Piecewise
Polynomial And Spline Approximation
Malik S. Al-Muhja
University of Al-Muthana, College of Sciences, Department of Mathematics and
Computer Application
Abstract
s  C r 1  1,1 be a spline of order k , on the interval  1,1 , and v = 1,2,3 . We estimate
the degree of spline approximation of s by peceiwise polynomial of degree  n . In our second result
Let
of this paper we show that inverse theorem interims of higher moduli of smoothness.
Key words : spline approximation , moduli of smoothness , algebraic polynomial approximation .
‫الخالصة‬
v  1,2,3 ‫ و‬ 1,1 ‫ من المشتقات المستمرة على التتىرة‬v  1 ‫ تمتلك‬k ‫ متعددة حدود ريشية من الدرجة‬s ‫لتكن‬
.
‫ فىي النتيجىة الااييىة لاىذا‬. n ‫في هذا البحث سنجد درجة التقريب لمتعددة الحدود الريشية باستخدام متعددات الحدود المتقطعة من الدرجة‬
.‫البحث حصلنا عل المتراجحة العكسية بداللة أعل مقياس يعومة‬
1. Introduction
Kopotun (2003), introduced a paper on k -monotone polynomial and spline
approximation in LP , 0  p   quasi norm . Al-Muhja (2009), discussed the errors
of approximation of k -monotone function by k -monotone interpolation . It turns out
that any two k -monotone functions f and g , whose graphs intersect each other at
certain ( sufficiently many ) points in a, b , have to be "close" to each other in the
sense that f  g P , has to be small .
S r z n 
be the space of all piecewise polynomial function of degree r (order
n
r  1 ) with the knots a partition z n  z i i 0 ,  1  z  z1  ...  z n1  z n  1 of the interval
Let
 1,1 . In other words
s  S r z n 
z i , z i 1  ,
0  i  n  1, s is in
 r , where  r denotes the space of algebraic polynomials of degree  r .
L J  0  p  
As usual , p
,
, denotes the space of all measurable functions f on J ,
such
f L

f
Lp  J 
that
 sup xJ f  x 
J 

if , on each interval
f
,
Lp  J 
where
. We also denote
f
p


p
   f x  dx 
J

: f
L p 1,1
1
p
if
p ,
and
. For k  1,2,3,... , let
 k k 
k i
   1 f x  k h 2  ih , if x  k h 2  J ,
k
 h  f , x, J    i 0  i 
0 ,
o.w,

k  f , x   kh  f , x,  1,1
be the k th symmetric difference , and h
. The k th modulus
f  L p J 
of smoothness of a function
is defined by
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Journal of Babylon University/Pure and Applied Sciences/ No.(3)/ Vol.(19): 2011
k  f , J  p  k  f , J , J p
, and
 k  f , t  p   k  f , t ,  1,1 p
.
The Ditzian-Totik modulus of smoothness which is defined for such an f , as
follows
 k,v  f , t P  sup  v kh   f ,
0 h
P
2
, where  x   1  x 
1
2
.
 0,v  f , t  p   f p
Also , note that
. One of recent results on spline of
approximation due to Zhou (2001), who proved the following theorem .

v
1.1 Theorem A.


2
Let Pn   n  1,1 , n  k  1 . Then for any t  0, n
r
r 
 r Pn , t   t r 1 Pn, t   ...  t Pn , 0  t  n 2 , with the equivalence constant
depending only on r .
Recently , Y. Hu. And Y. Liu (2005), showed that theorem A can be considerably
improved . Their result is stated as follows .
1.2Theorem B.
1
Let r  1 , n  1, 0  t  n and 0  p   . then for any Pn   n
 r Pn , t  p  t r1 Pn, t  , p  t 2 r2 Pn, t  2 , p  ...  t r Pnr   r , p
with the equivalence constants depending only on r and q .
1.1
2. The Main Result.
Using theorems A, B for equivalence of moduli of smoothness in the
0  p   , we shall prove the following theorems :
Lp
spaces ,
2.1Theorem I.
s  Sk I  ), and v  1,2,3 . If s v 1 is
Let n  N , k  1, s be a spline of order k ( i.e. ,
continuous , then for any 0  p   ,
 v s v 
p
 C  p, n  k s, n 1 , I p
2.1
The following Lemma is due to Kopotun (2003) . It will be used to polynomial
approximation of splines .
2.2Lemma C.
Let r  N , k  1, 0  p   , I  a, b , and P   r , and let s be a spline of order
P  s p  C p , r , k  k P  p
k , with at most C k  pieces in I . Then
.
2.3Theorem II.
s  S k I 
Let k  N be a spline of order k (i.e. ,
), and I   1,1 with c  S , for
some constant c . If s is continuous on I , then for any 0  p   ,
k 1 s, n 1 , I p  Ck , p En s p
To prove theorem I, we need the following result :
874
.
2.4Lemma D. [Liu 2005 ]
Let  ,   N be such that   7 , and 1  t  n  1 be a fixed . Then there
exists a polynomial Tt of degree  4  n , such that the following inequalities hold for
xI
Tt k   x   c   t  x  ht k 1  k  
,
.
PROOF OF THEOREM I :


 Q n   i S  x    i  x ; x  I
Pn  x   

; o.w.
Qn
Let
is piecewise polynomial of degree  n ( see
sx   Qn   i x   S k I   C v 1 I 
Al-Muhja 2009 ), and we need the spline
, so

v 
v 
v v 
v
v
 s
  Qn   i 
  Qn   i S x    i
p

p
 C nv I
v

p

S 3v  x    iv 
3
 Q     
v
n
v 0
v
i


,
 v 
not true S we write , we make use of Markov's inequality
3

v  k 1
 v s  v   C  p , v, n  n v I
 Qn   i p S  x    i  x 
p
 C  p, v, n  n 3 I
p
v 0
3 k 1
Qn   i
p
 C p ,v , n s
p
,
we choose a constant K to be sufficiently large , such that
 v s v   K kh,  s,  C sup kh,  s,  C  k s, n 1 ,  1,1 p
p
p

p
0 h  n 1

,
The Remez inequality (1936) is given in the following theorem .
2.5 Theorem E.
2
For any Pn   n , any measurable A   1,1 , with a Lebesgue measure 2  a n for
n2
0a
2 , and 0  p   , we have
some
Pn
p
 C Pn
2.2
Lp  A
Remark : If A   1,1 , with a Lebesgue measure . Note that , using possible
L
theorem I, to satisfying norm
p
 A
.
PROOF OF THEOREM II :
Let k  N , x  I , and
0h I
x
k  1h  I
q
, and suppose that n
peceiwise polynomial which was described in proof theorem I, such that q   s  ,
for some   I .
be such that
Then , for any x  I , we have
k 1 s, I  p  sup kh1 s,, I   2 k 1 kh1 s,, I 
0 h
p
875
p
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Journal of Babylon University/Pure and Applied Sciences/ No.(3)/ Vol.(19): 2011
 kh1 s  q n ,, I 
 ck , p  I s   q n
p
x

p
 2 k 1 I   s u   q n u  du 




1
p

p
,
References
L
Al-Muhja , Malik S. (2009) , On k -Monotone Approximation in p Spaces , M. Sc.
Thesis , University of Kufa .
Kopotun , K. A. (2003) , On k -Monotone Approximation by Free knot Splines ,
Society for Industrial and Applied Mathematics .
Kopotun , K. A. (1996) , Simultaneous Approximation by Algebraic Polynomials ,
Constr. Approx. , Vol., 12, pp.67-94 .
Liu, Y. Hu and Yo. (2005) , On Equivalence of Moduli of Smoothness of Polynomials
L
in p , 0  p   , J. Approx. Theory , Vol., 136, pp.182-197 .
Remez , E. J. (1936) , Surune Propriete Des Polynomes de Tchebycheff , Comm. Inst.
Sci. Kharkov , Vol.,13, pp.93-95 .
Zhou , S. P. (2001) , Some Remarks on Equivalence of Moduli of Smoothness , J.
Approx. Theory , Vol.113, pp.165-171 .
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