Published online: June 22, 2017
AN EXTENSION OF BEST L2 LOCAL APPROXIMATION
H ´ECTOR H. CUENYA, DAVID E. FERREYRA, AND CLAUDIA V. RIDOLFI
Abstract. We introduce two classes of functions, one containing the class ofL2 differentiable functions, and another containing the class ofL2 lateral
differentiable functions. For functions in these new classes we prove existence of best local approximation at several points. Moreover, we get results about the asymptotic behavior of the derivatives of the net of best approximations, which are unknown even forL2 differentiable functions.
1. Introduction
Let x1, x2, . . . , xk be k points in R and let a > 0 be such that the intervals Ia,i:= [xi−a, xi+a], 1≤i≤k, are pairwise disjoint. We denote byLthe space of
equivalence classes of Lebesgue measurable real functions defined onIa := k
S
i=1 Ia,i.
For each Lebesgue measurable setA⊂Ia, with|A|>0, we consider the semi-norms
onL,
kfkA:=
|A|−1
Z
A
|f(x)|2dx 1/2
,
where|A|denotes the measure of the setA.
If 0 < ≤ a, we use the notations I−,i = [xi −, xi], I+,i = [xi, xi +],
I,i = [xi −, xi +], and we write kfk±,i = kfkI±,i, kfk,i = kfkI,i. So,
kfk2
:=kfk2I =k
−1Pk
i=1
kfk2
,i.
In all this work, s is an arbitrary non negative integer. We denote by Πs the
linear space of polynomials of degree at mosts.
Henceforward, we fixn∈N∪ {0}andq∈Nsuch that n+ 1 =kq.
Iff ∈L2(I), it is well known that there exists a unique bestk·k-approximation
off from Πn, sayP, i.e., P∈Πn satisfies
kf−Pk≤ kf−Pk, P ∈Πn.
2010Mathematics Subject Classification. Primary 41A50; Secondary 41A10.
Key words and phrases. Best local approximation,L2 differentiability, algebraic polynomials.
Moreover, it is characterized by
Z
I
(f−P)(x)P(x)dx= 0, P ∈Πn. (1.1)
If lim
→0P exists, sayP0, it is called the best local approximation off on {xi : 1≤ i≤k}, fromΠn.
We recall that a functionf ∈L2(Ia,i) isL2 left (right) differentiable of order s
atxiif there exists Li(Ri)∈Πs such that
kf−Lik−,i=o(s) (kf −Rik+,i=o(s)). (1.2)
The class of these functions is denoted by t2
−s(xi) (t2+s(xi)). IfLi =Ri, we say
that f is L2 differentiable of order sat x
i and we writef ∈t2s(xi). This concept
was introduced by Calder´on and Zygmund in [1]. It is well known that there exists at most one polynomial satisfying (1.2) (see [6]).
The problem of best local approximation was formally introduced and studied in a paper by Chui, Shisha and Smith [2]. However, the initiation of this could be dated back to results of J. L. Walsh [7], who proved that the Taylor polynomial of an analytic functionf over a domain is the limit of the net of best polynomial approximations of a given degree, by shrinking the domain to a single point. In [6] this problem was considered whenf isL2differentiable of orderq−1 atx
i, 1≤i≤
k. In [4] the authors proved the existence of the best local approximation under weaker conditions, more precisely they assumed existence ofL2 lateral derivatives
of orderq−1 andL2 differentiability of orderq−2 at each point.
On the other hand, for k = 1, the authors in [3] considered a function f dif-ferentiable at 0 in the ordinary sense of order s, for somes≤n, and they proved that lim
→0(f −P)
(j)(0) = 0, 0≤j ≤s. In particular, if P is a cluster point of the
net{P}, as→0, thenP interpolates the derivatives off up to ordersat 0. In
[5] the authors obtained this property of interpolation for functions that satisify a more general condition thanL2differentiability, calledC2 condition.
The main purpose of this paper is to extend the results established in [3] and [4], assuming theC2condition andC2 lateral condition when we consider approx-imation onkpoints inRfrom Πn, for the casen+ 1 multiple ofk.
Definition 1.1. A function f ∈ L2(I
a,i) satisfies the C2 condition of order s at
xi, if there existsQi∈Πs such that
Z
I,i
(f−Qi)(x)(x−xi)jdx=o(s+j+1), 0≤j≤s, as→0. (1.3)
Analogously,f satisfies the left (right)C2 condition of ordersatx
i, if there exists
Li(Ri)∈Πssatisfying (1.3) withI−,i (I+,i) instead ofI,i.
We say thatf ∈c2
s(xi) if it satisfies (1.3). We denote withc2−s(xi) (c2+s(xi)) the
class of functions which satisfy the left (right)C2condition of ordersatxi. We also
denotec2
±s(xi) :=c2+s(xi)∩c2−s(xi). It is easy to see thatt2s(xi)⊆t2−s(xi)(t2+s(xi));
howeverc2
In [5] it was proved that if f ∈c2
s(xi), there exists a unique polynomialQi ∈Πs
satisfying (1.3). Moreover, t2
s(xi) ( c2s(xi). The polynomial in Πs that satisfies
kf−Qik,i=o(s) also satisfies (1.3). Analogouslyt2−s(xi)(c2−s(xi) andt2+s(xi)( c2
+s(xi).
In Section 2, given a function in c2
s(xi) (orc2±s(xi)),1 ≤i≤k, under suitable
conditions, we study the asymptotic behavior of the derivatives ofP, as → 0,
and we establish a relation between the derivatives of P up to order s with its
derivatives of greater order (see Theorem 2.3 and Theorem 2.9). This type of result is unknown, even forL2differentiable functions. We also prove that ifP is a
cluster point of the net{P}, thenP interpolates the derivatives ofQi (or Li+2Ri)
up to ordersat xi. Finally, we get existence of the best local approximation onk
points, for two new classes of functions.
2. The main results
Henceforward we will use the following notation. For Q ∈ Πn, we denote by
Ti,s(Q) its Taylor polynomial at xi of degree s. We write Ti,−1(Q) = 0 and we
recall thatq∈Nsatisfiesn+ 1 =kq.
To establish the results we need two auxiliary lemmas.
Lemma 2.1. Let l∈Z. If one of the following cases holds
a) q= 1,l=−1, andf ∈L2(I
a),
b) k= 1,q≥2,0≤l≤q−2, andf ∈L2(Ia),
c) k >1,q≥2,0≤l≤q−2, andf ∈t2l(xi),1≤i≤k,
then
Z
I,i
(f−P)(x)(x−xi)udx=o(2l+3), 0≤u≤l+ 1, 1≤i≤k.
Proof. We begin proving c). For each 0 ≤u≤l+ 1 and 1 ≤i≤k, we consider the polynomial Ru,i ∈ Π(l+2)k−1 such that R
(j)
u,i(xz) = δ(u,i),(j,z), 0 ≤ j ≤l+ 1,
1≤z≤k, whereδis the Kronecker delta function. From (1.1) we have
Z
I,i
(f−P)(x)Ru,i(x)dx=− k
X
z=1,z6=i
Z
I,z
(f −P)(x)Ru,i(x)dx
=−
k
X
z=1,z6=i
Z
I,z
(f−P)(x)
(l+2)k−1 X
r=0 R(u,ir)
r! (xz)(x−xz)
rdx
=−
k
X
z=1,z6=i
Z
I,z
(f−P)(x)
(l+2)k−1 X
r=l+2 R(u,ir)
r! (xz)(x−xz)
rdx
=−
k
X
z=1,z6=i
(l+2)k−1 X
r=l+2 R(u,ir)
r! (xz)
Z
I,z
(f −P)(x)(x−xz)rdx.
(2.1)
Since f ∈ t2l(xi) there exists Qi ∈ Πl such that kf −Qik,i = o(l), 1 ≤ i ≤ k.
LetS∈Π(l+2)k−1 defined byS(j)(x
i) =Q
(j)
kf−Sk,i≤ kf−Qik,i+kQi−Sk,i=o(l), sokf −Sk2 =k−1 k
P
i=1
kf−Sk2
,i=
o(2l), i.e.,kf −Sk
=o(l). Using H¨older’s inequality, we obtain
Z
I,z
(f −P)(x)(x−xz)rdx
≤
Z
I
|(f−P)(x)(x−xz)r|dx≤r+1kf −Pk
≤r+1kf −Sk=o(2l+3), l+ 2≤r≤(l+ 2)k−1, 1≤z≤k.
(2.2)
From (2.1) and (2.2) we get
Z
I,i
(f−P)(x)Ru,i(x)dx=o(2l+3). (2.3)
On the other hand,
Z
I,i
(f−P)(x)Ru,i(x)dx
=
Z
I,i
(f−P)(x)
(l+2)k−1 X
r=0 R(u,ir)
r! (xi)(x−xi)
rdx
=
Z
I,i
1
u!(f−P)(x)(x−xi)
udx
+
Z
I,i
(f −P)(x)
(l+2)k−1 X
r=l+2 R(u,ir)
r! (xi)(x−xi)
rdx
=
Z
I,i
1
u!(f−P)(x)(x−xi)
udx+o(2l+3),
(2.4)
where the last equality follows from (2.2). Finally, the lemma is a consequence of (2.3) and (2.4).
a) Ifk= 1 it is a direct consequence of (1.1). Supposek >1 and considerS= 0. Then r+1kfk
≤(a) 1
2r+1kfka =o(), 1≤r≤k−1, so (2.2) is true. Now, the proof follows analogously to that of c).
b) This is a direct consequence of (1.1).
Lemma 2.2. Let rw ∈ N∪ {0}, 1 ≤ w ≤m, be such that r1 < r2 <· · · < rm.
Given 1≤i≤k, let{U} be a net of polynomials in the linear space generated by
{(x−xi)rw: 1≤w≤m}. If
Z
I,i
U(x)(x−xi)rwdx=o(rm+rw+1), 1≤w≤m, (2.5)
Proof. We write U(x) = m
P
s=1
arw()(x−xi)
rw. Multiplying both sides of (2.5) by
arw(), and summing overwfrom 1 tom, we get Z
I,i
U(x)2dx=o(rm+1) m
X
w=1 arw()
rw =o(rm+1)
m
X
w=1
|arw()|
rw. (2.6)
Let ∆ be the space of polynomials generated by {xr1, . . . , xrm}. We consider the
normρon ∆ defined by
ρ(S) =
m
X
w=1
|drw|, S(x) =
m
X
w=1 drwx
rw,
and k · k[−1,1] the usual norm in L2([−1,1]). Let S[](x) := S(xi+x). By the
equivalence of the above norms on ∆, there existsK >0 such that
m
X
w=1
|arw()|
rw=ρ(U[]
)≤KkU[]k[−1,1]. (2.7)
Therefore by (2.6) we have thatkU[]k2[−1,1]=o(rm)kU
[]
k[−1,1]. So,kU []
k[−1,1] = o(rm). In consequence, from (2.7) we obtaina
rw() =o(
rm−rw), 1≤w≤m, i.e.,
U→0 as→0.
Ifq≥2 andl∈N∪ {0},l≤q−2, we consider
b,i,l=
(
1 + max{|P(v)(xi)|:l+ 2≤v≤n} ifl < q−2,
1 ifl=q−2. (2.8)
Forq= 1 andl=−1, we putb,i,l= 1.
The next theorem is one of our main results.
Theorem 2.3. Let l∈Z. If one of the following cases holds:
a) q= 1,l=−1, andf ∈c02(xi),1≤i≤k,
b) k= 1,q≥2,0≤l≤q−2, andf ∈c2l+1(x1),
c) q≥2,0≤l≤q−2,k >1, andf ∈cl2+1(xi)∩t2l(xi),1≤i≤k,
then
Ti,l+1(P)−Qi =o(1)b,i,l, 1≤i≤k, (2.9)
uniformly on compact sets, whereQi ∈Πl+1 was introduced in Definition 1.1. In
particular, ifP is a cluster point of the net {P}, as →0, then Ti,l+1(P) =Qi,
1≤i≤k.
Proof. Under the hypotheses a), b) or c), we havef ∈c2
l+1(xi), 1≤i≤k. So
Z
I,i
(f−Qi)(x)(x−xi)jdx=o(l+j+2), 0≤j≤l+ 1. (2.10)
By Lemma 2.1, we get
Z
I,i
From (2.10) and (2.11) we obtain
Z
I,i
(P−Qi)(x)(x−xi)jdx=o(l+j+2), 0≤j≤l+ 1. (2.12)
It immediately follows from the definition ofb,i,l that the set
Al,i,δ:={b−,i,l1 P
(v)
(xi) :l+ 2≤v≤n, 0< < δ},
is bounded for 0≤l < q−2 and 1≤i≤k. Ifk= 1, Aq−2,i,δis empty and therefore
bounded by convention. We will see next that Aq−2,i,δ is also bounded for k >1
andδsufficiently small. Indeed, assume that the net{P}is not uniformly bounded
as→0 and letH∈Πnbe the unique polynomial satisfyingH(v)(xu) =Q
(v)
u (xu),
0 ≤ v ≤ q−1, 1 ≤ u ≤ k. Then there exists a sequence m → 0 such that
kPm−Hk∞,Ia→ ∞. From (2.12) we get Z
I,i
(P−H)(x)(x−xi)jdx=o(q+j), 0≤j≤q−1. (2.13)
LetDm :=
Pm−H
kPm−Hk∞,Ia. Since{Dm} is uniformly bounded, (2.13) implies Z
Im,i
Ti,q−1(Dm)(x)(x−xi)
jdx=o(
mj+q)− n
X
v=q
D(vm)(xi) v!
Z
Im,i
(x−xi)j+vdx
=o(mj+q) +O(mj+q+1) =o(mj+q), 0≤j≤q−1.
(2.14)
By Lemma 2.2 we have that Ti,q−1(Dm)→ 0, 1 ≤i ≤k, asm → 0. Let ρbe
the norm on Πn defined byρ(P) := max
1≤i≤k 0≤maxv≤q−1|P (v)(x
i)|. Thenρ(Dm)→0, a
contradiction sincekDmk∞,Ia = 1. So, the net{P}is uniformly bounded.
From (2.12) we obtain
Z
I,i Ti,l+1
P
−Qi
b,i,l
(x)(x−xi)jdx=o(l+j+2)− n
X
v=l+2
P(v)(xi)
v!b,i,l
Z
I,i
(x−xi)j+vdx,
(2.15) where the sum is omitted if k = 1 and l = q−2. Since Al,i,δ is bounded for
0≤l≤q−2 andδsufficiently small, from (2.15) we get
Z
I,i Ti,l+1
P
−Qi
b,i,l
(x)(x−xi)jdx=o(l+j+2), 0≤j ≤l+ 1. (2.16)
Now, Lemma 2.2 withrw=w−1, 1≤w≤l+ 2, implies (2.9).
Finally, ifP = lim
m→0
Pm thenbm,i,l are bounded. So, from (2.9) it follows that
Ti,l+1(P) =Qi, 1≤i≤k.
Remark 2.5. LetP+ (P−) be the best approximation tof from Πn on k
S
i=1 I+,i
(
k
S
i=1
I−,i), and letb+,i,l (b−,i,l) be defined as in (2.8) with P+ (P−) instead of
P. We observe that Theorem 2.3 remains valid if we considerP+ (P−), and the
condition overf is satisfied on the right (left) at each point.
The following corollary is an immediate consequence of Theorem 2.3 with l =
q−2.
Corollary 2.6. If f ∈ c2
q−1(xi)∩t2q−2(xi), 1 ≤i ≤k, then there exists the best
local approximation off on{xi: 1≤i≤k}, fromΠn. It is the unique polynomial
H∈Πn satisfyingH(j)(x
i) =Q
(j)
i (xi),0≤j≤q−1,1≤i≤k, whereQi∈Πq−1
satisfies Definition 1.1. In the caseq= 1ork= 1we only assumef ∈c2
q−1(xi),1≤
i≤k.
Remark 2.7. We observe that c2s(xi) is not, in general, a subset of c2±s(xi). In
fact, next we give an example of a functionf ∈c2
0(0) such thatf /∈c2±0(0).
Example 2.8. Let{hm}and{m}be two sequences of nonnegative real numbers,
such that hm ↓ 0, m ↓ 0, m+1 < hm < m, hmm → ∞, and hmm+1 → ∞. For
example, we can takem= 2−2 2m
andhm= 2−2 2m+1
. We definef : (−1, 1)→R
by
f(x) =
(
1 ifx∈ ∪∞
m=1[hm, m),
0 ifx∈ ∪∞
m=1[m+1, hm),
f(0) = 0 andf(x) =−f(−x),x∈(−1,0). We have
kf −P+mk 2
+m≤ kf−1k 2 +m =
Z hm
0
(f(x)−1)2dx
m
≤hm
m
→0, (2.17)
kf−P+hmk 2
+hm ≤ kfk 2 +hm =
m+1 hm
Z m+1
0
f(x)2 dx
m+1
≤ m+1
hm
→0. (2.18)
From (2.17) we get kP+m −1k+m ≤ kP+m −fk+m+kf−1k+m → 0. Now,
P´olya’s inequality implies thatP+m→1. Analogously, from (2.18) we obtain that P+hm → 0. On the other hand, we know that iff ∈ c
2
+0(0) then lim
→0P+ exists
(see Remark 2.5), so f /∈ c2+0(0). Analogously,f /∈ c2−0(0). As f is odd, clearly
f ∈c2 0(0).
Let Pi(h) (Ii(h)) be the even (odd) part of the function hwith respect to xi,
i.e.,
Pi(h)(x) :=
h(x) +h(2xi−x)
2
Ii(h)(x) :=
h(x)−h(2xi−x)
2
.
It is easy to see that, h = Pi(h) +Ii(h), Pi(h)(xi +x) = Pi(h)(xi −x), and
Theorem 2.9. If one of the following cases holds: a) q= 1,l=−1, andf ∈c2
±0(xi),1≤i≤k,
b) k= 1,q≥2,0≤l≤q−2,f ∈c2
±(l+1)(x1)andT1,l(L1) =T1,l(R1),
c) q≥2,0≤l≤q−2, andf ∈c2
±(l+1)(xi)∩t 2
l(xi),1≤i≤k,
then
Ti,l+1(P)−
Li+Ri
2 =o(1)b,i,l, 1≤i≤k, (2.19) uniformly on compact sets, where Li, Ri∈Πl+1 were introduced in Definition 1.1.
In particular, if P is a cluster point of the net {P}, as →0, thenTi,l+1(P) =
Li+Ri
2 ,1≤i≤k.
Proof. Sincef ∈c2±(l+1)(xi), 1≤i≤k, we have
i) R
I+,i(f −Ri)(x)(x−xi)
jdx=o(l+j+2), 0≤j≤l+ 1,
ii) RI
−,i(f−Li)(x)(x−xi)
jdx=o(l+j+2), 0≤j≤l+ 1.
By means of a change of variable from i) and ii) we get
iii) R
I−,i(f−Ri)(2xi−x)(xi−x)
jdx=o(l+j+2), 0≤j ≤l+ 1,
iv) R
I+,i(f −Li)(2xi−x)(xi−x)
jdx=o(l+j+2), 0≤j≤l+ 1.
From i) it follows that
Z
I+,i
(f−Ti,l(Ri))(x)(x−xi)jdx=o(l+j+2) +
Z
I+,i
R(il+1)(xi)
(l+ 1)! (x−xi)
j+l+1dx
=o(l+j+2) +O(l+j+2) =o(l+j+1), 0≤j≤l.
Thus, f ∈c2
+l(xi). Analogously, from ii) it follows thatf ∈c2−l(xi). In addition,
as we have observed in the Introduction, t2
l(xi)⊂t2−l(xi)⊂c2−l(xi) andt2l(xi)⊂
t2
+l(xi)⊂c2+s(xi). If c) holds, by uniqueness of the polynomial in Πlthat satisfies
(1.3), we have thatTi,l(Ri) =Ti,l(Li), 1≤i≤k. Under the hypothesis a) or b) we
also getT1,l(R1) =T1,l(L1). We writeSi=Ti,l(Ri),Ri(x) =Si(x) +c1,i(x−xi)l+1
andLi(x) =Si(x) +c2,i(x−xi)l+1.
From i) and iv) we obtain
Z
I+,i
[f(x) +f(2xi−x)−(Si(x) +c1,i(x−xi)l+1
+Si(2xi−x) +c2,i(xi−x)l+1)](x−xi)jdx
=o(l+j+2), 0≤j≤l+ 1. (2.20) In a similar way, from ii) and iii) we get
Z
I−,i
[f(x) +f(2xi−x)−(Si(x) +c2,i(x−xi)l+1
+Si(2xi−x) +c1,i(xi−x)l+1)](x−xi)jdx
Now, we assumel odd. From (2.20) and (2.21) we get
Z
I,i
[f(x)+f(2xi−x)−(Ri(x)+Li(2xi−x))](x−xi)jdx=o(l+j+2), 0≤j≤l+1.
(2.22) By (2.22) we obtain
Z
I,i
Pi
f −Ri+Li 2
(x)(x−xi)jdx=o(l+j+2), 0≤j≤l+ 1. (2.23)
Sincef ∈t2l(xi), Lemma 2.1 implies
Z
I,i
Pi(f−P) (x)(x−xi)jdx=o(l+j+2), j = 0,2, . . . , l+ 1. (2.24)
As we have observed in the proof of Theorem 2.3,Al,i,δis bounded for 0≤l≤q−2
andδsufficiently small. In consequence, from (2.24) we have
Z
I,i
1
b,i,l
Pi(f−Ti,l+1(P))(x)(x−xi)jdx
=
n
X
v=l+2
P(v)(xi)
v!b,i,l
Z
I,i
Pi((x−xi)v) (x−xi)jdx+o(l+j+2)
=o(l+j+2), j= 0,2, . . . , l+ 1.
(2.25)
By (2.23) and (2.25), we get
Z
I,i
1
b,i,l
Pi
Ti,l+1(P)−
Ri+Li
2
(x)(x−xi)jdx
=o(l+j+2), j= 0,2, . . . , l+ 1. (2.26) Using Lemma 2.2 withrw= 2(w−1), 1≤w≤ l+12 + 1, we get
Pi
Ti,l+1(P)−
Ri+Li
2
=o(1)b,i,l, 1≤i≤k. (2.27)
Now, we assumel even. Adding to both sides of (2.21) the expression
−2
Z
I−,i
(c1,i(x−xi)l+1+c2,i(xi−x)l+1)(x−xi)jdx, 0≤j≤l+ 1,
we obtain
Z
I−,i
[(f(x) +f(2xi−x))−(Si(x) +c1,i(x−xi)l+1+Si(2xi−x)
+c2,i(xi−x)l+1)](x−xi)jdx=O(l+j+2), 0≤j≤l+ 1. (2.28)
As the integrand in (2.20) and (2.28) is the same, we have
Z
I,i
[(f(x) +f(2xi−x))−(Ri(x) +Li(2xi−x))](x−xi)jdx
It is easy to see thatR
I,i(Ri(x) +Li(2xi−x))(x−xi)
jdx=R
I,i(Li(x) +Ri(2xi− x))(x−xi)jdx, forj even. In consequence, from (2.29) we obtain
Z
I,i
Pi
f −Ri+Li 2
(x)(x−xi)jdx
=
Z
I,i
Pi(f)−
Ri(x) +Li(x) +Ri(2xi−x) +Li(2xi−x)
4
(x−xi)jdx
=o(l+j+1), j = 0,2, . . . , l.
(2.30)
Now, we proceed as in (2.25). Asf ∈t2
l(xi) andAl,i,δ, 0≤l≤q−2, is bounded
forδsufficiently small, Lemma 2.1 implies
Z
I,i
1
b,i,l
Pi(f −Ti,l+1(P)) (x)(x−xi)jdx=o(l+j+2), j= 0,2, . . . , l. (2.31)
By (2.30) and (2.31), we obtain
Z
I,i
1
b,i,l
Pi
Ti,l+1(P)−
Ri+Li
2
(x)(x−xi)jdx=o(l+j+1), j= 0,2, . . . , l.
(2.32) Since l is even, we have that b−,i,l1Pi Ti,l+1(P)−Ri+2Li
∈ Πl. In consequence, using Lemma 2.2 withrw= 2(w−1), 1≤w≤2l + 1, we get (2.27).
In a similar way to the previous proof, consideringj odd, we can prove that
Ii
Ti,l+1(P)−
Ri+Li
2
=o(1)b,i,l, 1≤i≤k, (2.33)
for arbitraryl. From (2.27) and (2.33) we obtain (2.19).
Finally, letP be a cluster point of the net{P}, as→0. Then there exists a
sequencem →0 such that P = lim m→0
Pm. Thus, bm,i,l is bounded. From (2.19)
it follows thatTi,l+1(P) = Ri+2Li, 1≤i≤k.
The following corollary is an immediate consequence of Theorem 2.9 with l =
q−2.
Corollary 2.10. Iff ∈c2±(q−1)(xi)∩t2q−2(xi),1≤i≤k, then there exists the best
local approximation off on{xi: 1≤i≤k}, fromΠn. It is the unique polynomial
H ∈ Πn satisfying H(j)(x
i) = (Li+2Ri)(j)(xi), 0 ≤ j ≤ q−1, 1 ≤i ≤k, where
Li, Ri ∈Πq−1 satisfy Definition 1.1. In the case q= 1 ork = 1 we only assume
f ∈c2
±(q−1)(xi),1≤i≤k.
This result was proved in [4, Theorem 2.9], forf ∈t2−(q−1)(xi)∩t2+(q−1)(xi)∩
t2q−2(xi), 1≤i≤k. We give an example to show that Corollary 2.10 extends that
Theorem 2.9 from [4]. More precisely, if q = 2, k = 1, andx1 = 0, we define a
functionf ∈c2
Example 2.11. Let m = m1, m ∈ N, hm = m− m16. It is easy to see that
m+1< hm< m,m≥2. We consider the functionf : [−21,12]→Rdefined by f(x) =
(
m ifx∈[hm, m]
0 ifx /∈ ∪m≥2[hm, m],
f(0) = 0 and f(x) = f(−x) ifx ∈[−1
2,0). Let 0 < ≤ 1
2. Suppose that there
existsm∈Nsuch that∈[hm, m]; then
0≤−3 Z
0
f(x)dx≤h−m3 Z m
0
f(x)dx=h−m3
∞
X
j=m
Z j
hj
f(x)dx
=h−m3
∞
X
j=m
Z j
hj
j dx=h−m3
∞
X
j=m
j(j−hj) =h−m3
∞
X
j=m
j−5 1
m.
(2.34)
If /∈ ∪j≥2[hj, j], letm∈Nbe such that 0< hm< m< < hm−1. Then
0≤−3 Z
0
f(x)dx≤−m3 Z m
0
f(x)dx≤h−m3 Z m
0
f(x)dx 1
m. (2.35)
Since 0≤R
0f(x)x dx≤ R
0f(x)dx, andf is even, from (2.34) and (2.35) we have
that f ∈ c2
±1(0). On the other hand, f is not L2 right differentiable of order 1
at 0. In fact, for S ∈ Π1 there exists a m0 ∈
N such that |S(x)| ≤ f(4x), for all x ∈ [hj, j], j ≥ m0. So, |f(x)−S(x)|2 ≥ f(x)(f(x)−2|S(x)|) ≥ f(x)
2
2 for all x∈[hj, j],j≥m0. Therefore, form≥m0 we have
−m3 Z m
0
|(f−S)(x)|2dx≥−3
m
∞
X
j=m
Z j
hj
|(f−S)(x)|2dx
≥1 2 −3 m ∞ X
j=m
Z j
hj
|f(x)|2dx=1
2 −3
m
∞
X
j=m
j2(j−hj) =
1 2 −3 m ∞ X
j=m
j−4 1 2.
Finally, we will prove that f is L2 differentiable of order 0 at 0. If either ∈
[hm, m] or∈[m, hm−1], we obtain
0≤ 1
Z
0
f(x)2dx≤h−m1 Z m
0
f(x)2dx=h−m1
∞
X
j=m
Z j
hj
f(x)2dx
=h−m1
∞
X
j=m
j2(j−hj) =h−m1
∞
X
j=m
j−4 1
m2.
References
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H. H. Cuenya
Dpto. de Matem´atica, FCEFQyN, Universidad Nacional de R´ıo Cuarto, (5800) R´ıo Cuarto, Argentina
D. E. FerreyraB
Dpto. de Matem´atica, FCEFQyN, Universidad Nacional de R´ıo Cuarto, (5800) R´ıo Cuarto, Argentina
C. V. Ridolfi
Instituto de Matem´atica Aplicada de San Luis, CONICET, Universidad Nacional de San Luis, (5700) San Luis, Argentina
