ISRN Mathematical AnalysisVolume 2011 (2011), Article ID 147026, 8 pagesdoi:10.5402/2011/147026

Research Article

Best Approximation of Data Distributed in a Band

H. Cuenya, F. Levis, and C. Rodriguez

Universidad Nacional de Río Cuarto, Ruta 36 km 601, Río Cuarto 5800, Argentina

Received 7 June 2011; Accepted 14 July 2011

Academic Editor: J. Jahn

Copyright © 2011 H. Cuenya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. 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 , 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 [5]).

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 [8].

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 [9]), 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 .

We also observe in the next example that (2.2) in Theorem 2.1 is essential.

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 .

Acknowledgments

This work was supported by Universidad Nacional de Rio Cuarto, CONICET, and ANPCyT.

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