Best Approximation of Data Distributed in a Band

We study the problem to approximate a data set which are a ﬀ ected in a such way that they present us as a band in the plane. We introduce a deviation measure, and we research the asymptotic behavior of the best approximants when the band shrink in some sense.


Introduction
In some situations, we find us with the problem to approximate a given function of physical origin which is contaminated by different causes. For example, it occurs when we receive a signal, and we observe at the screen of an electronic oscilloscope a band produced by noise or other factors. Here, a criterion of selecting is necessary in order to approximate to that band. More precisely, we must choose a measure of deviation from one band to a given approximant class. A way could be to approximate a segment value multivalued function using the Hausdorff metric in the plane see 1 ; another could be to consider the best simultaneous approximation to the set of every functions whose graphics live in the band determined by them see 2-4 . In this paper, we give an alternative deviation measure, and we establish a relation with the best simultaneous approximation.
Let 1 ≤ p < ∞, and F : a, b → 2 R , be a multivalued function with F x a Lebesgue measurable set for all x ∈ a, b . Given an approximant class S, we consider the following function as measure of deviation of F to Q ∈ S : It is easy to see that 1.1 is a special case to approximate a function I from a given class C with a norm | · | over the space of functions As usual, if 1 ≤ q ≤ ∞, A ⊂ a, b is a Lebesgue measurable set and w is a nonnegative integrable function on A, then L q w A denotes the space of Lebesgue measurable functions f satisfying with the usual understanding if q ∞. If w 1, we write L q A and f q,A for L q w A and f q,w,A , respectively.
Given two functions f, g : a, b → R, f ≤ g, we only consider in this paper the multivalued function F defined by F x f x , g x for each x ∈ a, b . Our main goal is to study the asymptotic behavior of those Q which minimize 1.1 when the band shrinks in some sense, and the approximant class S is a finite dimensional linear subspace. In this case, C is a finite dimensional linear subspace, and the existence of such a Q is well known see 5 .
We consider that the band shrinks to a curve in two situations: i The functions f and g are replaced by a family of functions f , g , where f , g converge to a function h, as tends to 0. That is, the band shrinks vertically.
ii The interval a, b is substituted by x 0 − , x 0 , where tends to 0; that is, the band shrinks horizontally.
If there exists the limit of Q minimizing 1.1 when the band shrinks to a curve, as such, it provides useful qualitative and approximation analytic information concerning the approximants on small bands, which is difficult to obtain from a strictly numerical treatment. The existence of the limit of Q is close to the best local approximation problem see 6-8 . In Section 2, we prove that if the band shrinks vertically to a given function, then the set of closure points of Q is contained in the set of best approximants to that function, with a suitable seminorm. Moreover, we see that the limit of Q exists when p > 1.
In Section 3, we prove that if the band shrinks horizontally, the limit of Q is the mean of the Taylor polynomials of f and g at x 0 . We also show that this approximation problem is related with the subject of best simultaneous local approximation which was studied in 8 .
We assume conditions about the functions f and g in order to be 1.1 finite for all Q ∈ S. Henceforward, f, g ∈ L q a, b , S ⊂ L q a, b , and g − f ∈ L q/ q−p a, b , p ≤ q. In this case, using Hölder's inequality, we have for all Q ∈ S. If q ≥ p 1, the condition g − f ∈ L q/ q−p a, b is automatically satisfied.

The Band Shrinks Vertically
Let g be a measurable nonnegative function in L q a, b , p ≤ q, and {f }, {g }, > 0, two net of measurable functions such that |f |, |g | ≤ g, a.e. on a, b . We write F Let Q ∈ S. By integral mean value theorem, for each x ∈ a, b , there exists such that J Q , |α − Q | p w and J Q, |ξ − Q| p w . Now, the Fubbini Theorem implies that J Q , and J Q, are measurable functions on a, b , and hence, α and ξ are measurable functions on a, b . Consequently, α , ξ ∈ L q a, b , because |α |, |ξ | ≤ g, a.e. on a, b .
On the other hand,

2.5
In consequence, we get By hypothesis, there is k > 0 satisfying |{w > k}| > 0. By the Egoroff Theorem see 9 , there exists a set A ⊂ {w > k}, |A| > 0, where we have that w * uniformly converges to w on A. So, we can choose a positive constant 0 such that for all x ∈ A and all 0 < < 0 , w * x ≥ k/2. Hence, According to 2.
As Q ∈ S is arbitrary, the theorem immediately follows.
Suppose that p > 1. If we have two functions W i : 0, ∞ → 0, ∞ and w i , i 1, 2, fulfilling the hypothesis of Theorem 2.1, we conclude that Q converges to the best approximant to f from S when we consider · p,w 1 , a,b and · p,w 2 , a,b , respectively. The next lemma shows that it is not surprising, because these norms differing by a constant. Proof. We denote by I i , i 1, 2, the subsets of Ω, where w i is finite, and we write K I 1 ∩ I 2 .
implies that W 1 /W 2 tends to zero if x x 1 , and it tends to infinite if x x 2 , a contradiction. So, we get μ J 1 ∩ J 2 > 0.
We also observe in the next example that 2.2 in Theorem 2.1 is essential.

2.15
Here, m g m − f m , is equal to 1 if m is even, and it is equal x in otherwise. Applying Theorem 2.1, we get Q 2m → 1/2 and Q 2m−1 → 2/3, as m → ∞.

The Band Shrinks Horizontally
Let x 0 ∈ a, b and I : x 0 − , x 0 ⊂ a, b , > 0. In this section, we assume that f and g have continuous derivatives up to order n at x 0 . From now on, S Π n is the space of algebraic polynomials of degree at most n, and Q denotes an element of Π n that minimizes Φ I F, Q , Q ∈ Π n .
We recall see 10, 11 the Newton divided difference formula for the interpolation polynomial of a function h of degree n at x 1 ≤ x 2 ≤ · · · ≤ x n 1 , for some ξ in the interval x 1 , x m 1 . It is well known that the m-th divided difference is a continuous function as function of their arguments x 1 , . . . , x m 1 .

ISRN Mathematical Analysis
For simplicity of notation, we write T f and T g the Taylor polynomials of f and g at x 0 of order n, respectively. Lemma 3.1. Let q ≥ 0, and let f i : a, b → R, 1 ≤ i ≤ 3 be continuous functions with then h is a continuous function.
Proof. If q > 0, the continuity of h follows from the continuity of f i , 1 ≤ i ≤ 3 and the uniform continuity of t q sgn t on compact sets. If q 0, we have

3.4
Thus, the continuity of h is immediate.

Theorem 3.2.
If p ≥ 1, then any net Q converges to T f T g /2, as → 0.