Research Article | Open Access
Best Approximation of Data Distributed in a Band
We study the problem to approximate a data set which are affected 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.
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 ); 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 , and , be a multivalued function with a Lebesgue measurable set for all . Given an approximant class , we consider the following function as measure of deviation of to
Let . It is easy to see that (1.1) is a special case to approximate a function from a given class with a norm over the space of functions defined on . In fact, if , where , then , where and
As usual, if , is a Lebesgue measurable set and is a nonnegative integrable function on , then denotes the space of Lebesgue measurable functions satisfying with the usual understanding if . If , we write and for and , respectively.
Given two functions , , we only consider in this paper the multivalued function defined by for each .
Our main goal is to study the asymptotic behavior of those which minimize (1.1)when the band shrinks in some sense, and the approximant class is a finite dimensional linear subspace. In this case, is a finite dimensional linear subspace, and the existence of such a is well known (see ).
We consider that the band shrinks to a curve in two situations: (i)The functions and are replaced by a family of functions , , where , converge to a function , as tends to 0. That is, the band shrinks vertically.(ii)The interval is substituted by , where tends to 0; that is, the band shrinks horizontally.
If there exists the limit of 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 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 is contained in the set of best approximants to that function, with a suitable seminorm. Moreover, we see that the limit of exists when .
In Section 3, we prove that if the band shrinks horizontally, the limit of is the mean of the Taylor polynomials of and at . We also show that this approximation problem is related with the subject of best simultaneous local approximation which was studied in .
We assume conditions about the functions and in order to be (1.1) finite for all . Henceforward, , and ,. In this case, using Hölder's inequality, we have for all . If , the condition is automatically satisfied.
2. The Band Shrinks Vertically
Let be a measurable nonnegative function in , , and , , , two net of measurable functions such that , a.e. on . We write , . Given a finite dimensional lineal subspace, let which minimizes , .
Theorem 2.1. Assume that there are two functions and such that almost everywhere on with . If and converge to almost everywhere on , then the set of closure points of is a nonempty set, and it is contained in the set of best approximants to from with the seminorm . In particular, if , the net converges to the unique best approximant to , as .
Proof. For , we denote , where . Let
Let . By integral mean value theorem, for each , there exists
such that and . Now, the Fubbini Theorem implies that and are measurable functions on , and hence, and are measurable functions on . Consequently, , , because , a.e. on .
On the other hand, In consequence, we get As a.e. on , from (2.1) and the Hölder inequality, we have From (2.2) and the Lebesgue Dominated Convergence Theorem, it follows that By hypothesis, there is satisfying . By the Egoroff Theorem (see ), there exists a set , , where we have that uniformly converges to on . So, we can choose a positive constant such that for all and all , . Hence, According to (2.6)–(2.9), is a uniformly bounded net. Then, have a subsequence that we again denote by converging to . Again, the Lebesgue dominated convergence theorem implies . Finally, from (2.6) and (2.8), we conclude that As is arbitrary, the theorem immediately follows.
Suppose that . If we have two functions and , , fulfilling the hypothesis of Theorem 2.1, we conclude that converges to the best approximant to from when we consider and , respectively. The next lemma shows that it is not surprising, because these norms differing by a constant.
Lemma 2.2. Let be a finite measure space. Let be a net of nonnegative measurable functions. Assume that there are two functions , such that almost everywhere on is finite. If and , , then there exists satisfying , a.e. on .
Proof. We denote by , , the subsets of , where is finite, and we write . Clearly, . Let .
If , then and . We take and . The equality implies that tends to zero if , and it tends to infinite if , a contradiction. So, we get .
For all , we have Next, we prove that . If the above argument implies that , and if , we obtain . In either case, it contradicts (2.13). Therefore, and It immediately follows that (2.14) holds a.e. on .
Example 2.3. Let , , and . We consider the space of constant functions, Here, , is equal to 1 if is even, and it is equal in otherwise. Applying Theorem 2.1, we get and , as .
3. The Band Shrinks Horizontally
Let and , . In this section, we assume that and have continuous derivatives up to order at . From now on, is the space of algebraic polynomials of degree at most , and denotes an element of that minimizes , .
We recall (see [10, 11]) the Newton divided difference formula for the interpolation polynomial of a function of degree at , Here, denotes the th order Newton divided difference. If has continuous derivatives up to order , on an interval containing to , then the th divided difference can be expressed as for some in the interval . It is well known that the -th divided difference is a continuous function as function of their arguments .
For simplicity of notation, we write and the Taylor polynomials of and at of order , respectively.
Lemma 3.1. Let , and let , be continuous functions with . If then is a continuous function.
Proof. If , the continuity of follows from the continuity of , and the uniform continuity of on compact sets. If , we have Thus, the continuity of is immediate.
Theorem 3.2. If , then any net converges to , as .
Proof. It is easy to see that is characterized by where By Lemma 3.1, is continuous. Therefore, interpolates to zero in at least different points of the interval , say . In fact, if has different zeros, , we can find an element such that for all , . It contradicts (3.5). So, It follows that is a best constant approximant to the identity function with norm , . A straightforward computation shows that , ; that is, interpolates to the function at , . From (3.1) and (3.2), it follows that where , , . Taking limit for in (3.8) and using the continuity of the derivatives of the functions and , we get the theorem.
Corollary 3.3. Let . Suppose that and have continuous derivatives up to order at and . Then, for sufficiently small , is the best -simultaneous approximant in ; that is, minimizes
Proof. By hypothesis, there is such that on . Let be the best -simultaneous approximant to and in . From Theorem 3.2, and [8, Theorem 3.4], there exists such that On the other hand, we have From (3.10) and (3.11), we get for all . So, for all .
Remark 3.4. If then implies that is the best -simultaneous approximant in , for all .
This work was supported by Universidad Nacional de Rio Cuarto, CONICET, and ANPCyT.
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