ORDERS OF COAPPROXIMATION OF
FUNCTIONS BY ALGEBRAIC POLYNOMIALS
Ali H. Batoor
Department of mathematics, College of Education, Kufa University.
Eman S. Bhayh
Department of mathematics, College of Education, Babylon University.
Abstract. In 2003 we proved a direct theorem for k-monotone approximation of
functions in L p , p  0 . Here we using this theorem to obtain the equivalence
 
 
between En  f  p   n  and Enk  f  p   n  , for 0    k  1 and p  0 .
Also we show that the rate of the best algebraic approximation of a k-monotone
function with bounded (k-2)nd derivative in L p ,0  p  1 , is o n  k 1 p  .


1.Introduction and Main Results
Let L p a, b, p  0 denote the space of all measurable functions on
a, bsuch that
 b

p
  f ( x) dx  0  p  ,

f p : f L a , b  :  a

p

p  ,
sup xa , b  f  x 
is finite. As usual the integral modulus of smoothness of order m is given by
 m  f ,   p : sup 0h mh  f , ,
p
where
1
 m
m  
i m 

  f  x    i h ,



1
m
 h  f , x  : 
2  
i  
 i 0
0


If m=0, we set  0  f ,   p : f
p
x
m
h  a, b,
2
otherwise.
. We also believe hat for estimating the rate of
best approximation of a function f  L p  1,1 the measure of smoothness
 m  f ,   p introduced by Ditzian and Totik [1]
 m  f ,   p : sup 0  h  mh   f ,
p
We now recall the definition of k-monotone functions. A function
f : a, b  R is said to be k-monotone, k  1 , on a, bif for all choices of (k+1)
distinct points x0 , x1 ,..., xk in a, b the inequality f x0 , x1 ,...xk   0 holds, where
f x0 , x1 ,...xk  :
k
 f x / x  ,
j
j
j 0
denote the k-th divided difference of f at x0 , x1 ,..., xk , and
  x  :  kj 0 x  x j .
Note that 0-monotone, 1-monotone and 2-monotone functions are just
nonnegative, nondecreasing and convex functions, respectively. We denote the
class of all k-monotone functions on a, bby k a, b . We are interested in
approximation of such functions by polynomials from k a, b, i.e. so called “kmonotone approximation”. Also let  n denote the set of all algebraic
polynomials of degree  n . Recall that the rates of best unconstrained and kmonotone polynomial approximation are given respectively by
En  f  p : inf Pn  n f  pn
and
Enk  f  p : inf P
n  n
 k a , b 
2
p
,
f  pn
p
.
c are positive constants which are not necessarily the same even when
they occur in the same line. In order to emphasize that c depends only on those
parameters v1 ,...,vk the notation cv1 ,...,vk  is used.
Also we write f  x   O g  x  as x  A , A is a constant if
f x
 K,
g x
with constant K as x  A , and f  x  : o g  x  as x  A , if
f x
0
g x
as
x  A.
The following Theorem A was proved by Kopotun [2]
Theorem A. Let 0    3 , and let a function f C  1,1 changes sign r  
times at the points Yr : 1  y1  ...  y r  1 in [-1,1]. Then
 
 
En  f   O n   En0  f ,Yr   O n 
where
En0  f , Yr  : inf P
n  n
 0 Yr 
f  pn

and
0 Yr  the set of all functions that have r   sign changes at the points Yr .
In this paper we generalize Theorem A for 0    k  1 and any kmonotone function, k  1 in L p , p  0 .
Theorem I. For any f  L p  1,1  k  1,1 and 0    k  1 , we have
 
 
En  f  p  O n   Enk  f  p  O n  .
The second result in this paper shows that the rate of the best
algebraic approximation of a k-monotone function (with bounded (k-2)nd
derivative) in L p ,0  p  1 is o n  k 1 p  .


3
Theorem II. Let f  k  1,1 be such that f k  2   C  1,1 . Then there exists
a polynomial p n   n , such that
f  pn
p

 cn k 1 p  k f , n 1

f k 2 
p

1 p

,
for 0<p<1, where c depends only on p. In particular


Enk  f  p  o n  k 1 p  ,
0  p  1.
2.Proof of Theorem I
For the proof of this theorem we need the following assertions from [3]
Theorem D. Let f  k  1,1, k  1 , be such that f m  L p  1,1, p  0 . For
some m, 0  m  k  1 . Then there exists p n a polynomial of degree  n , such
that
f  j   p n j 
p


 c k j f  j  , n 1 p ,
for all j=0,1,…,m, where c depends only on k and p (if p<1).
Lemma 1. For Pn   n , m=1,2,… and p>0, we have
 m  pn ,  p  cn m pn
p
,
where c depends on k and p (if p<1) only.
Now we are ready to prove theorem I.
1
1
For   0 define  by 2    2 1 , and for n  , let P2i   2 be


polynomial of best approximation of f
(i.e. E i  f  p  f  p2 i ). Then
2
theorem D with j=0 yields
Enk  f  pp  c( p) k  f ,  pp

 c( p)( k f  p2 ,2 

p
p

  k p2 ,2 
4
)
p
p
p


 c( p) k


p
 c ( p ) f  p 2
p

 p
2i
 p2i 1
i 0
p

 
,2

p

where p2 1  p0  0 .
In view of Lemma 1 we have
Enk
f 
p
p
 c ( p ) E 2  f   c ( p )
p
p

 2 
i  kp
p 2i  p 2i 1
i 0
 c p 2
kp

2
ikp
p
p
E 2i  f  pp .
i 0
On the other hand, there exists a constant c=c(k,p) depending on k and p only
such that [4]
2i
 m  1
2ikp  c
kp 1
.
m  2 i 1 1
Whence
Enk
f 
n
p
p
 c( p, k )
kp
m  1
kp1
Em  f  pp ,
m0
and
Enk  f  p
1/ p
 n

k
 c( p, k )
m kp1p 


 m1


 
 O n k
3. Proof of Theorem II
5
For the proof of Theorem II we use the following lemma from [5]
Lemma 2. Let 2  1,1. Then for any   0
 2  f ,  1  c 2 f

,
where c is an absolute constant .
The following theorem about differentiability of k-monotone functions [6] will
be useful
Theorem E. Suppose for some k  2 that f : a, b  R is k-monotone. Then
f  j   x , the derivative of order j, exists on (a,b) for j  k  2 and is (k-j)-
monotone. In particular f k  2   x  exists and convex.
Now let us prove our theorem.
Suppose that p n is a polynomial of degree  n , satisfies Theorem D for
j=0 and p   , respectively i.e.
f  pn


 c k f ,n 1


(4.6)


f  pn 1  c k f , n 1 1
f 
q
pn p
 1 1 1
q   
 2  p q

q
f  pn dx ,
1
where q>1, such that 1/q+p=1. Then
f 
q
pn p
 1 1 1
q   
 2  p q

f  pn
q 1
f  pn dx
1
 c p  f  pn  f  pn 1 .
Together with (4.6) and Lemma 2 this implies
q 1
6
f  pn
q
p

 c p  k f , n 1

q 1 
 k

 c p n  k  2 k f , n 1

 c p n  k  k f , n 1
f ,n 
1
1

q 1 
 2

q 1

f 
f k  2 
k  2


, n 1 1
.
Hence
f  pn
p
 c p n
 k 1 p 


k f , n
1

1

1
q
f k  2 
1 p

.
This completes our proof ♠
REFERENCES
[1]. Z.Ditzian and V.Totik (1987): Moduli of smoothness, SpringerVerlag. New Yourk.
[2]. K.Kopotun (1995): On copositive approximation
polynomials, Analysis Mathematia, 21; 269-283.
by algebraic
[3]. E. Samir(2003): On the constrained
and
unconstrained
approximation, (Ph.D. Thesis), University of Baghdad.
[4]. D. Dryanov (1989): One-sided approximation by trigonometric
polynomials in L p ,norm 0  p  1 , Banach Center publications, 22;
99-110.
[5]. K.Kopotun (1995): On k-monotone polynomial
and spline
approximation in L p 0  p    ,(quasi) norm, J. Approx. Theory
8(1); 295-302.
[6]. A..Robert and D.Vverberg (1973): Convex
Press, New York.
functions, Academic
‫رتب التقريب بقيود باستخدام‬
‫متعددات الحدود الجبرية‬
7
‫علي حسين بتور‪ /‬قسم الرياضيات‪ /‬كلية التربية‪ /‬جامعة الكوفة‬
‫إيمان سمير بهية‪ /‬قسم الرياضيات‪ /‬كلية التربية‪ /‬جامعة بابل‬
‫المستخلص‬
‫يف ع ا ‪٢٠٠٣‬برهن ا يةرماار يف اااري يف‬
‫فض ا ا‬
‫‪Lp‬‬
‫مل ل رمن‬
‫عنا اال ‪ . 0  p‬يف ه ا ا‬
‫رم ا‬
‫آلين لاال‬
‫‪ k‬ط اريفي يف‬
‫يفحا اات م ا ا كل ن رلا ااه نةرما اار ه ا ا إ‬
‫‪ ‬‬
‫‪ ‬‬
‫‪Enk  f  p  O n ‬‬
‫‪En  f  p  O n ‬‬
‫كا ني ع عناال ‪ . 0    k  1‬كا ه لناالن علااة يةرماار ريفااع إ رريفاار‬
‫لال‬
‫‪ k‬طاريفي ا ت لاه ‪ k  2‬شا ر يالي يف فضا‬
‫ابم كل مل عليف‬
‫جل مر هي ‪. on k 1 p  ‬‬
‫‪8‬‬
‫‪ L p‬عنال‬
‫رما‬
‫‪1 p‬‬