International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 147026 | 8 pages | https://doi.org/10.5402/2011/147026

Best Approximation of Data Distributed in a Band

Academic Editor: J. Jahn
Received07 Jun 2011
Accepted14 Jul 2011
Published13 Sep 2011

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 1≤𝑝<∞, and 𝐹∶[ğ‘Ž,𝑏]→2ℝ, be a multivalued function with 𝐹(𝑥) a Lebesgue measurable set for all 𝑥∈[ğ‘Ž,𝑏]. Given an approximant class 𝒮, we consider the following function as measure of deviation of 𝐹 to 𝑄∈𝒮∶Φ[ğ‘Ž,𝑏](𝐹,𝑄)=ğ‘ğ‘Žî€œğ¹(𝑥)|𝑦−𝑄(𝑥)|𝑝𝑑𝑦𝑑𝑥1/𝑝.(1.1)

Let 𝐷∶={(𝑥,𝑦)∶𝑥∈[ğ‘Ž,𝑏],𝑦∈𝐹(𝑥)}. It is easy to see that (1.1) is a special case to approximate a function 𝐼 from a given class ğ’ž with a norm |‖⋅‖| over the space of functions defined on 𝐷. In fact, if ğ’ž={𝐻𝑄∶𝑄∈𝒮}, where 𝐻𝑄(𝑥,𝑦)=𝑄(𝑥), then Φ[ğ‘Ž,𝑏](𝐹,𝑄)=|‖𝐼−𝐻𝑄‖|, where 𝐼(𝑥,𝑦)=𝑦 and||||=‖𝐺‖𝐷|𝐺(𝑥,𝑦)|𝑝𝑑𝑦𝑑𝑥1/𝑝.(1.2)

As usual, if 1â‰¤ğ‘žâ‰¤âˆž, 𝐴⊂[ğ‘Ž,𝑏] is a Lebesgue measurable set and 𝑤 is a nonnegative integrable function on 𝐴, then ğ¿ğ‘žğ‘¤(𝐴) denotes the space of Lebesgue measurable functions 𝑓 satisfying â€–ğ‘“â€–ğ‘ž,𝑤,𝐴∶=𝐴||||𝑓(𝑥)ğ‘žî‚¶ğ‘¤(𝑥)𝑑𝑥1/ğ‘ž<∞,(1.3) with the usual understanding if ğ‘ž=∞. If 𝑤=1, we write ğ¿ğ‘ž(𝐴) and â€–ğ‘“â€–ğ‘ž,𝐴 for ğ¿ğ‘žğ‘¤(𝐴) and â€–ğ‘“â€–ğ‘ž,𝑤,𝐴, respectively.

Given two functions 𝑓,𝑔∶[ğ‘Ž,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∈[ğ‘Ž,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, ğ’ž 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 𝑓𝜖, 𝑔𝜖, where 𝑓𝜖, 𝑔𝜖 converge to a function ℎ, as 𝜖 tends to 0. That is, the band shrinks vertically.(ii)The interval [ğ‘Ž,𝑏] is substituted by [𝑥0−𝜖,𝑥0+𝜖], where 𝜖 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 𝑥0. 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,ğ‘”âˆˆğ¿ğ‘ž([ğ‘Ž,𝑏]),ğ’®âŠ‚ğ¿ğ‘ž([ğ‘Ž,𝑏]), and ğ‘”âˆ’ğ‘“âˆˆğ¿ğ‘ž/(ğ‘žâˆ’ğ‘)([ğ‘Ž,𝑏]),ğ‘â‰¤ğ‘ž. In this case, using Hölder's inequality, we haveΦ𝑝[]ğ‘Ž,𝑏(𝐹,𝑄)â‰¤ğ‘ğ‘Žî€½||||,||||max𝑓(𝑥)−𝑄(𝑥)𝑔(𝑥)−𝑄(𝑥)𝑝≤‖‖||||,||||‖‖(𝑔(𝑥)−𝑓(𝑥))𝑑𝑥maxğ‘“âˆ’ğ‘„ğ‘”âˆ’ğ‘„ğ‘ğ‘ž,[ğ‘Ž,𝑏]â€–ğ‘”âˆ’ğ‘“â€–ğ‘ž/(ğ‘žâˆ’ğ‘),[ğ‘Ž,𝑏]<∞,(1.4) for all 𝑄∈𝒮. If ğ‘žâ‰¥ğ‘+1, the condition ğ‘”âˆ’ğ‘“âˆˆğ¿ğ‘ž/(ğ‘žâˆ’ğ‘)([ğ‘Ž,𝑏]) is automatically satisfied.

2. The Band Shrinks Vertically

Let 𝑔 be a measurable nonnegative function in ğ¿ğ‘ž([ğ‘Ž,𝑏]), ğ‘â‰¤ğ‘ž, and {𝑓𝜖}, {𝑔𝜖}, 𝜖>0, two net of measurable functions such that |𝑓𝜖|,|𝑔𝜖|≤𝑔, a.e. on [ğ‘Ž,𝑏]. We write 𝐹𝜖(𝑥)=[𝑓𝜖(𝑥),𝑔𝜖(𝑥)], 𝑥∈[ğ‘Ž,𝑏]. Given ğ’®âŠ‚ğ¿ğ‘ž([ğ‘Ž,𝑏]) a finite dimensional lineal subspace, let 𝑄𝜖∈𝒮 which minimizes Φ[ğ‘Ž,𝑏](𝐹𝜖,𝑄), 𝑄∈𝒮.

Theorem 2.1. Assume that there are two functions 𝑊(𝜖)∶(0,∞)→(0,∞) and â„Žâˆˆğ¿ğ‘ž/(ğ‘žâˆ’ğ‘)([ğ‘Ž,𝑏]) such that almost everywhere on [ğ‘Ž,𝑏]𝑔𝜖−𝑓𝜖𝑊(𝜖)≤ℎ,(2.1)lim𝜖→0𝑔𝜖−𝑓𝜖𝑊(𝜖)=𝑤,(2.2) with |{𝑤>0}|>0. If 𝑓𝜖 and 𝑔𝜖 converge to 𝑓 almost everywhere on [ğ‘Ž,𝑏], then the set of closure points of 𝑄𝜖 is a nonempty set, and it is contained in the set of best approximants to 𝑓 from 𝒮 with the seminorm ‖⋅‖𝑝,𝑤,[ğ‘Ž,𝑏]. In particular, if 𝑝>1, the net 𝑄𝜖 converges to the unique best approximant to 𝑓, as 𝜖→0.

Proof. For 𝜖>0, we denote 𝑤∗𝜖=𝑤𝜖/𝑊(𝜖), where 𝑤𝜖=𝑔𝜖−𝑓𝜖. Let 𝐽𝑄,𝜖(𝑥)∶=𝑔𝜖𝑓(𝑥)𝜖(𝑥)||||𝑦−𝑄(𝑥)𝑝[]𝑑𝑦,ğ‘¥âˆˆğ‘Ž,𝑏,𝑄∈𝒮.(2.3) Let 𝑄∈𝒮. By integral mean value theorem, for each 𝑥∈[ğ‘Ž,𝑏], there exists 𝛼𝜖(𝑥),𝜉𝜖𝑓(𝑥)∈𝜖(𝑥),𝑔𝜖(𝑥),(2.4) such that 𝐽𝑄𝜖,𝜖=|𝛼𝜖−𝑄𝜖|𝑝𝑤𝜖 and 𝐽𝑄,𝜖=|𝜉𝜖−𝑄|𝑝𝑤𝜖. Now, the Fubbini Theorem implies that 𝐽𝑄𝜖,𝜖 and 𝐽𝑄,𝜖 are measurable functions on [ğ‘Ž,b], and hence, 𝛼𝜖 and 𝜉𝜖 are measurable functions on [ğ‘Ž,𝑏]. Consequently, 𝛼𝜖, ğœ‰ğœ–âˆˆğ¿ğ‘ž([ğ‘Ž,𝑏]), because |𝛼𝜖|,|𝜉𝜖|≤𝑔, a.e. on [ğ‘Ž,𝑏].
On the other hand, ‖‖𝑓−𝑄𝜖‖‖𝑝,𝑤𝜖,[ğ‘Ž,𝑏]−‖‖𝛼𝜖‖‖−𝑓𝑝,𝑤𝜖,[ğ‘Ž,𝑏]≤‖‖𝛼𝜖−𝑄𝜖‖‖𝑝,𝑤𝜖,[ğ‘Ž,𝑏]=Φ[ğ‘Ž,𝑏]𝐹𝜖,𝑄𝜖≤Φ[ğ‘Ž,𝑏]𝐹𝜖=‖‖𝜉,𝑄𝜖‖‖−𝑄𝑝,𝑤𝜖,[ğ‘Ž,𝑏].(2.5) In consequence, we get ‖‖𝑓−𝑄𝜖‖‖𝑝,𝑤∗𝜖,[ğ‘Ž,𝑏]≤‖‖𝛼𝜖‖‖−𝑓𝑝,𝑤∗𝜖,[ğ‘Ž,𝑏]+‖‖𝜉𝜖‖‖−𝑄𝑝,𝑤∗𝜖,[ğ‘Ž,𝑏].(2.6) As |𝑓|≤𝑔 a.e. on [ğ‘Ž,𝑏], from (2.1) and the Hölder inequality, we have |𝛼𝜖−𝑓|𝑝𝑤∗𝜖≤(2𝑔)ğ‘â„Žâˆˆğ¿1([]ğ‘Ž,𝑏),|𝜉𝜖−𝑄|𝑝𝑤∗𝜖||𝑄||)≤(𝑔+ğ‘â„Žâˆˆğ¿1([]ğ‘Ž,𝑏).(2.7) From (2.2) and the Lebesgue Dominated Convergence Theorem, it follows that lim𝜖→0‖‖𝛼𝜖‖‖−𝑓𝑝,𝑤∗𝜖,[ğ‘Ž,𝑏]+‖‖𝜉𝜖‖‖−𝑄𝑝,𝑤∗𝜖,[ğ‘Ž,𝑏]=‖𝑓−𝑄‖𝑝,𝑤,[ğ‘Ž,𝑏].(2.8) 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∗𝜖 uniformly converges to 𝑤 on 𝐴. So, we can choose a positive constant 𝜖0 such that for all 𝑥∈𝐴 and all 0<𝜖<𝜖0, 𝑤∗𝜖(𝑥)≥𝑘/2. Hence, ‖‖𝑓−𝑄𝜖‖‖𝑝,𝐴≤𝑘2−1/𝑝‖‖𝑓−𝑄𝜖‖‖𝑝,𝑤∗𝜖,[ğ‘Ž,𝑏],0<𝜖<𝜖0.(2.9) According to (2.6)–(2.9), {𝑄𝜖} is a uniformly bounded net. Then, {𝑄𝜖} have a subsequence that we again denote by {𝑄𝜖} converging to 𝑇∈𝒮. Again, the Lebesgue dominated convergence theorem implies lim𝜖→0‖𝑓−𝑄𝜖‖𝑝,𝑤∗𝜖,[ğ‘Ž,𝑏]=‖𝑓−𝑇‖𝑝,𝑤,[ğ‘Ž,𝑏]. Finally, from (2.6) and (2.8), we conclude that ‖𝑓−𝑇‖𝑝,𝑤,[ğ‘Ž,𝑏]≤‖𝑓−𝑄‖𝑝,𝑤,[ğ‘Ž,𝑏].(2.10) As 𝑄∈𝒮 is arbitrary, the theorem immediately follows.

Suppose that 𝑝>1. If we have two functions 𝑊𝑖∶(0,∞)→(0,∞) and 𝑤𝑖, 𝑖=1,2, fulfilling the hypothesis of Theorem 2.1, we conclude that 𝑄𝜖 converges to the best approximant to 𝑓 from 𝒮 when we consider ‖⋅‖𝑝,𝑤1,[ğ‘Ž,𝑏] and ‖⋅‖𝑝,𝑤2,[ğ‘Ž,𝑏], respectively. The next lemma shows that it is not surprising, because these norms differing by a constant.

Lemma 2.2. Let (Ω,Σ,𝜇) be a finite measure space. Let 𝑤𝜖∶Ω→ℝ,𝜖>0 be a net of nonnegative measurable functions. Assume that there are two functions 𝑊𝑖∶(0,∞)→(0,∞), 𝑖=1,2 such that almost everywhere on Ωlim𝜖→0𝑤𝜖(𝑥)𝑊𝑖(𝜖)=𝑤𝑖(𝑥)(2.11) is finite. If 𝐴𝑖∶={𝑥∈Ω∶𝑤𝑖(𝑥)>0} and 𝜇(𝐴𝑖)>0, 𝑖=1,2, then there exists 𝑘>0 satisfying 𝑤1=𝑘𝑤2, a.e. on Ω.

Proof. We denote by 𝐼𝑖, 𝑖=1,2, the subsets of Ω, where 𝑤𝑖 is finite, and we write 𝐾=𝐼1∩𝐼2. Clearly, 𝜇(𝐾)=𝜇(Ω). Let 𝐽𝑖=𝐴𝑖∩𝐾.
If 𝜇(𝐽1∩𝐽2)=0, then 𝜇(𝐽1−𝐽2)=𝜇(𝐴1)>0 and 𝜇(𝐽2−𝐽1)=𝜇(𝐴2)>0. We take 𝑥1∈𝐽1−𝐽2 and 𝑥2∈𝐽2−𝐽1. The equality 𝑤𝜖(𝑥)𝑊2=𝑤(𝜖)𝜖(𝑥)𝑊1𝑊(𝜖)1(𝜖)𝑊2(𝜖),𝑥∈Ω(2.12) implies that 𝑊1(𝜖)/𝑊2(𝜖) tends to zero if 𝑥=𝑥1, and it tends to infinite if 𝑥=𝑥2, a contradiction. So, we get 𝜇(𝐽1∩𝐽2)>0.
For all 𝑥∈𝐽1∩𝐽2, we have 0<𝑘∶=lim𝜖→0𝑊1(𝜖)𝑊2(=𝑤𝜖)2(𝑥)𝑤1(.𝑥)(2.13) Next, we prove that 𝜇(𝐽1−𝐽2)=𝜇(𝐽2−𝐽1)=0. If 𝜇(𝐽1−𝐽2)>0 the above argument implies that 𝑘=0, and if 𝜇(𝐽2−𝐽1)>0, we obtain 𝑘=∞. In either case, it contradicts (2.13). Therefore, 𝜇(𝐽1∩𝐽2)=𝜇(𝐽1∪𝐽2)=𝜇(𝐴1)=𝜇(𝐴2) and 𝑤2(𝑥)=𝑘𝑤1(𝑥),a.e.on𝐴1∪𝐴2.(2.14) It immediately follows that (2.14) holds a.e. on Ω.

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

Example 2.3. Let ğ‘Ž=0, 𝑏=1, and 𝑝=2. We consider 𝒮 the space of constant functions, 𝑓𝑚(𝑥)=𝑥,𝑔𝑚=⎧⎪⎨⎪⎩1𝑥+𝑚𝑥,if𝑚iseven,𝑥+𝑚,if𝑚isodd.(2.15) Here, 𝑚(𝑔𝑚−𝑓𝑚), is equal to 1 if 𝑚 is even, and it is equal 𝑥 in otherwise. Applying Theorem 2.1, we get 𝑄2𝑚→1/2 and 𝑄2𝑚−1→2/3, as ğ‘šâ†’âˆž.

3. The Band Shrinks Horizontally

Let 𝑥0∈[ğ‘Ž,𝑏] and 𝐼𝜖∶=[𝑥0−𝜖,𝑥0+𝜖]⊂[ğ‘Ž,𝑏], 𝜖>0. In this section, we assume that 𝑓 and 𝑔 have continuous derivatives up to order 𝑛 at 𝑥0. From now on, 𝒮=Π𝑛 is the space of algebraic polynomials of degree at most 𝑛, and 𝑄𝜖 denotes an element of Π𝑛 that minimizes Φ𝐼𝜖(𝐹,𝑄), 𝑄∈Π𝑛.

We recall (see [10, 11]) the Newton divided difference formula for the interpolation polynomial of a function ℎ of degree 𝑛 at 𝑥1≤𝑥2≤⋯≤𝑥n+1,𝑃𝑥(𝑥)=ℎ1+𝑥−𝑥1î€¸â„Žî€ºğ‘¥1,𝑥2+⋯+𝑥−𝑥1î€¸â‹¯î€·ğ‘¥âˆ’ğ‘¥ğ‘›î€¸â„Žî€ºğ‘¥1,…,𝑥𝑛+1.(3.1) Here, ℎ[𝑥1,…,𝑥𝑚+1] denotes the 𝑚-th order Newton divided difference. If ℎ has continuous derivatives up to order 𝑚, on an interval [ğ‘Ž,𝑏] containing to 𝑥1,…,𝑥𝑚+1, then the 𝑚-th divided difference can be expressed asâ„Žî€ºğ‘¥1,…,𝑥𝑚+1=ℎ(𝑚)(𝜉),𝑚!(3.2) for some 𝜉 in the interval [𝑥1,𝑥𝑚+1]. It is well known that the 𝑚-th divided difference is a continuous function as function of their arguments 𝑥1,…,𝑥𝑚+1.

For simplicity of notation, we write 𝑇(𝑓) and 𝑇(𝑔) the Taylor polynomials of 𝑓 and 𝑔 at 𝑥0 of order 𝑛, respectively.

Lemma 3.1. Let ğ‘žâ‰¥0, and let 𝑓𝑖∶[ğ‘Ž,𝑏]→ℝ, 1≤𝑖≤3 be continuous functions with 𝑓1≤𝑓2. If ℎ(𝑥)=𝑓2𝑓(𝑥)1(𝑥)||𝑦−𝑓3||(𝑥)ğ‘žî€·sgn𝑦−𝑓3(𝑥)𝑑𝑦,ğ‘Žâ‰¤ğ‘¥â‰¤ğ‘,(3.3) then ℎ is a continuous function.

Proof. If ğ‘ž>0, the continuity of ℎ follows from the continuity of 𝑓𝑖, 1≤𝑖≤3 and the uniform continuity of ğ‘¡ğ‘žsgn(𝑡) on compact sets. If ğ‘ž=0, we have âŽ§âŽªâŽ¨âŽªâŽ©ğ‘“â„Ž(𝑥)=1(𝑥)−𝑓2(𝑥),if𝑓3(𝑥)≥𝑓2𝑓(𝑥),2(𝑥)−𝑓1(𝑥),if𝑓3(𝑥)≤𝑓1𝑓(𝑥),1(𝑥)+𝑓2(𝑥)−2𝑓3(𝑥),if𝑓1(𝑥)≤𝑓3(𝑥)≤𝑓2(𝑥).(3.4) Thus, the continuity of ℎ is immediate.

Theorem 3.2. If 𝑝≥1, then any net 𝑄𝜖 converges to (𝑇(𝑓)+𝑇(𝑔))/2, as 𝜖→0.

Proof. It is easy to see that 𝑄𝜖 is characterized by î€œğ¼ğœ–â„Ž(𝑥)𝑄(𝑥)𝑑𝑥=0,𝑄∈Π𝑛,(3.5) where ℎ(𝑥)=𝑔(𝑥)𝑓(𝑥)||𝑦−𝑄𝜖||(𝑥)𝑝−1sgn𝑦−𝑄𝜖(𝑥)𝑑𝑦.(3.6) By Lemma 3.1, ℎ is continuous. Therefore, ℎ interpolates to zero in at least 𝑛+1 different points of the interval 𝐼𝜖, say 𝑥1<𝑥2<⋯<𝑥𝑛+1. In fact, if ℎ has 𝑚 different zeros, 𝑚≤𝑛, we can find an element 𝑄∈Π𝑛 such that ℎ(𝑥)𝑄(𝑥)>0 for all 𝑥≠𝑥𝑖, 1≤𝑖≤𝑚. It contradicts (3.5). So, 𝑔(𝑥𝑖)𝑓(𝑥𝑖)|𝑦−𝑄𝜖𝑥𝑖|𝑝−1sgn𝑦−𝑄𝜖𝑥𝑖𝑑𝑦=0,1≤𝑖≤𝑛+1.(3.7) It follows that 𝑄𝜖(𝑥𝑖) is a best constant approximant to the identity function with norm ‖⋅‖𝑝,[𝑓(𝑥𝑖),𝑔(𝑥𝑖)], 1≤𝑖≤𝑛+1. A straightforward computation shows that 𝑄𝜖(𝑥𝑖)=(𝑓(𝑥𝑖)+𝑔(𝑥𝑖))/2, 1≤𝑖≤𝑛+1; that is, 𝑄𝜖 interpolates to the function (𝑓+𝑔)/2 at 𝑥𝑖, 1≤𝑖≤𝑛+1. From (3.1) and (3.2), it follows that 𝑄𝜖𝑥(𝑥)=ℎ1+𝑥−𝑥1ℎ(1)𝜉1+⋯+𝑥−𝑥1î€¸â‹¯î€·ğ‘¥âˆ’ğ‘¥ğ‘›î€¸â„Ž(𝑛)𝜉𝑛,𝑛!(3.8) where ℎ=(𝑓+𝑔)/2, 𝜉𝑖∈𝐼𝜖, 1≤𝑖≤𝑛. Taking limit for 𝜖→0 in (3.8) and using the continuity of the derivatives of the functions 𝑓 and 𝑔, we get the theorem.

Corollary 3.3. Let 𝑝≥1. Suppose that 𝑓 and 𝑔 have continuous derivatives up to order 𝑛+1 at 𝑥0 and 𝑓(𝑥0)<𝑔(𝑥0). Then, for sufficiently small 𝜖, 𝑄𝜖 is the best 𝑙𝑝+1-simultaneous approximant in 𝐿𝑝+1(𝐼𝜖); that is, 𝑄𝜖 minimizes ‖𝑓−𝑄‖𝑝+1𝑝+1,𝐼𝜖+‖𝑔−𝑄‖𝑝+1𝑝+1,𝐼𝜖1/(𝑝+1),𝑄∈Π𝑛.(3.9)

Proof. By hypothesis, there is 𝜖0>0 such that 𝑓<𝑔 on 𝐼𝜖0. Let 𝑃𝜖 be the best 𝑙𝑝+1-simultaneous approximant to 𝑓 and 𝑔 in 𝐿𝑝+1(𝐼𝜖). From Theorem 3.2, and [8, Theorem  3.4], there exists 0<𝜖1<𝜖0 such that 𝑄𝜖(𝑥),𝑃𝜖[](𝑥)∈𝑓(𝑥),𝑔(𝑥),𝑥∈𝐼𝜖,𝜖<𝜖1.(3.10) On the other hand, we have (𝑝+1)Φ𝑝𝐼𝜖(𝐹,𝑄)=𝐼𝜖||||𝑓(𝑥)−𝑄(𝑥)𝑝+1+sgn(𝑄(𝑥)−𝑓(𝑥))𝑑𝑥𝐼𝜖|𝑔(𝑥)−𝑄(𝑥)|𝑝+1sgn(𝑔(𝑥)−𝑄(𝑥))𝑑𝑥,𝑄∈Π𝑛,(3.11) From (3.10) and (3.11), we get ‖‖𝑓−𝑄𝜖‖‖𝑝+1𝑝+1,𝐼𝜖+‖‖𝑔−𝑄𝜖‖‖𝑝+1𝑝+1,𝐼𝜖=‖‖𝑓−𝑃𝜖‖‖𝑝+1𝑝+1,𝐼𝜖+‖‖𝑔−𝑃𝜖‖‖𝑝+1𝑝+1,𝐼𝜖,(3.12) for all 𝜖<𝜖1. So, 𝑄𝜖=𝑃𝜖 for all 𝜖<𝜖1.

Remark 3.4. If Φ[ğ‘Ž,𝑏](𝐹,𝑄)=sup𝑥∈[ğ‘Ž,𝑏]sup𝑦∈𝐹(𝑥)|𝑦−𝑄(𝑥)|, then Φ[ğ‘Ž,𝑏](𝐹,𝑄)=sup𝑥∈[ğ‘Ž,𝑏]||𝑓||,||𝑔||max(𝑥)−𝑄(𝑥)(𝑥)−𝑄(𝑥)(3.13) implies that 𝑄𝜖 is the best ğ‘™âˆž-simultaneous approximant in ğ¿âˆž([ğ‘Ž,𝑏]), for all 𝜖>0.

Acknowledgments

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

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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.

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