Best Approximation of Data Distributed in a Band

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∶[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∶Φ_[𝑎,𝑏](𝐹,𝑄)=^𝑏𝑎_𝐹(𝑥)|𝑦−𝑄(𝑥)|^𝑝𝑑𝑦𝑑𝑥^1/𝑝.(1.1)

Let 𝐷∶={(𝑥,𝑦)∶𝑥∈[𝑎,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 𝒞 with a norm |‖⋅‖| over the space of functions defined on 𝐷. In fact, if 𝒞={𝐻_𝑄∶𝑄∈𝒮}, where 𝐻_𝑄(𝑥,𝑦)=𝑄(𝑥), then Φ_[𝑎,𝑏](𝐹,𝑄)=|‖𝐼−𝐻_𝑄‖|, where 𝐼(𝑥,𝑦)=𝑦 and||‖𝐺‖||=_𝐷|𝐺(𝑥,𝑦)|^𝑝𝑑𝑦𝑑𝑥^1/𝑝.(1.2)

As usual, if 1≤𝑞≤∞, 𝐴⊂[𝑎,𝑏] is a Lebesgue measurable set and 𝑤 is a nonnegative integrable function on 𝐴, then 𝐿^𝑞𝑤(𝐴) denotes the space of Lebesgue measurable functions 𝑓 satisfying ‖𝑓‖_{𝑞,𝑤,𝐴}∶=_𝐴||𝑓(𝑥)||^𝑞𝑤(𝑥)𝑑𝑥^1/𝑞<∞,(1.3) with the usual understanding if 𝑞=∞. If 𝑤=1, we write 𝐿^𝑞(𝐴) and ‖𝑓‖_𝑞,𝐴 for 𝐿^𝑞𝑤(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_𝜖, 𝑔_𝜖, where 𝑓_𝜖, 𝑔_𝜖 converge to a function ℎ, as 𝜖 tends to 0. That is, the band shrinks vertically.(ii)The interval [𝑎,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,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 haveΦ^{𝑝[]𝑎,𝑏}(𝐹,𝑄)≤≤‖‖^𝑏𝑎mmaxax||𝑓||𝑓−(𝑥𝑄)||−,||𝑔𝑄−(𝑥𝑄)||||,||‖‖𝑔(𝑥)−_{𝑝𝑞,[𝑎,𝑏]}𝑄(𝑥)||^𝑝(𝑔(𝑥)−𝑓(𝑥))𝑑𝑥‖𝑔−𝑓‖_{𝑞/(𝑞−𝑝),[𝑎,𝑏]}<∞,(1.4) for all 𝑄∈𝒮. If 𝑞≥𝑝+1, the condition 𝑔−𝑓∈𝐿^{𝑞/(𝑞−𝑝)}([𝑎,𝑏]) is automatically satisfied.

2. The Band Shrinks Vertically

Let 𝑔 be a measurable nonnegative function in 𝐿^𝑞([𝑎,𝑏]), 𝑝≤𝑞, and {𝑓_𝜖}, {𝑔_𝜖}, 𝜖>0, two net of measurable functions such that |𝑓_𝜖|,|𝑔_𝜖|≤𝑔, a.e. on [𝑎,𝑏]. We write 𝐹_𝜖(𝑥)=[𝑓_𝜖(𝑥),𝑔_𝜖(𝑥)], 𝑥∈[𝑎,𝑏]. Given 𝒮⊂𝐿^𝑞([𝑎,𝑏]) a finite dimensional lineal subspace, let 𝑄_𝜖∈𝒮 which minimizes Φ_[𝑎,𝑏](𝐹_𝜖,𝑄), 𝑄∈𝒮.

Theorem 2.1. Assume that there are two functions 𝑊(𝜖)∶(0,∞)→(0,∞) and ℎ∈𝐿^{𝑞/(𝑞−𝑝)}([𝑎,𝑏]) such that almost everywhere on [𝑎,𝑏]𝑔_𝜖−𝑓_𝜖lim_𝜖→0𝑊(𝜖)≤ℎ,(2.1)𝑔_𝜖−𝑓_𝜖𝑊(𝜖)=𝑤,(2.2) with |{𝑤>0}|>0. If 𝑓_𝜖 and 𝑔_𝜖 converge to 𝑓 almost everywhere on [𝑎,𝑏], then the set of closure points of 𝑄_𝜖 is a nonempty set, and it is contained in the set of best approximants to 𝑓 from 𝒮 with the seminorm ‖⋅‖_{𝑝,𝑤,[𝑎,𝑏]}. In particular, if 𝑝>1, the net 𝑄_𝜖 converges to the unique best approximant to 𝑓, as 𝜖→0.

Proof. For 𝜖>0, we denote 𝑤^∗𝜖=𝑤_𝜖/𝑊(𝜖), where 𝑤_𝜖=𝑔_𝜖−𝑓_𝜖. Let 𝐽_𝑄,𝜖(𝑥)∶=^{𝑔𝜖𝑓(𝑥)}_𝜖(𝑥)||𝑦−𝑄(𝑥)||^𝑝𝑑𝑦,𝑥∈[𝑎,𝑏],𝑄∈𝒮.(2.3) Let 𝑄∈𝒮. By integral mean value theorem, for each 𝑥∈[𝑎,𝑏], there exists 𝛼_𝜖(𝑥),𝜉_𝜖(𝑥)∈𝑓_𝜖(𝑥),𝑔_𝜖(𝑥),(2.4) such that 𝐽_{𝑄𝜖,𝜖}=|𝛼_𝜖−𝑄_𝜖|^𝑝𝑤_𝜖 and 𝐽_𝑄,𝜖=|𝜉_𝜖−𝑄|^𝑝𝑤_𝜖. Now, the Fubbini Theorem implies that 𝐽_{𝑄𝜖,𝜖} and 𝐽_𝑄,𝜖 are measurable functions on [𝑎,b], and hence, 𝛼_𝜖 and 𝜉_𝜖 are measurable functions on [𝑎,𝑏]. Consequently, 𝛼_𝜖, 𝜉_𝜖∈𝐿^𝑞([𝑎,𝑏]), because |𝛼_𝜖|,|𝜉_𝜖|≤𝑔, a.e. on [𝑎,𝑏].
On the other hand, ‖‖𝑓−𝑄_𝜖‖‖_{𝑝,𝑤𝜖,[𝑎,𝑏]}−‖‖𝛼_𝜖−𝑓‖‖_{𝑝,𝑤𝜖,[𝑎,𝑏]}≤≤Φ‖‖𝛼_𝜖−_[𝑎,𝑏]𝑄_𝜖‖‖_{𝑝,𝑤𝜖,[𝑎,𝑏]}=Φ_[𝑎,𝑏]𝐹_𝜖,𝑄_𝜖𝐹_𝜖,𝑄=‖‖𝜉_𝜖−𝑄‖‖_{𝑝,𝑤𝜖,[𝑎,𝑏]}.(2.5) In consequence, we get ‖‖𝑓−𝑄_𝜖‖‖_{𝑝,𝑤∗𝜖,[𝑎,𝑏]}≤‖‖𝛼_𝜖−𝑓‖‖_{𝑝,𝑤∗𝜖,[𝑎,𝑏]}+‖‖𝜉_𝜖−𝑄‖‖_{𝑝,𝑤∗𝜖,[𝑎,𝑏]}.(2.6) As |𝑓|≤𝑔 a.e. on [𝑎,𝑏], from (2.1) and the Hölder inequality, we have |𝛼_𝜖−𝑓|^𝑝𝑤^∗𝜖≤(2𝑔)^𝑝ℎ∈𝐿¹([𝑎,𝑏]),|𝜉_𝜖−𝑄|^𝑝𝑤^∗𝜖≤(𝑔+||𝑄||)^𝑝ℎ∈𝐿¹([𝑎,𝑏]).(2.7) From (2.2) and the Lebesgue Dominated Convergence Theorem, it follows that lim_𝜖→0‖‖𝛼^𝜖−𝑓‖‖_{𝑝,𝑤∗𝜖,[𝑎,𝑏]}+‖‖𝜉_𝜖−𝑄‖‖_{𝑝,𝑤∗𝜖,[𝑎,𝑏]}=‖𝑓−𝑄‖^{𝑝,𝑤,[𝑎,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−𝑄_𝜖‖‖_𝑝,𝐴≤𝑘2^−1/𝑝‖‖𝑓−𝑄_𝜖‖‖_{𝑝,𝑤∗𝜖,[𝑎,𝑏]},0<𝜖<𝜖₀.(2.9) According to (2.6)–(2.9), {𝑄_𝜖} is a uniformly bounded net. Then, {𝑄_𝜖} have a subsequence that we again denote by {𝑄_𝜖} converging to 𝑇∈𝒮. Again, the Lebesgue dominated convergence theorem implies lim_𝜖→0‖𝑓−𝑄_𝜖‖_𝑝,𝑤^∗𝜖_,[𝑎,𝑏]=‖𝑓−𝑇‖_{𝑝,𝑤,[𝑎,𝑏]}. Finally, from (2.6) and (2.8), we conclude that ‖𝑓−𝑇‖_{𝑝,𝑤,[𝑎,𝑏]}≤‖𝑓−𝑄‖_{𝑝,𝑤,[𝑎,𝑏]}.(2.10) 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_𝜖 converges to the best approximant to 𝑓 from 𝒮 when we consider ‖⋅‖_{𝑝,𝑤1,[𝑎,𝑏]} and ‖⋅‖_{𝑝,𝑤2,[𝑎,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 𝑤_𝜖∶Ω→ℝ,𝜖>0 be a net of nonnegative measurable functions. Assume that there are two functions 𝑊_𝑖∶(0,∞)→(0,∞), 𝑖=1,2 such that almost everywhere on Ωlim_𝜖→0𝑤_𝜖(𝑥)𝑊_𝑖(𝜖)=𝑤^𝑖(𝑥)(2.11) is finite. If 𝐴_𝑖∶={𝑥∈Ω∶𝑤_𝑖(𝑥)>0} and 𝜇(𝐴_𝑖)>0, 𝑖=1,2, then there exists 𝑘>0 satisfying 𝑤₁=𝑘𝑤₂, a.e. on Ω.

Proof. We denote by 𝐼_𝑖, 𝑖=1,2, the subsets of Ω, where 𝑤_𝑖 is finite, and we write 𝐾=𝐼₁∩𝐼₂. Clearly, 𝜇(𝐾)=𝜇(Ω). Let 𝐽_𝑖=𝐴_𝑖∩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 𝑤_𝜖(𝑥)𝑊₂(𝜖)=𝑤^𝜖(𝑥)𝑊₁(𝜖)𝑊¹(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∶=lim_𝜖→0𝑊₁(𝜖)𝑊₂(𝜖)=𝑤²(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 𝑎=0, 𝑏=1, and 𝑝=2. We consider 𝒮 the space of constant functions, 𝑓_𝑚(𝑥)=𝑥,𝑔_𝑚=⎧⎪⎨⎪⎩𝑥𝑥++𝑚𝑥1,if𝑚iseven,𝑚,if𝑚isodd.(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→∞.

3. The Band Shrinks Horizontally

Let 𝑥₀∈[𝑎,𝑏] and 𝐼_𝜖∶=[𝑥₀−𝜖,𝑥₀+𝜖]⊂[𝑎,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=Π^𝑛 is the space of algebraic polynomials of degree at most 𝑛, and 𝑄_𝜖 denotes an element of Π^𝑛 that minimizes Φ_𝐼𝜖(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 𝑥₁≤𝑥₂≤⋯≤𝑥_n+1,𝑃(𝑥)=ℎ𝑥₁+𝑥−𝑥₁ℎ𝑥₁,𝑥₂+⋯+𝑥−𝑥₁⋯𝑥−𝑥_𝑛ℎ𝑥₁,…,𝑥_𝑛+1.(3.1) Here, ℎ[𝑥₁,…,𝑥_𝑚+1] denotes the 𝑚-th order Newton divided difference. If ℎ has continuous derivatives up to order 𝑚, on an interval [𝑎,𝑏] containing to 𝑥₁,…,𝑥_𝑚+1, then the 𝑚-th divided difference can be expressed asℎ𝑥₁,…,𝑥_𝑚+1=ℎ^(𝑚)(𝜉)𝑚!,(3.2) for some 𝜉 in the interval [𝑥₁,𝑥_𝑚+1]. It is well known that the 𝑚-th divided difference is a continuous function as function of their arguments 𝑥₁,…,𝑥_𝑚+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 𝑞≥0, and let 𝑓_𝑖∶[𝑎,𝑏]→ℝ, 1≤𝑖≤3 be continuous functions with 𝑓₁≤𝑓₂. If ℎ(𝑥)=^{𝑓2𝑓(𝑥)}_1(𝑥)||𝑦−𝑓₃(𝑥)||^𝑞sgn𝑦−𝑓₃(𝑥)𝑑𝑦,𝑎≤𝑥≤𝑏,(3.3) 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^𝑞sgn(𝑡) on compact sets. If 𝑞=0, we have ℎ(𝑥)=⎧⎪⎨⎪⎩𝑓¹𝑓(𝑥)−𝑓₂(𝑥),if𝑓₃(𝑥)≥𝑓₂(𝑥),₂𝑓(𝑥)−𝑓₁(𝑥),if𝑓₃(𝑥)≤𝑓₁(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 𝑄_𝜖 converges to (𝑇(𝑓)+𝑇(𝑔))/2, as 𝜖→0.

Proof. It is easy to see that 𝑄_𝜖 is characterized by _𝐼𝜖ℎ(𝑥)𝑄(𝑥)𝑑𝑥=0,𝑄∈Π^𝑛,(3.5) where ℎ(𝑥)=^{𝑔(𝑥)𝑓(𝑥)}||𝑦−𝑄_𝜖(𝑥)||^𝑝−1sgn𝑦−𝑄_𝜖(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 𝐼_𝜖, say 𝑥₁<𝑥₂<⋯<𝑥_𝑛+1. In fact, if ℎ has 𝑚 different zeros, 𝑚≤𝑛, we can find an element 𝑄∈Π^𝑛 such that ℎ(𝑥)𝑄(𝑥)>0 for all 𝑥≠𝑥_𝑖, 1≤𝑖≤𝑚. It contradicts (3.5). So, ^{𝑔(𝑥𝑖)𝑓(𝑥}_𝑖)|𝑦−𝑄_𝜖𝑥_𝑖|^𝑝−1sgn𝑦−𝑄_𝜖𝑥_𝑖𝑑𝑦=0,1≤𝑖≤𝑛+1.(3.7) It follows that 𝑄_𝜖(𝑥_𝑖) 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 𝑄_𝜖(𝑥_𝑖)=(𝑓(𝑥_𝑖)+𝑔(𝑥_𝑖))/2, 1≤𝑖≤𝑛+1; that is, 𝑄_𝜖 interpolates to the function (𝑓+𝑔)/2 at 𝑥_𝑖, 1≤𝑖≤𝑛+1. From (3.1) and (3.2), it follows that 𝑄_𝜖(𝑥)=ℎ𝑥₁+𝑥−𝑥₁ℎ⁽¹⁾𝜉₁+⋯+𝑥−𝑥₁⋯𝑥−𝑥_𝑛ℎ^(𝑛)𝜉_𝑛𝑛!,(3.8) where ℎ=(𝑓+𝑔)/2, 𝜉_𝑖∈𝐼_𝜖, 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, 𝑄_𝜖 is the best 𝑙^𝑝+1-simultaneous approximant in 𝐿^𝑝+1(𝐼_𝜖); that is, 𝑄_𝜖 minimizes ‖𝑓−𝑄‖^{𝑝+1𝑝+1,𝐼}_𝜖+‖𝑔−𝑄‖^{𝑝+1𝑝+1,𝐼}_𝜖^1/(𝑝+1),𝑄∈Π^𝑛.(3.9)

Proof. By hypothesis, there is 𝜖₀>0 such that 𝑓<𝑔 on 𝐼_𝜖0. Let 𝑃_𝜖 be the best 𝑙^𝑝+1-simultaneous approximant to 𝑓 and 𝑔 in 𝐿^𝑝+1(𝐼_𝜖). From Theorem 3.2, and [8, Theorem  3.4], there exists 0<𝜖₁<𝜖₀ such that 𝑄_𝜖(𝑥),𝑃_𝜖(𝑥)∈[𝑓(𝑥),𝑔(𝑥)],𝑥∈𝐼_𝜖,𝜖<𝜖₁.(3.10) On the other hand, we have (𝑝+1)Φ^𝑝𝐼_𝜖(𝐹,𝑄)=+_𝐼𝜖||𝑓(𝑥)−𝑄(𝑥)||^𝑝+1sgn(𝑄(𝑥)−𝑓(𝑥))𝑑𝑥_𝐼𝜖|𝑔(𝑥)−𝑄(𝑥)|^𝑝+1sgn(𝑔(𝑥)−𝑄(𝑥))𝑑𝑥,𝑄∈Π^𝑛,(3.11) From (3.10) and (3.11), we get ‖‖𝑓−𝑄_𝜖‖‖^{𝑝+1𝑝+1,𝐼}_𝜖+‖‖𝑔−𝑄_𝜖‖‖^{𝑝+1𝑝+1,𝐼}_𝜖=‖‖𝑓−𝑃_𝜖‖‖^{𝑝+1𝑝+1,𝐼}_𝜖+‖‖𝑔−𝑃_𝜖‖‖^{𝑝+1𝑝+1,𝐼}_𝜖,(3.12) for all 𝜖<𝜖₁. So, 𝑄_𝜖=𝑃_𝜖 for all 𝜖<𝜖₁.

Remark 3.4. If Φ_[𝑎,𝑏](𝐹,𝑄)=sup_{𝑥∈[𝑎,𝑏]}sup_{𝑦∈𝐹(𝑥)}|𝑦−𝑄(𝑥)|, then Φ_[𝑎,𝑏](𝐹,𝑄)=sup_{𝑥∈[𝑎,𝑏]}max||𝑓(𝑥)−𝑄(𝑥)||,||𝑔(𝑥)−𝑄(𝑥)||(3.13) implies that 𝑄_𝜖 is the best 𝑙^∞-simultaneous approximant in 𝐿^∞([𝑎,𝑏]), for all 𝜖>0.

Acknowledgments

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