Inverse and direct theorems for
monotone approximation
Sukeina Abdulla Al-Bermani
Babylon University
Science College For Women
Computer Science Department
Abstract
We prove that if f is increasing function on > 1,1@ then for
each n 1,2,... , there is an increasing algebraic polynomial pn of
1
degree 8n such that f pn p d c p Z 2M §¨ f , ·¸ ,
©
n ¹p
Where Z 2M is the second order Ditizian – Totik modulus of
smoothness. Also a converse theorem for this direct theorem
were obtained. These results complement the classical
pointwise estimates of the same type for unconstrained
polynomial approximation.
1 .Introduction and Main Results
Several
results
show
that
in
some
sense
monotone
approximation by algebraic polynomials performs as well as
unconstrained approximation. For example Lorentz and Zeller,
[7] have shown that for each increasing function f in C I ( the
27
space of all continuous functions on I > 1,1@ ) there is an
increasing polynomial pn of degree n that satisfies
1
d c Z §¨ f , ·¸ ,
© n¹
f pn
n 1,2 ,...
(1.1)
where Z is the modulus of continuity of f .
A general result for (2.1.1) for any k 0,1,
there are
increasing pn that satisfies
f pn
d cn
k
1·
§
Z ¨ f k , ¸ , n 1,2,...
n¹
©
(1.2)
this result of Lorentz [6], where as the general case was proved
by DeVore [3], the cases k 0,1 are much easier to prove than
the general cases. Since they can be proved using linear
method, in contrast, the proof in [3] uses rather involved non
linear techniques. It is well known that for unconstrained
approximation much improvement can be made in estimates of
the form (1.1) where x is near the end points of I .
In this thesis, we are interested in pointwise estimates for
monotone approximation, the only result of this type that we
know of is by Beatson [1]. He proved that the estimate
f x p n x d cZ f , 'n x , x  I , n
' n x 1,2 ,...
1 x2 1
2 , holds for suitable increasing polynomials pn
n
n
whenever f is increasing. Devore and Yn have shown that if f
28
is increasing function on I > 1,1@ , then for each n 1,2,... there is
an increasing polynomial pn of degree n such that
§
1 x2
f x pn x d cZ 2 ¨ f ,
¨
n
©
·
¸,
¸
¹
where Z 2 is the second order moduli of smoothness. Among
other things, we shall show that this can be improved to allow
second
order
modulus
of
smoothness
fo r
the
spaces
L p ,0 p 1 .
Theorem I:
If f is an increasing function in L p I ,0 p 1
then for each n 1,2,... there is an increasing polynomial in
L p I , of degree (8n) satisfying
f pn
p
1
d c( p )Z 2M §¨ f , ·¸ .
© n ¹p
(1.3)
Using this theorem we can obtain our second Inverse
inequality:
Theorem II:
Let
f
be an increasing function in
L p I ,0 p 1 , then
Z 2M f , n 1
pp d cpEn1 f pp cpn 2 p
29
1
¦ m 1 p1 En f pp .
m !n
2 .Auxiliary Lemmas
Before we prove our theorems we need the following
notions and lemmas. Our proof is based on a two stage
approximation. We first approximate
f
by an increasing
piecewise linear function S n . We then approximate S n by an
increasing algebraic polynomial . S n is the piecewise linear
function that interpolates f at [ k , k n,..., n , if we let s j be the
slops
sj
Then
ca n
Sn
f [ j 1 f [ j [ j 1 [ j
be
j
,
n,..., n 1
represented
by
.
using
(2.1)
the
function
) j x max^x [ j ,0`as
S n x f 1 s n x 1 n 1
¦ s
j n 1
j
s j 1 ) j x , [4]
(2.2)
It is clear that S n is increasing if f is also.
We shall now contract a polynomial R j , j n,..., n 1 , as in [4]
that approximate the function ) j x . The construction of R j
begins
with
trigonometric
polynomial
30
T j , j 1,...,2n with
tj
jS
,
2n
j
0,1,...,2n .
Let
denote
Kn
the
Jackson
kernel
8
nt ·
§
¨ sin ¸
2 ¸ ,
an ¨
¨ sin t ¸
¨
¸
©
2 ¹
K n t (2.3)
where an is a constant depending on n chosen so that
S
³ K n t dt 1 .
S
Here and throughout c p , c , denote absolute constants
depending on p and c p , c 's,
values may vary with each
occurrence on the same line.
Define
T j t ³ tttt j K n u du , j 0 ,1,...,2 n ,[4]
j
and define
^
` d j t max n dist t , t j , t j ,1 ,[4]
(2.4)
Now let
r j x Tm j t ,
x
cos t .
And for x  > 1,1@ define
R j x x
³ r j u du , j
1
n ,..., n ,[4]
(2.5)
In particular R n x x 1 )x and Rn x 0 , the points [ j are
defined by the equations
1 [ j
R j 1, j
n,..., n . If f  L p I we define
31
n 1
f 1 s n Rn ¦ s j s j 1 R j .
Ln f with
sj
sj t 0 , j
defined
(2.6)
n 1
by
(2 . 1 )
f
if
is
increasing,
the
n,..., n 1 and since we can also write
Ln f n 1
f 1 ¦ s j R j R j 1 .
n
Now from the definition of the polynomials T j we have
Tn j Tn j 1 t 0 ,
hence
r j r j 1 t 0 ,
and
therefore R j R j 1
is
increasing , it follows that Ln f is increasing.
We now estimate
E x S n x Ln f , x ¦ s
j
j
s j 1 ) j x R j x .
(2.7)
Now for j n,..., n 1, x cos t with 0 d t d S , we have
) j x R j x d cn 1 sin t n j t t n j d n j t Lemma 2.9.
Ln f p d c p f
p
Proof: We have
S n x Ln f , x n 1
¦ s
j n 1
j
s j 1 ) j x R j x .
Then
Ln f , x S n x d S n x n 1
¦ s
j n 1
n 1
¦ s
j n 1
j
j
s j 1 ) j x R j x s j 1 ) j x R j x .
Definition of S n implies
32
5
,[20]
(2.8)
S n x d f 1 s n x 1 n1
¦
j n 1
s j s j 1 ) j x .
Thus
Sn d c f x .
And
L n f , x d c f x n1
¦
j n 1
s j s j 1 ) j R j .
Then (2.2.1) implies
sj d
c f x Gj
d
c f x G j1
where G j [ j 1 [ j .
Then
L n f , x d c f x c f x and by G j [ j 1 [ j
n1
¦
j n 1
G j 1 ) j R j ,
c
, we have G j1 d cn .
n
So
L n f , x d c f x c f x d c f x c f x n1
¦
j n 1
n1
¦
1 n t tn j dn j t 5
1
j n 1 n
4
Then by the following [4]
Lemma 2.10
If gc is absolutely continuous and gcc d M
almost every where on I. Then for each n 1,2,.. and each x  I ,
we have
§ 1 x2
gx Ln g , x d cM ¨
¨ n
©
33
2
·
¸ .
¸
¹
We can prove
Lemma 2.11
If g c is absolutely continuous and g cc d M
almost every where on I, then for each n 1,2 ,.. , and each
x  I we have
g Ln g p d
c p n2
.
3 .proof of theorem I
Firstly let us introduce the so called Ditzian Totik functional
definition as
1 ·
~ §
K 2 ,M ¨ f , 2 ¸
© n ¹p
§
inf ¨ f g
g ©
p
1
·
M 2 gcc ¸ ,
p¹
n
2
for f  L p I .0 p d f .
We have
~ § 1 ·
Z 2M f , n 1 p | K 2,M ¨ f , 2 ¸ [2].
© n ¹p
Given x  I , then from the result above there is a g satisfies
f g
p
d c p Z 2M f , n 1
p
and
1 2
M g cc d c p Z 2M f , n 1
2
n
p
f Ln f p d f g
p
p
( 3.1)
p
p
p
g Ln g p Ln g Ln f p .
Then by the linearity, and the boundedness of Ln f , we obtain
p
f Ln f p d f g
p
p
p
g Ln g p Ln
34
p
p
f g
p
p
p
p
d 1 Ln
f g
p
p
p
g Ln g p .
p
Lemma (2.11) implies g Ln g p d
c p n2
.
Using (3.1), Lemma( 2.11) and the linearity of Ln f , we have
p
p
p
f Ln f p d 1 Ln
d 1 Ln
cp Z f , n M
cpZ f , n cpZ f , n .
p
p
c p
n2
1 p
p
M
2
2
g cc
1 p
p
M
2
p
p
M
2
1 p
p
By virtue of Lemma ( 2.9) we have
p
f Ln f p d c p 1 f
p
p
Z f , n d c p Z 2M f , n 1
1 p
p
M
2
Z 2M f , n 1
p
p
.
p
p
Since L n f is an increasing polynomial of degree d 8n we
have proved theorem , ♣
4 .proof of Theorem II
For
^
given by
`
n , we expand pn x by
max i : 2 i n ,2
p x p x p x p x p n x p0 x n
2
We recall that for m n
Z 2M f , n 1 p d c p f pn
p
p
p
2
pn pm
2
p
p
d c p where
p2i is
f p
p
p
c p n p Z1M pn , n 1
c p n
2p
p1 x p0 x .
d c p Em1 f p
d c p En1 f pp c p n 2 p pcn
En1
1
p
p
p
¦ pc2 i
i 1
p
p
,
p
an algebraic polynomial of best monotone
approximation of degree not greater than or equal 2 it mean
35
p
f p 2i
p
E21i f p .
(4.1)
p
Then
Z
M
2
f ,n 1 p
p
d c p En1
f p
p
c p n
2p
¦ pc2 i pc2 i 1
i 1
d c p En1 f pp c p n 2 p ¦ pc2 i pc2 i 1
i 1
p
p
p
p
.
Then by Bernstein inequality we have
Z 2M f , n 1 p d c p En f pp c p n
1
p
2p
f
p E
i
¦2 n
i 1
1
2i n
f pp .
Applying the inequality
2ivp d c p, v ¦ m 1
vp 1
,v N
m 2 i 1 1
[5]
We get
Z2M f , n 1 p d c p En1 f p c p n 2 p ¦ m 1 Em1 f p ♣
p
f
p
p 1
p
m!n
References
[1]
R.
Beatson(2008):
Shape
preserving
convolution
operators(preprint).
[2] E. S. Bhaya (2003): On constrained and unconstrained
approximation. Ph. D. Thesis. Baghdad university, College of
Education , bn – Al- Haitham.
[3] R. A. DeVore (1977): Monotone approximation by polynomials. SIAMJ. Math. Anal., 8; 906-921.
[4] R. A. DeVore, X. M. Yu (1985): Pointwise estimates for
monotone polynomial approximation. Constr. Approx., 1;
323-331.
36
[5] D. Dryanov (1989): One sided approximation by trigonometric polynomials in L p -norm, 0 p 1 . Banach Centre Publications, 22; 99-110
[6] G. Lorentz(1972): Monotone approximation. New York:
Academic press, 201-215.
[7] G. Lorentz, Z. L. Zeller (1968): Degree of approximation by
monotone polynomials II. J. Approx. Theory, 2; 265-269
37