On constrained uniform approximation

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ON CONSTRAINED UNIFORM APPROXIMATION

M. A. BOKHARI

Received 22 April 2001 and in revised form 8 October 2001

The problem of uniform approximants subject to Hermite interpolatory constraints is considered with an alternate approach. The uniqueness and the convergence aspects of this problem are also discussed. Our approach is based on the work of P. Kirchberger (1903) and a generalization of Weierstrass approximation theorem.

2000 Mathematics Subject Classiﬁcation: 41A05, 41A10, 41A29, 41A52.

1. Introduction. Letπm denote the set of all polynomials of degree less than or equal tom. Let[a,b]be a closed ﬁnite interval andC[a,b]the space of real continuous functions on[a,b]with uniform norm

h∞= max

x∈[a,b]h(x). (1.1)

Here we discuss uniform approximation of a prescribedf∈C[a,b]by the polyno-mials that are also Hermite interpolants to a set of given data at a ﬁnite number of preassigned points in the interval[a,b]. More precisely, we consider the following problem due to Loeb et al. [6].

Problem1.1. Suppose thatk,m, andn_i, wherei=1,2,...,k, are positive integers

withm≥(k_i=1ni)−1 and that{ui}ki=1is a subset of[a,b]satisfyinga≤u1< u2< ···< uk≤b. Then the problem is to ﬁnd a best uniform approximant to a given f∈C[a,b]from the class

Φm,β=φ∈πm:φ(j)ui=βij, 1≤i≤k,0≤j≤ni−1, (1.2)

where

Sβ:=βij: 1≤i≤k,0≤j≤ni−1 (1.3)

is a subset of the real numbers withβi0=f (ui).

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Hermite interpolatory conditions. In fact, they discussedProblem 1.1through the no-tions of then-dimensional extended Haar subspace of orderνofC[a,b](see [4]) and the generalized weight functions (see [7]). They also established a convergence result by giving a generalization of a theorem of de La Vallée Poussin [1, page 77].

In the present paper we continue the study ofProblem 1.1. Our approach for its solution is based on the work of Kirchberger [5] that deals with extreme values of the error function. Uniqueness and convergence problems are also addressed in our work taking into account an extension of Weierstrass approximation theorem [10].

2. Notations and reformulation ofProblem 1.1. For the sake of convenience, we setIk:= {1,2,...,k},Ni:= {0,1,...,ni−1},s:=i∈Ikni, and

W (x):= i∈Ik

x−uini_. _(2.1)

The notationH_s−1(x,Sβ), whereSβis given in (1.3), stands for the polynomial of degree less than or equal tos−1 that satisﬁes the following conditions:

H_s−(j)1

ui,Sβ=βij, ∀i∈Ik,∀j∈Ni. (2.2)

For everyf∈C[a,b], we deﬁne

fH,β(x):=f (x)−Hs−1x,Sβ. (2.3)

The error function corresponding tof ,g∈C[a,b]and the set of its extreme points in[a,b]will be denoted, respectively, as follows:

ef ,g(x):=f (x)−g(x),

critef ,g:=x∈[a,b]:ef ,g(x)=ef ,g_∞. (2.4)

Letπ∗

mdenote the (m−s+1)-dimensional subspace ofπmgenerated by the polyno-mialsxj_{W (x),}_j₌_0,1,2,...,m₋_{s, where}_W _{is given by (}_2.1_{). The following remark} gives an explicit representation of the elements ofΦm,β(see (1.2)).

Remark_2.1. _{A typical element}φ∗_{in the approximating class}Φ_m,β_{is of the form}

φ∗_(x)₌_Hs− 1

x,Sβ+q∗_(x), _(2.5)

where q∗ _∈_π∗

m. This shows thatφ∗∈Φm,β is a best approximant to f from the classΦm,βif and only ifq∗∈πm∗ is a best approximant tofH,βfrom the classπm∗. In particular, ifm=s−1 thenH_s−1(x,Sβ)will be the best approximation tof. For this obvious reason we assume thatm≥sin the rest of the paper.

In view of the above remark,Problem 1.1can be reformulated as follows.

Problem2.2. For a given functionf∈C[a,b], ﬁnd a best approximation tofH,β

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3. Characterization of best approximation. This section deals with a necessary and suﬃcient condition for a solution ofProblem 2.2. We note that everyq∈π∗

mcan be expressed as

q(x)=W (x)Rq(x), (3.1)

whereRq(x)is a polynomial of degree at mostm−s. For anf∈C[a,b]and aq∈π∗ m, we set

EfH,β,q(x):=

fH,β(x)

W (x) −Rq(x). (3.2) An alternate form of the characterization theorem [6, Theorem 3.1] that solves

Problem 1.1may be stated as follows.

Theorem_3.1. _Letf∈C[a,b]_{such that}f_H,β∉π_m∗_{. Then}q∗_{is a best uniform}

ap-proximant tofH,βfrom the classπ∗if and only if there existNpointsαi∈crit(efH,β,q∗)

satisfying the following conditions: (a) N=m−s+2;

(b) a≤α1< α2<···< αN≤b;

(c) sgn(Ef_H,β,q∗(αi))=(−1)i+1sgn(Ef_H,β,q∗(α1)), for alli=2,3,...,N.

Our method of proof is based on the following lemma which may be found in the standard texts of approximation theory, for example, [9, Lemma 7.1].

Lemma_3.2. _LetY _{be a linear subspace of}C[a,b]_{and let}h∈C[a,b]. Theng∗∈Y

is a best uniform approximant tof inY if and only if there does not exist anyg∈Y such that

h(x)−g∗_(x)_{g(x) >}₀ _(3.3)

for allx∈crit(ef ,g∗)(see (2.3)).

Remark3.3. If we setY=π_m∗andh=f_H,βin the above lemma, then the necessary

and suﬃcient condition forq∗_∈_π∗

mto be a best approximation tofH,βis that there does not exist anyp∈π∗

msuch that

Ef_H,β,q∗(x)Rp(x) >0 (3.4)

for allx∈crit(efH,β,q∗)whereRp(x)andEfH,β,q∗ are, respectively, given in (3.1) and

(3.2). To justify this, it is enough to note that

fH,β(x)−q∗_(x)_p(x)₌_Ef

H,β,q∗(x)Rp(x)W2(x). (3.5)

Remark3.4. Iff_H,β∉π_m∗, thenW (x)is a nonvanishing function on the compact

set crit(ef_H,β,p)regardless of the choice ofp∈πm∗. Consequently,fH,β is continuous as well as nowhere zero on crit(efH,β,p).

4. Proof ofTheorem 3.1. Ifq∗_{is not a best approximation to}_{fH,β, then by}_Remark

3.3, there existsp∈πm∗ such that

Ef_H,β,q∗(x)Rp(x) >0 (4.1)

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from (4.1) thatEf_H,β,q∗(x)cannot change sign more than (m−s) times asx ranges over crit(ef_H,β,q∗). This contradictsTheorem 3.1(c) asN=m−s+2.

Conversely, assume that there areNpointsαi∈crit(efH,β,q∗),i=1,2,...,N,

satis-fyingTheorem 3.1(b) and (c) butN≤m−s+1. For eachi=1,2,3,...,N−1, ﬁx a point wi∈(αi,αi+1)such that

uj∉ wi,αi+1

, j=1,2,...,k,

_w

i,αi+1,critefH,β,q∗

_{are disjoint,}

sgnEfH,β,q∗ _w

i=sgnEfH,β,q∗ _α

i+1.

(4.2)

The choice ofwi, as required above, directly follows fromRemark 3.4. Now we set

p(x):=sgnEfH,β,q∗

α1

W (x) N−1

i=1

wi−x. (4.3)

Thenp∈π∗

mwithRp(x)=sgn(EfH,β,q∗(α1))N−_i=11(wi−x). We claim that

Ef_H,β,q∗(α)R_p(α) >0 ∀α∈critef_H,β,q∗. (4.4)

This can be seen by restrictingαto each set(αi,αi+1]∩crit(efH,β,q∗)fori=0,1,...,N−

1, whereα0=a, and then usingTheorem 3.1(c) along with (4.2). Hence byRemark 3.3, we note thatq∗_{cannot be a best uniform approximant to}_fH,β_{from the class}_π∗

m. This completes the proof.

Remark _4.1. _{An immediate consequence of} _{Theorem 3.1} _{is that} Hs−₁(x,Sβ)+

q∗_(x)_{is a best uniform approximant to}_{f (x)}_{from the class}_Φ_m,β_.

5. Uniqueness. We retain the setting of the previous sections in order to establish the uniqueness of the solution ofProblem 2.2. More precisely, we prove the following theorem.

Theorem5.1. There is exactly one best uniform approximantp∗tofH,βfromπ_m∗.

Ifq∗_{is another best uniform approximant to}_f_H,β_from_π∗

m, then there existNpoints αi∈crit(ef_H,β,q∗),i=1,2,...,N, satisfyingTheorem 3.1(a), (b), and (c). Using the prop-erties of best approximant tofH,β, we observe that|fH(αi)−p∗(αi)| ≤ efH,β,p∗∞=

efH,β,q∗∞= |fH,β(αi)−q∗(αi)|for eachi=1,2,...,N, and consequently,

Ef_H,β,p∗αi≤Ef_H,β,q∗αi. (5.1)

We setD(x):=Rp∗(x)−Rq∗(x). ThenD∈πm−s and

Dαi=EfH,β,q∗ _α

i−EfH,β,p∗ _α

i, i=1,2,...,N. (5.2)

Note that ifD(αi)≠0 for anyi=1,2,...,N, then by (5.2), sgn(D(αi))=sgn(EfH,β,q∗(αi)).

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6. Convergence. In this section, we discuss the convergence of the sequence of best uniform approximants{q∗

k}∞k=s−1tofH,βwith the conditions thatfissuﬃciently diﬀerentiableand the setSβ(seeProblem 1.1) is replaced by

Sf=f(j)ui:j∈Ni, i∈Ik. (6.1)

In this case, we writefH,f andΦm,f instead offH,βandΦm,β(see (2.3)).

Theorem6.1. Assume thatf∈Cn∗[a,b]withn∗=(maxi∈I_kni)−1and that the

setSβ (see (1.3)) is replaced bySf. Ifq∗

m∈πm∗ is the best approximant tofH,f in the sense ofTheorem 3.1, then

lim m→∞q

∗

m−fH,f∞=0. (6.2)

Consequently, the sequence{q∗

m+Hs−1(·,Sf)}∞m=s−1will converge uniformly tof. The crux of the proof of this theorem is in an extension of a result based on the Weierstrass approximation theorem [10, page 160]. We state it in the next lemma without proof as it is a routine exercise.

Lemma_6.2. _{For any}f∈Cr[a,b], and for a givenε >_{0, there exists a polynomial}

psuch that

f(j)₋_p(j)

∞< ε (6.3)

for allj=0,1,2,...,r.

Proof ofTheorem_6.1. _{In the notations of (}_2.3_{) and (}_6.1_{), we can write}

fH,f(x):=f (x)−Hs−1

x,Sf. (6.4)

The polynomialHs−1(x,Sf)due to its interpolation properties may be expressed as

Hs−1

x,Sf= l∈Ik

n∈Nl

f(n)_ul_Ln,l(x), _(6.5)

whereLn,l(x)are the fundamental polynomials of degrees−1 satisfying the condi-tions

L(j)_n,lui=   

1 forl=i, n=j,

0 otherwise. (6.6)

For a givenε >0, we can ﬁx a polynomialpof degreer > ssuch that (seeLemma 6.2)

f(j)₋_p(j) ∞<

ε

2τ, j=0,1,...,n∗, (6.7)

whereτ=max{1,λ}withλ=max_x∈[a,b]_l∈I_k_n∈N_l|Ln,l(x)|. From (6.1), (6.5), and (6.7) it follows that

max x∈[a,b]Hs−1

_x,S

f−Hs−1x,Sp< ε

2. (6.8)

We setq(x):=p(x)−H_s−1(x,Sp). Thenq∈πrandq(j)(ui)=0 forj∈Niandi∈Ik. Hence,W as deﬁned in (2.1) is a factor of the polynomialq. This shows thatq∈π∗

r. Using (6.4), (6.5), and (6.8) it can be seen that

_fH,f₋_q

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Now consider the best uniform approximantq∗

r tofHfromπr∗(seeTheorem 3.1) and note that

ε >fH,f−q∞≥fH,f−q∗r∞≥fH,f−q∗m∞ (6.10)

for allm≥r. The last inequality follows from the relationπ∗

r ⊆πm∗. This proves the desired result.

Acknowledgment. The author acknowledges with gratitude the research

facili-ties available at King Fahd University of Petroleum and Minerals, Saudi Arabia, and the University of Alberta, Canada, during the preparation of this paper.

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M. A. Bokhari : Department of Mathematical Sciences, King Fahd University of