1 xi-ɛ λ ɛ

 m P ∈ π . Then the limit P0 (f) = lim ɛ→0 Pɛ(f) exist and it is characterized as the unique Q0 ∈ πm, which minimizes

 ( ) k |f(xi) --Q-(xi)| inf{ λ > 0 : Σi=12φ λ ≤ 1},

Q ∈ πm.

We point out that to prove the existence of the polynomial Q0 (f) in Theorem 6.3 still remains an open problem when φ is just a strictly convex function and the existence of the function ^φ is not required.

References

[1]    A. Alzamel, J.M. Wolfe. Interpolation and Best Lp Local Approximation. Journal of Approximation Theory 32(1981), 96-102 .        [ Links ]

[2]    P. Billingsley. Convergence of Probability Measures. John Wiley & and Sons, Inc., 1968.        [ Links ]

[3]    A.P. Calderón, A. Zygmund. Local Properties of Solutions of Elliptic Partial Differential Equations. Studia Math. 20 (1961), 171-225.        [ Links ]

[4]    C.K. Chui, H. Diamond, L. Raphael. On Best Data Approximation. Approximation Theory and its Applications. 1 (1984), 37-56.        [ Links ]

[5]    C.K. Chui, H. Diamond, L. Raphael. Best Local Approximation in Several Variables. Journal of Approximation Theory 40 (1984), 343-350.        [ Links ]

[6]    C. K. Chui, O. Shisha, P. W. Smith. Best Local Approximation. Journal of Approximation Theory 15 (1975), 371-381.        [ Links ]

[7]    C. K. Chui, P. W. Smith, J.D. Ward. Best L2 Approximation. Journal of Approximation Theory 22 (1978), 254-261.        [ Links ]

[8]    M. Cotlar, C. Sadosky. On Quasi-homogeneous Bessel Potential Operators. Proceedings of Symposia in Pure Mathematics of the AMS vol.X (1967), 275-287.        [ Links ]

[9]    H. Cuenya, M. Lorenzo, C. Rodriguez. Weighted Inequalities and Applications to Best Local Approximation in Luxemburg norm. Analysis in Theory and Applications 20 (3)(2004), 265-280.        [ Links ]

[10]    H.H.Cuenya, M.D. Lorenzo, C.N. Rodriguez. A unified approach to certain problems of best local approximation. Analysis in Theory and Applications 23 (2) (2007), 162-170.        [ Links ]

[11]    H. H. Cuenya, F. Zó. Best Approximations on Small Regions. A general approach, in Procedings of the Second International School. Advanced Courses of Mathematical Analysis II. Edited by M. V. Velasco and A. Rodríguez- Palacios. World Scientific (2007), 193- 213.        [ Links ]

[12]    E. B. Fabes, N. M. Rivière. Singular Integrals with Mixed Homogeneity. Studia Math. 27 (1966), 19-38.        [ Links ]

[13]    S.Favier. Convergence of Functions Averages in Orlicz Spaces. Numer. Funct. Anal. and Optimiz. 15 (3&4) (1994), 263-278.        [ Links ]

[14]    S. Favier, C. Fernandez and F. Zó. The Taylor Polinomial on Best Local Approximation in Rectangles. Revista de la Unión Matemática Argentina 32 (1986), 254-262.        [ Links ]

[15]    S. Favier, C. Ridolfi. Weighted Best Local Approximation in Orlicz Spaces. To appear in Analysis in Theory and Applications.        [ Links ]

[16]    W. Feller . An Introduction to Probability and Its Applications. Wiley , Vol.II (1966), New York.        [ Links ]

[17]    V. B. Headly and R.A. Kerman. Best Local Approximation in Lp(μ) . Journal of Approximation Theory 62 (1990).        [ Links ]

[18]    R. Macías, F. Zó. Weighted Best Local Lp Approximation. Journal of Approximation Theory 42 (1984), 181-192.        [ Links ]

[19]    H. Maehly, Ch. Witzgall. Tschebyscheff - Approximationen in kleinen Intervallen I. Approximation durch Polynome. Numerische Mathematik 2 (1960), 142-150.        [ Links ]

[20]    M. Marano. Mejor Aproximación Local. Phd Thesis, Universidad Nacional de San Luis (1986).        [ Links ]

[21]    N. M. Rivière. Singular Integrals and Multiplier Operators . Ark. Mat. 9 (1971), 243-278.        [ Links ]

[22]    C. Sadosky. On Some Properties of a Class of Singular Integrals. Studia Math. 27 (1966), 105-118.        [ Links ]

[23]    J. L. Walsh. On Approximation to an Analytic Function by Rational Functions of Best Approximation. Mathematische Zeitschrift (1934), 163-176.        [ Links ]

[24]    J. M. Wolfe. Interpolation and Best L p Local Approximation. Journal of Approximation Theory 32 (1981), 96-102.        [ Links ]

[25]    J.M. Wolfe. Best Multipoint Local Lp Approximation. Journal of Approximation Theory 62 (2)(1990), 96-102 .        [ Links ]

[26]    F. Zó . On Inequalities Arising From Best Local Approximations In Rectangles. Topics in Multivariate Approximation. Academic Press, Inc., 265-263.        [ Links ]

[27]    A. Zygmund. Trigonometric Series. Vols. I,II third edition Cambridge Mathematical Library, Cambridge University Press, 2002.        [ Links ]

 

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