On convergence of subspaces generated by dilations of polynomials. An application to best local approximation

We study the convergence of a net of subspaces generated by dilations of polynomials in a finite dimensional subspace. As a consequence, we extend the results given by Zó and Cuenya [Advanced Courses of Mathematical Analysis II (Granada, 2004), 193–213, World Scientific, 2007] on a general approach to the problems of best vector-valued approximation on small regions from a finite dimensional subspace of polynomials.


Introduction
Suppose that {a j } is a data set. These data are values of a function and its derivatives at a point. If we want to approximate these data using a polynomial of degree at most l, which will be the best algorithm to use? A Taylor polynomial of degree l is probably the most natural procedure to use.
The problem of finding an optimal algorithm to approximate a finite number of data corresponding to a function is developed in the best local approximation theory.
In 1934, Walsh proved in [11] that the Taylor polynomial of degree l for an analytic function f can be obtained by taking the limit as → 0 of the best Chebyshev approximation to f from Π l on the disk |z| ≤ . This paper was the first association between the best local approximation to a function f from Π l in 0 and the Taylor polynomial for f at the origin. However, the concept of best local approximation has been introduced and developed more recently by Chui, Shisha, and Smith in [2]. Later, several authors [3,4,5,6,8,9,10,12] have studied this problem.
We consider a family of function seminorms { · } >0 , acting on Lebesgue measurable functions F : B ⊂ R n → R k , where B is the unit ball centered at the origin in R n . We will use the notation F (x) = F ( x) and F * = F . For l ∈ N ∪ {0}, 50 F. E. LEVIS AND C. V. RIDOLFI we will denote by Π l the class of algebraic polynomials in n variables of degree at most l, and Π l k the set {P = (p 1 , . . . , p k ) : p s ∈ Π l }. Let A be a subspace of Π l k and let {P } >0 be a net of best approximants to F from A with respect to · * , i.e., F − P * ≤ F − P * , for all P ∈ A. (1.1) If the net {P } >0 has a limit in A as → 0, this limit is called the best local approximation to F from A in 0. According to (1.1), we observe that P is a polynomial in A := {P : P ∈ A} ⊂ Π l k (1.2) of best approximation to F by elements of the class A , with respect to the seminorm · . We write it briefly as P ∈ P A , (F ). Note that A is a subspace generated by dilations of polynomials in A.
From now on, we assume the following properties for the family of function seminorms · , 0 ≤ ≤ 1.
An important point to note here is that there exist positive constants C = C(m, k) and (m, k) such that for every 0 < ≤ (m, k), In order to give an example of norms · , 0 ≤ ≤ 1, with the properties (1)-(3), we recall a definition of convergence of measures given in [6]. See also [1] for the notion of weak convergence of measures in general. Definition 1.1. Let µ , 0 ≤ ≤ 1, be a family of probability measures on B. We say that the measures µ converge weakly in the proper sense to the measure µ 0 if we have and µ 0 (B ) > 0 for any ball B ⊂ B.
The assumption on the measure µ 0 implies that is actually a norm on C k (B) for = 0 and 1 ≤ p < ∞, where · stands for any monotone norm on R k . We use a monotone norm on R k to ensure property (1) for the family of seminorms · , 0 ≤ ≤ 1.
Let F be in C k (B); it is readily seen, by using the definition of weak convergence of measures, that there exists For more examples of nets of seminorms fulfilling conditions (1)-(3), we refer the reader to [13].
We say that F : B ⊂ R n → R k has a Taylor polynomial of degree m at 0 if there exists P ∈ Π m k such that It is well known that if a Taylor polynomial exists, it is unique [13, Proposition 3.3]; we denote it by T m = T m (F ). We write F ∈ t m if the function F has the Taylor polynomial of degree m at 0. Moreover, if F ∈ t m and T m (F ) = |α|≤m C α x α , then the Taylor polynomial of degree l ≤ m for F at 0 is given by T l (F ) = |α|≤l C α x α [13,Proposition 3.5], where α = (α 1 , . . . , α n ) ∈ R n with α i ≥ 0 and |α| = α 1 + α 2 + · · · + α n . We set ∂ α F (0) for the vector α!C α with The problem of best local approximation with a family of function seminorms { · } >0 satisfying (1)-(3) was considered in [13] for two types of approximation class A fulfilling Π m k ⊂ A ⊂ Π l k and (c1) A = A, for each > 0, or (c2) if P ∈ A and T m+1 (P ) = 0, then P = 0.
Firstly, the authors studied the asymptotic behavior of a normalized error function as → 0 [13, Theorems 4.2 and 4.5]. Secondly, they showed that there exists the best local approximation to F in 0 and is associated with a Taylor polynomial for F in 0 [13, Theorem 5.1]. In particular, if A = Π m k and F ∈ t m , they proved that P → T m (F ) as → 0 [13,Theorem 3.1].
In this work we generalize the results found in [13], without the restrictions (c1) or (c2) given above. For this, it is essential to study the convergence of the net {A } as → 0. This paper is organized as follows. In Section 2, we investigate the asymptotic behavior of {A }. In Section 3, we study the asymptotic behavior of the error function −m−1 (F − P ) for a suitable integer, and we show some results about the best local approximation in the origin which generalizes those of [13].

Asymptotic behavior of the net {A }
In this section, we study the asymptotic behavior of the net {A } given in (1.2). We begin with the following definition. Next, we show a simple example of A = A.
From now on, for any Lebesgue measurable function F : The following sets will be needed throughout the paper. Let A be a non-zero subspace of Π l k . We define We can now formulate our main result which describes the limit set B.
Furthermore, under the above notation the following holds: be the linear operator given by According to (2.6) and (2.7) we have P = 0, and (iii) is proved. Using (i)-(iii), we deduce that the subspace is a direct sum isomorphic to A. The proof concludes by proving We observe that if P ∈ S si \ {0}, then T si (P ) = 0 and T si−1 (P ) = 0 by (2.3). So, Proposition 2.5 implies that T si (P ) ∈ B. On the other hand, if P ∈ A s N \ {0}, we get T s N (P ) = 0. Moreover, we have T s N +1 (P ) = 0. In fact, on the contrary, we see that P ∈ A s N +1 = {0}. Proposition 2.5 now gives T s N +1 (P ) ∈ B. Therefore, On the other hand, if P ∈ B, there exists {P } ⊂ A such that and {w ir } di r=1 bases of A s N and S si , respectively. It is easy to check that for each According to (2.4), we have that there exist real numbers D l, and C i,r, such that From (2.9) it follows that Thus, a straightforward computation yields (2.12) From (2.8) and (2.10), we deduce that T s0 (P )( r=1 is a basis of T s0 (S s0 ), there are real numbers C 0,r , 1 ≤ r ≤ d 0 , such that C 0,r, → C 0,r as → 0. According to (2.8) and (2.11) it follows that d1 r=1 C 1,r, T s1 (w 1,r )(x) is convergent as → 0. Hence, there are real numbers C 1,r , 1 ≤ r ≤ d 1 , such that C 1,r, → C 1,r as → 0, because r=1 is a basis of T si (S si ), 0 ≤ i ≤ N , (2.8) and (2.10)-(2.12) show that there are real numbers D l and C i,r such that D l, , → D l and C i,r, → C i,r as → 0. In consequence, The following corollary follows immediately from the proof of Theorem 2.6. Proof. We first claim that T is an injective operator. Indeed, if T (P ) = T (Q) for P, Q ∈ A, then T s (P − Q) = 0 and so P − Q ∈ A s . Since A s = {0}, we have P = Q.
Since A is isomorphic to T (A), the proof concludes by proving Let A j be the sets defined in (2.2). Since Therefore Π m k is isomorphic to B m ⊕ · · · ⊕ B 0 . On the other hand, since s 0 = 0, r 0 = s and J \ {r 0 } = {0, 1, . . . , m}, by Proposition 2.6 (a), From the proof of Theorem 2.6, we obtain that B m ⊕ · · · ⊕ B 0 is isomorphic to T m (B m ) ⊕ · · · ⊕ T 0 (B 0 ), and consequently Π m

An application to best local approximation
Let {P } be a net of best approximants to F from A with respect to · * , and let E be the error function (1), and T m+1 is the Taylor polynomial of F of degree m + 1 at 0. Moreover, λP ∈ P A , (λF ) and P + P ∈ P A , ((P + F ) ), for P ∈ A.
The following proposition may be proved in much the same way as [13, Proposition 4.1]. However, we repeat the proof for completeness.

Proposition 3.1. Let A be a non-zero subspace of Π l k with l > m, and let {P } be a net of best approximants of F from
Next, we give a new result about the asymptotic behavior of the error without the conditions (c1) or (c2), which generalizes Theorems 4.2 and 4.5 of [13].

Theorem 3.2. Let A be a non-zero subspace of
Proof. By Proposition 3.1, In fact, let P ∈ B. By the definition of B, there exists a net {Q } ⊂ A such that By Property (3), φ m+1 − P → φ m+1 − P 0 , as → 0. Hence, using the triangle inequality we have as → 0. Now, according to (3.3) we get (3.2).
In consequence, there exists a subsequence of {P }, which is denoted in the same way, and P 0 ∈ Π l k such that P → P uniformly on B, as → 0. Since We observe that P ∈ B by Corollary 2.7. Therefore, by Proposition 3.1, is proved.
The following result provides us with a useful and important property for a net of best approximants to F from A. Proof. We observe that , and consequently, (3.8) According to (1.3), there exist constants 0 , M > 0 such that for all 0 < ≤ 0 . The equivalence of the norms in Π l k implies that the net is uniformly bounded on B. So, there exists a subsequence of , which is denoted in the same way, and a polynomial P 0 such that (P − T m ) m+1 converges to P 0 , uniformly on B, as → 0.
On the other hand, if P 0 ∈ C, there is a sequence ↓ 0 such that (P −Tm) m+1 → P 0 . Since T m ∈ A, we have P − T m ∈ A, and so P 0 ∈ B by Corollary 2.7. Finally, from Property (3) and (3.8) we conclude that i.e., P 0 is a solution of (3.7).
The following theorem is an extension of [13, Theorem 5.1].

Theorem 3.4. Let
A be a non-zero subspace of Π l k with l > m, and let {P } be a net of best approximants of F from A with respect to · * . Assume m + 1 = min {j : 0 ≤ j ≤ l and A j = {0}}, F ∈ t m+1 with T m ∈ A, and set φ m+1 = T m+1 − T m . If the minimization problem (3.7) has a unique solution P 0 , then P → T m +P , where P ∈ A is uniquely determined by the condition T m+1 (P ) = P 0 − T m (P 0 ).
Proof. Since (3.7) has a unique solution P 0 , Theorem 3.3 implies that Let T : A → Π m+1 k be the linear operator defined by T (P ) = T m+1 (P ). As A m+1 = {0}, an analysis similar to that in the proof of Corollary 2.9 shows that T is an injective operator. Since T (A) is a closed subspace and {T m+1 (P − T m )} ⊂ T (A), (3.9) implies that there exists a unique P ∈ A such that T m+1 (P ) = R. Hence T m+1 (P − T m − P ) → 0 as → 0. As A m+1 = {0} we see that Q := T m+1 (Q) 0 is a norm on A, and so P → T m + P as → 0. Finally, by Theorem 2.6, B ⊂ Π m+1 k , and consequently P 0 − T m (P 0 ) = T m+1 (P 0 ) − T m (P 0 ) = R. The proof is complete.
Remark 3.5. If A satisfies the condition (c2), then A = Π m k ⊕ A m with A m+1 = {0}. By Corollary 2.9, B = Π m k ⊕ T m+1 (A m ) and each element P ∈ A is uniquely determined by T m+1 (P ). So, we can rewrite the problem (3.7) in the following (equivalent) form: The following result has been proved in [13, Theorem 5.1] and it is a consequence of Theorem 3.4. Corollary 3.6. Let Π m k ⊂ A ⊂ Π l k be a non-zero subspace that satisfies the condition (c2) and let {P } be a net of best approximants of F from A with respect to · * . Assume F ∈ t m+1 . If the minimization problem (3.10) has a unique solution P 0 , then P → T m + P , where P ∈ A is uniquely determined by the condition T m+1 (P ) = P 0 − T m (P 0 ).
In the following example we present a function F ∈ ∞ m=0 t m such that T 2 (F ) / ∈ A and the net {T i (P )} does not converge for the same i > m + 1. where T 0 (x) = 0 and T s (x) = x. In consequence, F ∈ t m for all m ∈ N ∪ {0}, and Since − x − 7 5 2 x 3 x i dx = 0, i = 0, 2, 3, then P (x) = 7 5 2 x 3 is the best approximant to F from A with respect to · * . Therefore T i (P )(x) → 0, for i = 0, 1, 2, but T 3 (P )(x) does not converge, as → 0. So, the best local approximation to F from A in 0 does not exist, and E (F ) = F − P * 3 = 2 √ 6 15 2 → ∞, as → 0. We now give another example which shows that the condition T m ∈ A is not necessary for the existence of the best local approximation. Example 3.8. Set B, · * and F as in Example 3.7, and we consider the subspace A = span{1, x 2 }. It is clear that A 0 = A 1 = span{x 2 }, A 2 = {0} and B = A. Moreover, we have F ∈ t 2 , T 1 / ∈ A, and A does not satisfy the condition (c2) since T 1 (x 2 ) = 0. As − (x − 0) x i dx = 0, i = 0, 2, then P (x) = 0 is the best approximant to F from A with respect to · * . Therefore, the polynomial 0 is the best local approximation to F from A in 0.