An extension of best $L^2$ local approximation

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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:= [xia, xi+a], 1≤ik, 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 setAIa, 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 kf,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.

IffL2(I), it is well known that there exists a unique bestk·k-approximation

off from Πn, sayP, i.e., P∈Πn satisfies

kfPk≤ kfPk, P ∈Πn.

2010Mathematics Subject Classification. Primary 41A50; Secondary 41A10.

Key words and phrases. Best local approximation,L2 differentiability, algebraic polynomials.

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Moreover, it is characterized by

Z

I

(fP)(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≤ ik}, fromΠn.

We recall that a functionfL2(Ia,i) isL2 left (right) differentiable of order s

atxiif there exists Li(Ri)∈Πs such that

kfLik−,i=o(s) (kfRik+,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 writeft2s(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 somesn, and they proved that lim

→0(fP)

(j)(0) = 0, 0j 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 fL2(I

a,i) satisfies the C2 condition of order s at

xi, if there existsQi∈Πs such that

Z

I,i

(fQi)(x)(xxi)jdx=o(s+j+1), 0≤js, 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 thatfc2

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

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In [5] it was proved that if fc2

s(xi), there exists a unique polynomialQi ∈Πs

satisfying (1.3). Moreover, t2

s(xi) ( c2s(xi). The polynomial in Πs that satisfies

kfQik,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) (orcs(xi)),1 ≤ik, 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 thatqNsatisfiesn+ 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, andfL2(I

a),

b) k= 1,q≥2,0≤lq−2, andfL2(Ia),

c) k >1,q≥2,0≤lq−2, andft2l(xi),1≤ik,

then

Z

I,i

(fP)(x)(xxi)udx=o(2l+3), 0≤ul+ 1, 1≤ik.

Proof. We begin proving c). For each 0 ≤ul+ 1 and 1 ≤ik, we consider the polynomial Ru,i ∈ Π(l+2)k−1 such that R

(j)

u,i(xz) = δ(u,i),(j,z), 0 ≤ jl+ 1,

1≤zk, whereδis the Kronecker delta function. From (1.1) we have

Z

I,i

(fP)(x)Ru,i(x)dx=− k

X

z=1,z6=i

Z

I,z

(fP)(x)Ru,i(x)dx

=−

k

X

z=1,z6=i

Z

I,z

(fP)(x)

(l+2)k−1 X

r=0 R(u,ir)

r! (xz)(xxz)

rdx

=−

k

X

z=1,z6=i

Z

I,z

(fP)(x)

(l+2)k−1 X

r=l+2 R(u,ir)

r! (xz)(xxz)

rdx

=−

k

X

z=1,z6=i

(l+2)k−1 X

r=l+2 R(u,ir)

r! (xz)

Z

I,z

(fP)(x)(xxz)rdx.

(2.1)

Since ft2l(xi) there exists Qi ∈ Πl such that kfQik,i = o(l), 1 ≤ ik.

LetS∈Π(l+2)k−1 defined byS(j)(x

i) =Q

(j)

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kfSk,i≤ kfQik,i+kQiSk,i=o(l), sokfSk2 =k−1 k

P

i=1

kfSk2

,i=

o(2l), i.e.,kf Sk

=o(l). Using H¨older’s inequality, we obtain

Z

I,z

(fP)(x)(xxz)rdx

Z

I

|(fP)(x)(xxz)r|dxr+1kfPk

r+1kfSk=o(2l+3), l+ 2≤r≤(l+ 2)k−1, 1≤zk.

(2.2)

From (2.1) and (2.2) we get

Z

I,i

(fP)(x)Ru,i(x)dx=o(2l+3). (2.3)

On the other hand,

Z

I,i

(fP)(x)Ru,i(x)dx

=

Z

I,i

(fP)(x)

(l+2)k−1 X

r=0 R(u,ir)

r! (xi)(xxi)

rdx

=

Z

I,i

1

u!(fP)(x)(xxi)

udx

+

Z

I,i

(fP)(x)

(l+2)k−1 X

r=l+2 R(u,ir)

r! (xi)(xxi)

rdx

=

Z

I,i

1

u!(fP)(x)(xxi)

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≤rk−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 ≤ wm, be such that r1 < r2 <· · · < rm.

Given 1≤ik, let{U} be a net of polynomials in the linear space generated by

{(xxi)rw: 1≤wm}. If

Z

I,i

U(x)(xxi)rwdx=o(rm+rw+1), 1≤wm, (2.5)

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Proof. We write U(x) = m

P

s=1

arw()(xxi)

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(

rmrw), 1wm, i.e.,

U→0 as→0.

Ifq≥2 andlN∪ {0},lq−2, we consider

b,i,l=

(

1 + max{|P(v)(xi)|:l+ 2≤vn} 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, andfc02(xi),1≤ik,

b) k= 1,q≥2,0≤lq−2, andfc2l+1(x1),

c) q≥2,0≤lq−2,k >1, andfcl2+1(xi)∩t2l(xi),1≤ik,

then

Ti,l+1(P)−Qi =o(1)b,i,l, 1≤ik, (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≤ik.

Proof. Under the hypotheses a), b) or c), we havefc2

l+1(xi), 1≤ik. So

Z

I,i

(fQi)(x)(xxi)jdx=o(l+j+2), 0≤jl+ 1. (2.10)

By Lemma 2.1, we get

Z

I,i

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From (2.10) and (2.11) we obtain

Z

I,i

(PQi)(x)(xxi)jdx=o(l+j+2), 0≤jl+ 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≤vn, 0< < δ},

is bounded for 0≤l < q−2 and 1≤ik. 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 ≤ vq−1, 1 ≤ uk. Then there exists a sequence m → 0 such that

kPmHk∞,Ia→ ∞. From (2.12) we get Z

I,i

(PH)(x)(xxi)jdx=o(q+j), 0≤jq−1. (2.13)

LetDm :=

PmH

kPmHk∞,Ia. Since{Dm} is uniformly bounded, (2.13) implies Z

Im,i

Ti,q−1(Dm)(x)(xxi)

jdx=o(

mj+q)− n

X

v=q

D(vm)(xi) v!

Z

Im,i

(xxi)j+vdx

=o(mj+q) +O(mj+q+1) =o(mj+q), 0≤jq−1.

(2.14)

By Lemma 2.2 we have that Ti,q−1(Dm)→ 0, 1 ≤ik, asm → 0. Let ρbe

the norm on Πn defined byρ(P) := max

1≤ik 0≤maxvq−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)(xxi)jdx=o(l+j+2)− n

X

v=l+2

P(v)(xi)

v!b,i,l

Z

I,i

(xxi)j+vdx,

(2.15) where the sum is omitted if k = 1 and l = q−2. Since Al,i,δ is bounded for

0≤lq−2 andδsufficiently small, from (2.15) we get

Z

I,i Ti,l+1

P

Qi

b,i,l

(x)(xxi)jdx=o(l+j+2), 0≤jl+ 1. (2.16)

Now, Lemma 2.2 withrw=w−1, 1≤wl+ 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≤ik.

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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 fc2

q−1(xi)∩t2q−2(xi), 1 ≤ik, then there exists the best

local approximation off on{xi: 1≤ik}, fromΠn. It is the unique polynomial

H∈Πn satisfyingH(j)(x

i) =Q

(j)

i (xi),0≤jq−1,1≤ik, whereQi∈Πq−1

satisfies Definition 1.1. In the caseq= 1ork= 1we only assumefc2

q−1(xi),1≤

ik.

Remark 2.7. We observe that c2s(xi) is not, in general, a subset of cs(xi). In

fact, next we give an example of a functionfc2

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

kfP+mk 2

+m≤ kf−1k 2 +m =

Z hm

0

(f(x)−1)2dx

m

hm

m

→0, (2.17)

kfP+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+mfk+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 iffc

2

+0(0) then lim

→0P+ exists

(see Remark 2.5), so f /c2+0(0). Analogously,f /c20(0). As f is odd, clearly

fc2 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(2xix)

2

Ii(h)(x) :=

h(x)−h(2xix)

2

.

It is easy to see that, h = Pi(h) +Ii(h), Pi(h)(xi +x) = Pi(h)(xix), and

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Theorem 2.9. If one of the following cases holds: a) q= 1,l=−1, andfc2

±0(xi),1≤ik,

b) k= 1,q≥2,0≤lq−2,fc2

±(l+1)(x1)andT1,l(L1) =T1,l(R1),

c) q≥2,0≤lq−2, andfc2

±(l+1)(xi)∩t 2

l(xi),1≤ik,

then

Ti,l+1(P)−

Li+Ri

2 =o(1)b,i,l, 1≤ik, (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≤ik.

Proof. Sincefc2±(l+1)(xi), 1≤ik, we have

i) R

I+,i(fRi)(x)(xxi)

jdx=o(l+j+2), 0jl+ 1,

ii) RI

−,i(fLi)(x)(xxi)

jdx=o(l+j+2), 0jl+ 1.

By means of a change of variable from i) and ii) we get

iii) R

I−,i(fRi)(2xix)(xix)

jdx=o(l+j+2), 0j l+ 1,

iv) R

I+,i(fLi)(2xix)(xix)

jdx=o(l+j+2), 0jl+ 1.

From i) it follows that

Z

I+,i

(fTi,l(Ri))(x)(xxi)jdx=o(l+j+2) +

Z

I+,i

R(il+1)(xi)

(l+ 1)! (xxi)

j+l+1dx

=o(l+j+2) +O(l+j+2) =o(l+j+1), 0≤jl.

Thus, fc2

+l(xi). Analogously, from ii) it follows thatfc2−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≤ik. 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(xxi)l+1

andLi(x) =Si(x) +c2,i(xxi)l+1.

From i) and iv) we obtain

Z

I+,i

[f(x) +f(2xix)−(Si(x) +c1,i(xxi)l+1

+Si(2xix) +c2,i(xix)l+1)](xxi)jdx

=o(l+j+2), 0≤jl+ 1. (2.20) In a similar way, from ii) and iii) we get

Z

I−,i

[f(x) +f(2xix)−(Si(x) +c2,i(xxi)l+1

+Si(2xix) +c1,i(xix)l+1)](xxi)jdx

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Now, we assumel odd. From (2.20) and (2.21) we get

Z

I,i

[f(x)+f(2xix)−(Ri(x)+Li(2xix))](xxi)jdx=o(l+j+2), 0≤jl+1.

(2.22) By (2.22) we obtain

Z

I,i

Pi

fRi+Li 2

(x)(xxi)jdx=o(l+j+2), 0≤jl+ 1. (2.23)

Sinceft2l(xi), Lemma 2.1 implies

Z

I,i

Pi(fP) (x)(xxi)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≤lq−2

andδsufficiently small. In consequence, from (2.24) we have

Z

I,i

1

b,i,l

Pi(fTi,l+1(P))(x)(xxi)jdx

=

n

X

v=l+2

P(v)(xi)

v!b,i,l

Z

I,i

Pi((xxi)v) (xxi)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)(xxi)jdx

=o(l+j+2), j= 0,2, . . . , l+ 1. (2.26) Using Lemma 2.2 withrw= 2(w−1), 1≤wl+12 + 1, we get

Pi

Ti,l+1(P)−

Ri+Li

2

=o(1)b,i,l, 1≤ik. (2.27)

Now, we assumel even. Adding to both sides of (2.21) the expression

−2

Z

I−,i

(c1,i(xxi)l+1+c2,i(xix)l+1)(xxi)jdx, 0≤jl+ 1,

we obtain

Z

I−,i

[(f(x) +f(2xix))−(Si(x) +c1,i(xxi)l+1+Si(2xix)

+c2,i(xix)l+1)](xxi)jdx=O(l+j+2), 0≤jl+ 1. (2.28)

As the integrand in (2.20) and (2.28) is the same, we have

Z

I,i

[(f(x) +f(2xix))−(Ri(x) +Li(2xix))](xxi)jdx

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It is easy to see thatR

I,i(Ri(x) +Li(2xix))(xxi)

jdx=R

I,i(Li(x) +Ri(2xix))(xxi)jdx, forj even. In consequence, from (2.29) we obtain

Z

I,i

Pi

fRi+Li 2

(x)(xxi)jdx

=

Z

I,i

Pi(f)−

Ri(x) +Li(x) +Ri(2xix) +Li(2xix)

4

(xxi)jdx

=o(l+j+1), j = 0,2, . . . , l.

(2.30)

Now, we proceed as in (2.25). Asft2

l(xi) andAl,i,δ, 0≤lq−2, is bounded

forδsufficiently small, Lemma 2.1 implies

Z

I,i

1

b,i,l

Pi(fTi,l+1(P)) (x)(xxi)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)(xxi)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≤w2l + 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≤ik, (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≤ik.

The following corollary is an immediate consequence of Theorem 2.9 with l =

q−2.

Corollary 2.10. Iffc2±(q1)(xi)∩t2q−2(xi),1≤ik, then there exists the best

local approximation off on{xi: 1≤ik}, fromΠn. It is the unique polynomial

H ∈ Πn satisfying H(j)(x

i) = (Li+2Ri)(j)(xi), 0 ≤ jq−1, 1 ≤ik, where

Li, Ri ∈Πq−1 satisfy Definition 1.1. In the case q= 1 ork = 1 we only assume

fc2

±(q−1)(xi),1≤ik.

This result was proved in [4, Theorem 2.9], forft2(q1)(xi)∩t2+(q1)(xi)∩

t2q2(xi), 1≤ik. 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

functionfc2

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Example 2.11. Let m = m1, m ∈ N, hm = mm16. 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)dxhm3 Z m

0

f(x)dx=hm3

X

j=m

Z j

hj

f(x)dx

=hm3

X

j=m

Z j

hj

j dx=hm3

X

j=m

j(jhj) =hm3

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)dxm3 Z m

0

f(x)dxhm3 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 fc2

±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], jm0. So, |f(x)−S(x)|2 ≥ f(x)(f(x)−2|S(x)|) ≥ f(x)

2

2 for all x∈[hj, j],jm0. Therefore, formm0 we have

m3 Z m

0

|(fS)(x)|2dx−3

m

X

j=m

Z j

hj

|(fS)(x)|2dx

≥1 2 −3 m ∞ X

j=m

Z j

hj

|f(x)|2dx=1

2 −3

m

X

j=m

j2(jhj) =

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)2dxhm1 Z m

0

f(x)2dx=hm1

X

j=m

Z j

hj

f(x)2dx

=hm1

X

j=m

j2(jhj) =hm1

X

j=m

j−4 1

m2.

References

[1] Calder´on, A.P. and Zygmund, A.Local properties of solution of elliptic partial differential equation, Studia Math.20(1961), 171–225. MR 0136849.

[2] Chui, C.K., Shisha, O. and Smith, P.W.Best local approximation, J. Approx. Theory 15

(1975), 371–381. MR 0433101.

[3] Chui, C.K., Smith, P.W. and Ward, I.D.BestL2local approximation, J. Approx. Theory22

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[4] Cuenya, H.H. and Rodriguez, C.N.Differentiability and best local approximation, Rev. Un. Mat. Argentina,54(2013), no. 1, 15–25. MR 3114610.

[5] Cuenya, H.H. and Ferreyra, D.E. Cp Condition and the best local approximation. Anal. Theory Appl.31(2015), 67–76. MR 3338788.

[6] Marano, M. Mejor aproximaci´on local, Ph. D. dissertation, Universidad Nacional de San Luis, 1986.

[7] Walsh, J.L.On approximation to an analitic function by rational functions of best approxi-mation, Math. Z.38(1934), 163–176. MR 1545445.

H. H. Cuenya

Dpto. de Matem´atica, FCEFQyN, Universidad Nacional de R´ıo Cuarto, (5800) R´ıo Cuarto, Argentina

hcuenya@exa.unrc.edu.ar

D. E. FerreyraB

Dpto. de Matem´atica, FCEFQyN, Universidad Nacional de R´ıo Cuarto, (5800) R´ıo Cuarto, Argentina

deferreyra@exa.unrc.edu.ar

C. V. Ridolfi

Instituto de Matem´atica Aplicada de San Luis, CONICET, Universidad Nacional de San Luis, (5700) San Luis, Argentina

ridolfi@unsl.edu.ar

Figure

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