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]→2ℝ, 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)=(∫ab∫F(x)|y-Q(x)|pdydx)1/p.

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={HQ:Q∈𝒮}, where HQ(x,y)=Q(x), then Φ[a,b](F,Q)=|∥I-HQ∥|, where I(x,y)=y and|‖G‖|=(∬D|G(x,y)|pdydx)1/p.

As usual, if 1≤q≤∞, A⊂[a,b] is a Lebesgue measurable set and w is a nonnegative integrable function on A, then Lwq(A) denotes the space of Lebesgue measurable functions f satisfying ‖f‖q,w,A:=(∫A|f(x)|qw(x)dx)1/q<∞,
with the usual understanding if q=∞. If w=1, we write Lq(A) and ∥f∥q,A for Lwq(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

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:

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.

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∈Lq([a,b]),𝒮⊂Lq([a,b]), and g-f∈Lq/(q-p)([a,b]),p≤q. In this case, using Hölder's inequality, we haveΦ[a,b]p(F,Q)≤∫abmax{|f(x)-Q(x)|,|g(x)-Q(x)|}p(g(x)-f(x))dx≤‖max{|f-Q|,|g-Q|}‖q,[a,b]p‖g-f‖q/(q-p),[a,b]<∞,
for all Q∈𝒮. If q≥p+1, the condition g-f∈Lq/(q-p)([a,b]) is automatically satisfied.

2. The Band Shrinks Vertically

Let g be a measurable nonnegative function in Lq([a,b]), p≤q, and {fϵ}, {gϵ}, ϵ>0, two net of measurable functions such that |fϵ|,|gϵ|≤g, a.e. on [a,b]. We write Fϵ(x)=[fϵ(x),gϵ(x)], x∈[a,b]. Given 𝒮⊂Lq([a,b]) a finite dimensional lineal subspace, let Qϵ∈𝒮 which minimizes Φ[a,b](Fϵ,Q), Q∈𝒮.

Theorem 2.1.

Assume that there are two functions W(ϵ):(0,∞)→(0,∞) and h∈Lq/(q-p)([a,b]) such that almost everywhere on [a,b]gϵ-fϵW(ϵ)≤h,limϵ→0gϵ-fϵW(ϵ)=w,
with |{w>0}|>0. If fϵ and gϵ converge to f almost everywhere on [a,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,[a,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 wϵ*=wϵ/W(ϵ), where wϵ=gϵ-fϵ. Let
JQ,ϵ(x):=∫fϵ(x)gϵ(x)|y-Q(x)|pdy,x∈[a,b],Q∈S.
Let Q∈𝒮. By integral mean value theorem, for each x∈[a,b], there exists
αϵ(x),ξϵ(x)∈[fϵ(x),gϵ(x)],
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 [a,b], and hence, αϵ and ξϵ are measurable functions on [a,b]. Consequently, αϵ, ξϵ∈Lq([a,b]), because |αϵ|,|ξϵ|≤g, a.e. on [a,b].

On the other hand,
‖f-Qϵ‖p,wϵ,[a,b]-‖αϵ-f‖p,wϵ,[a,b]≤‖αϵ-Qϵ‖p,wϵ,[a,b]=Φ[a,b](Fϵ,Qϵ)≤Φ[a,b](Fϵ,Q)=‖ξϵ-Q‖p,wϵ,[a,b].
In consequence, we get
‖f-Qϵ‖p,wϵ*,[a,b]≤‖αϵ-f‖p,wϵ*,[a,b]+‖ξϵ-Q‖p,wϵ*,[a,b].
As |f|≤g a.e. on [a,b], from (2.1) and the Hölder inequality, we have
|αϵ-f|pwϵ*≤(2g)ph∈L1([a,b]),|ξϵ-Q|pwϵ*≤(g+|Q|)ph∈L1([a,b]).
From (2.2) and the Lebesgue Dominated Convergence Theorem, it follows that
limϵ→0(‖αϵ-f‖p,wϵ*,[a,b]+‖ξϵ-Q‖p,wϵ*,[a,b])=‖f-Q‖p,w,[a,b].
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 wϵ* 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, wϵ*(x)≥k/2. Hence,
‖f-Qϵ‖p,A≤(k2)-1/p‖f-Qϵ‖p,wϵ*,[a,b],0<ϵ<ϵ0.
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,wϵ*,[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].
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,[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 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)
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(ϵ)=wϵ(x)W1(ϵ)W1(ϵ)W2(ϵ),x∈Ω
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(ϵ)=w2(x)w1(x).
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.
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,
fm(x)=x,gm={x+1m,ifmiseven,x+xm,ifmisodd.
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 m→∞.

3. The Band Shrinks Horizontally

Let x0∈[a,b] and Iϵ:=[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=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)=h(x1)+(x-x1)h[x1,x2]+⋯+(x-x1)⋯(x-xn)h[x1,…,xn+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 [a,b] containing to x1,…,xm+1, then the m-th divided difference can be expressed ash[x1,…,xm+1]=h(m)(ξ)m!,
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 q≥0, and let fi:[a,b]→ℝ, 1≤i≤3 be continuous functions with f1≤f2. If
h(x)=∫f1(x)f2(x)|y-f3(x)|qsgn(y-f3(x))dy,a≤x≤b,
then h is a continuous function.

Proof.

If q>0, the continuity of h follows from the continuity of fi, 1≤i≤3 and the uniform continuity of tqsgn(t) on compact sets. If q=0, we have
h(x)={f1(x)-f2(x),if f3(x)≥f2(x),f2(x)-f1(x),if f3(x)≤f1(x),f1(x)+f2(x)-2f3(x),if f1(x)≤f3(x)≤f2(x).
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
∫Iϵh(x)Q(x)dx=0,Q∈Πn,
where
h(x)=∫f(x)g(x)|y-Qϵ(x)|p-1sgn(y-Qϵ(x))dy.
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,
∫f(xi)g(xi)|y-Qϵ(xi)|p-1sgn(y-Qϵ(xi))dy=0,1≤i≤n+1.
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)=h(x1)+(x-x1)h(1)(ξ1)+⋯+(x-x1)⋯(x-xn)h(n)(ξn)n!,
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+1,Iϵp+1+‖g-Q‖p+1,Iϵp+1)1/(p+1),Q∈Πn.

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.
On the other hand, we have
(p+1)ΦIϵp(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∈Πn,
From (3.10) and (3.11), we get
‖f-Qϵ‖p+1,Iϵp+1+‖g-Qϵ‖p+1,Iϵp+1=‖f-Pϵ‖p+1,Iϵp+1+‖g-Pϵ‖p+1,Iϵp+1,
for all ϵ<ϵ1. So, Qϵ=Pϵ for all ϵ<ϵ1.

Remark 3.4.

If Φ[a,b](F,Q)=supx∈[a,b]supy∈F(x)|y-Q(x)|, then
Φ[a,b](F,Q)=supx∈[a,b]max{|f(x)-Q(x)|,|g(x)-Q(x)|}
implies that Qϵ 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.

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