The Second Conference on Mathematical Sciences (CMS'2008), Zarqa Private University, Jordan
Estimates for Comonotone Polynomial
Approximation in Lp
By Eman Samir Bhaya,
Mathematics Department, College of Education,
Babylon University, Iraq.
Email: emanbhaya@yahoo.com
Abstract
Lp , ∞ > p ≥ 1
When we approximate a function f in
which changes its monotonicity finitely many,
say s time, in [−1,1] , we wish some times that the approximating polynomial follow these changes of
monotonicity. However it is well known that this requirement restricts very much the degree of
approximation that the polynomials can achieve, namely the rate of ω2 ⎛⎜ f , n −1 ⎞⎟ . In [1] we prove
⎝
⎠p
that relaxing the comonotonicity requirements in very small intervals about the interior extremes and
near the end points, what we called nearly comonotone approximation allows the polynomials to
achieve a pointwise approximation rate of ω 3⎛⎜ f , n −1 ⎞⎟ . Also we proved that even when we relax the
⎝
⎠p
requirement of monotonicity of the polynomials on sets of measures approaching 0, ω4 is not
reachable. We prove here that when f belongs to the Sobolev space, allow estimates involving the kth
modulus of smoothness of f ′ for k ≥ 1 .
1. Introduction
Let
f ∈ L p [− 1,1]] , change monotonicity
s ≥ 1 times on
I = [− 1,1] . Let
Ys = {yi }i =1 set of points so that − 1 < y s < ... < y1 < 1 . Denote by ∆(1) (Ys ) the
s
collection of functions which change monotonicity at the points yi . A
polynomial p n ∈ Ρn the space of polynomials of degree not exceeding n , is
said to be comonotone with f , on a set E ⊂ I = [− 1,1] if and only if
p n′ ( x ) Π( x − y i ) ≥ 0
s
i =1
∀x ∈ E .
The degree of approximation of f ∈ ∆(1) (Ys )I L p [− 1,1] by comonotone
polynomials is measured in the L p norm for 1 ≤ p < ∞ , and defined by
E n1 ( f , Ys ) p :=
inf
pn∈Ρn I ∆(1) (Ys )
f − pn
p
.
We also denote by W pk [a, b] , the set of all functions f on [a, b] such that
f (k −1) is absolutely continuous , and f (k ) ∈ L p , such a space is called the
Sobolev space.The rth symmetric difference of f is given by
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The Second Conference on Mathematical Sciences (CMS'2008), Zarqa Private University, Jordan
⎧ r ⎛r⎞
rh
rh
⎞
r −i ⎛
⎪ ∑ ⎜⎜ ⎟⎟(− 1) f ⎜ x − + ih ⎟ , x ± ∈ [a, b]
r
r
∆ h ( f , x, [a, b]) := ∆ h ( f , x ) := ⎨i =0 ⎝ i ⎠
2
2
⎠
⎝
⎪0,
o.w.
⎩
Then the rth usual modulus of smoothness of f ∈ L p [a, b] is defined by
ω r ( f , δ , [a, b]) p := sup ∆rh ( f ,.)
0< h≤δ
L p [a ,b ]
, δ ≥ 0. [2]
We will also use the so called τ − mdulus (or Sendov-Popov modulus), an
averaged modulus of smoothness, defined for bounded measurable
functions on [a, b ] by
τ r ( f , δ , [a, b]) p := ω r ( f ,., δ ) L [a ,b ] , [5]
p
where
⎧
ω r ( f , x, δ ) := sup ⎨ ∆rh ( f , y ) : y ±
⎩
rh ⎡
rδ
rδ ⎤
⎫
∈ ⎢ x − , x + ⎥ I [a, b]⎬
2 ⎣
2
2⎦
⎭
is the rth local modulus of smoothness of f . From the definition one can
easily see
τ r ( f , δ , [a, b])∞ := ωr ( f , δ , [a, b])∞ .
The following relationship between the ω and τ moduli holds for any
f ∈ W p1 [a, b],1 ≤ p ≤ ∞ .
τ r ( f , δ , [a, b]) p ≤ c(r )δωr −1 ( f ′, δ , [a, b]) p . [6]
( 1.1)
If the interval I = [− 1,1], is used in any of the above notation it will be
omitted for the sack of simplicity, for example:
ωr ( f , δ ) p := ωr ( f , δ , [− 1,1]) p ,
and we will also denote
ωr ( f , δ ) := ω r ( f , δ , [− 1,1])∞ .
The moduli ω and τ , measure the smoothness of f over the interval
uniformly. It is well known that polynomials approximate better
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The Second Conference on Mathematical Sciences (CMS'2008), Zarqa Private University, Jordan
near the end points of the interval than in the middle, and this leads
to either pointwise estimates ( if p = ∞ ) or the introduction of
nonuniform moduli of smoothness the non uniform modulus that
we use is the rth Ditizian Totik [3] modulus of smoothness defined
for f ∈ L p (I )
ω rϕ ( f , δ , I ) p := sup ∆rhϕ (⋅) ( f ,.)
0< h≤δ
Lp (I )
,
with ϕ ( x ) = 1 − x 2 . We have
1
p
ω rϕ ( f , δ ) p ≤ ω r ( f , δ ) p ≤ τ r ( f , δ ) p ≤ 2 ω r ( f , δ )∞ ,
1 ≤ p ≤ ∞ .[6]
However In Lemma 2.2.5 in [1] we proved that the moduli ω r and
ω rϕ for an f defined on J = [a, b] ⊂ [− 1,1] are equivalent if J ≈ ∆ n (a )
with
∆ n (a ) = n −1 1 − a 2 + n −2 :
Lemma 1.1 Let [a, b] ⊂ [− 1,1] be such that (b − a ) ≤ ρ n (a ) , where c1 ≥ 1 is an
absolute constant. Then for any nonnegative integer r there is a constant
c(r ) such that
ω rϕ f , n −1 , [a, b ] p ≥ c(r )ω r ( f , ρ n (a ), [a, b ]) p .
(
)
Now let us turn to the comonotone approximation
In [1]( Corollary 2.1.4, p52) we proved that
Theorem 1.2 If f ∈ L p (I ) ∩ ∆(1) (Ys ) , then there is a constant A(s ) such that
for n >
A(s )
, there is a polynomial p n ∈ Ρn I ∆(1) (Ys ) satisfies
d (Ys )
f − pn
p
(
≤ c (s )ω 2ϕ f , n −1
)
p
(
≤ c(s )τ 2 f , n −1
)
p
where d (Ys ) = min{1 + y1 , y 2 ,..., y s − y s −1 ,1 − y s }. The constant c(s ) depends only
on s. On the other hand one cannot replace ω 2ϕ in theorem 1.2 by ω 3 , where
ω k denotes the modulus of smoothness of order k. It is quite natural to ask
whether one can strengthen theorem 1.2 in the sense of being able to
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The Second Conference on Mathematical Sciences (CMS'2008), Zarqa Private University, Jordan
replace ω 2 by moduli of smoothness of higher order , if one willing to
allow p n not to be comonotone with f on a rather small subset of I, what
we called nearly comonotone approximation. In theorem 3.1.2 p.72 , in [1]
we proved that in such a case it is possible to achieve the estimates
f − p n p ≤ c (s )τ 3 ( f , n −1 ) p .
However this improvement can not be extend to ω 4 or τ 4 . In theorem 4.1.1
p. 87 [1] we show that even when we relax the requirement of
monotonicity of the polynomials on sets of measures approaching zero, ω 4
or τ 4 is not reachable.
We let
O(h, Ys ) = I I U ( y i −∆ n ( yi ), y i + ∆ n ( yi ))
s
i =1
In this paper we prove that if we assume f belongs to the Sobolev space
W p1 (I ) ., then we can obtain estimates involving moduli of higher orders
Theorem 1.3. For each A > 0 , there is a constant c(k , s, A) c for which if
f ∈ ∆(1) (Ys ) ∩ W p1 (I ) then for every n ≥ k a polynomial p n ∈ Ρn which is
comonotone with f on I \ Ο( A / n, Ys ) exists such that
f − pn
p
≤ c(k , s, A)
( (
1 ϕ
ω k f ′, n −1
n
)
p
(
+ τ k f ′, n −1
) ).
p
(1.2)
2. The proof of the main result
Let k ≥ 1 be fixed. We construct a piecewise polynomial
(2.1)
S ∈ Σ k +1,O (Y ,n ) ,
which is comonotone with f on I \ O(Ys , nm ) , and sufficiently close to it. To
this end we introduce the following
Lemma 2.1 [1] If f is monotone function in W p1 [a, b], h < 1 , the there is a
monotone polynomial pr −1 ∈ Pr −1 interpolating f at 0 and h, such that
f − p r −1 L [0,h ] ≤ chω r −1 ( f ′, h, [0, h]) p .
s
p
For I j .n ⊄ O(Ys , n ) , there is a polynomial p = p of degree ≤ k , which
interpolates f at both end points of I j.n for which
j
f − pj
[ ]
L p I j ,n
j ,n
(
≤ c I j ,n ω k f ′, I j ,n , I j , n
)
p
(2.2)
Then from Theorem 1.2 since ω k ( f ′, I j ,n , I j ,n )p ≤ cω kϕ ( f ′, n −1 ) p , so we have
f − pj
[ ]
Lp I j ,n
(
≤ cn −1ω kϕ f ′, n −1
441
)
p
(2.3)
The Second Conference on Mathematical Sciences (CMS'2008), Zarqa Private University, Jordan
Here and for the rest of the proof constants c are absolute constants and
may depend on k and s or k, and m, they may differ at occurrences, even if
they appear in the same line.
Then
Let
( j − 1)(b − a ) , a + j (b − a )⎤, j = 1,2,..., m ,
⎡
I j = ⎢a +
m ⎥⎦
m
⎣
and set
( j − 2)(b − a ) , a + ( j + 1)(b − a )⎤ [a, b] .
⎡
Oˆ i = Oˆ i ,m (Ys ) = ⎢ a +
⎥⎦I
m
m
⎣
If
( j − 1)(b − a ) , a + j (b − a )⎤ y ∈ 1 ≤ i ≤ s , and Oˆ = s Oˆ .
⎡
a
+
U
i
i
⎢⎣
m
m ⎥⎦
i =1
For each Ôv , we obtain by Lemma 2.1 , a polynomial q v of degree
≤ k which is comonotone with f on Oˆ v / O(Ys , nm) , such that
f − qv
(
ˆ
ˆ ˆ
( ) ≤ c Ov ω k f ′, Ov , Ov
L p Oˆ v
)
p
≤
1⎞
c ϕ⎛
ω k ⎜ f ′, ⎟ .
n ⎝
n⎠p
(2.4)
The piecewise polynomial
⎧⎪q ( x )
Sˆ ( x ) = ⎨ v
⎪⎩ p j ( x )
if
x ∈ Oˆ v ,
if
x ∈ I j ,n \ O
,
has discontinuities at the end points of Ôv , and we alter it to obtain
continuous piecewise S which is comonotone with f on I \ O(Ys , nm ) ,
satisfies (2.1), and by virtue of (2.3) and (2.4)
f −S
p
≤
c4 ϕ ⎛
1⎞
ω k ⎜ f ′, ⎟ .
n
n⎠p
⎝
(2.5)
Since S is a polynomial of degree at most k on each Ôv , then it changes
monotonicity at most k-1 there. Hence there is Y∗ ⊂ O(Ys , nm ) , containing
s∗ ≤ (k − 1)s points such that either S ∈ ∆(1) (Y∗ ) , or − S ∈ ∆(1) (Y∗ ) . Since m ≥ 2 , it
follows that O(Ys ,2n ) ⊂ O(Ys , n ) , whence
S ∈ Σ k +1,O (Y , 2 n ) .
In our proof of this theorem we make use of the following lemma 3.2.22 ,
p. 81 from [1]
Lemma 2.2. If S ∈ Σ k1,O (Y ,n ) I ∆(1) (Ys ) , then
Ec(1n) (S , Ys ) ≤ c(k , s )τ m (S , n −1 ) p .
In particular
∗
∗
1
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The Second Conference on Mathematical Sciences (CMS'2008), Zarqa Private University, Jordan
(
)
E c(11n) (S , Ys ) ≤ c(k , s )n −1ω r −1 f ′, n −1 p ,
where c1 = c1 (k , s ) .
Then applying Lemma 2.2, and conclude that there is a polynomial p of
degree ≤ c7 n , which is comonotone with S everywhere on I, and such that
S−P
p
⎛ 1 ⎞
≤ c8τ k +1 ⎜ S , ⎟ .
⎝ 2n ⎠ p
(2.6)
It is readily seen that p is comonotone with f on I \ O(Ys , nm ) , and combining
(2.5) and (2.6) we have
f −p
p
≤ f −S
p
≤ f −S
≤ f −S
p
p
+ S−p
p
⎛ 1 ⎞
+ c8τ k +1 ⎜ S , ⎟
⎝ 2n ⎠ p
1 ⎞
1 ⎞
⎛
⎛
+ c8τ k +1 ⎜ S − f , ⎟ + c8τ k +1 ⎜ f , ⎟
2n ⎠ p
⎝
⎝ 2n ⎠ p
(
≤ cn −1ω kϕ f ′, n −1
)
p
1 ⎞
⎛
+ c8τ k +1 ⎜ f , ⎟ .
⎝ 2n ⎠ p
Then by the relationship (1.1), we have
( (
)
(
))
1 ϕ
ω k f ′, n −1 p + ω k f , n −1 p .
n
Given n ≥ c7 , and m ≥ 2 . Applying the above to n1 = [n / c7 ] and m1 ≥ mc7 so
f −p
p
≤ c(r )
big that nm divides n1m1 . We obtain a polynomial p n of degree ≤ n , which
is comonotone with f on I \ O(Ys , nm ) such that (1.2) satisfied. Thus a proper
choice of m = m( A) yields our theorem for n ≥ c7 and for k ≤ n ≤ c7 , our
theorem follows from Lemma 2.1. This completes the proof ♠
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The Second Conference on Mathematical Sciences (CMS'2008), Zarqa Private University, Jordan
References
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[3] Z. Ditizian, V. Totik, Moduli of smoothness, Springer-Verlag, New
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