Gen. Math. Notes, Vol. 21, No. 2, April 2014, pp. 95-103
ISSN 2219-7184; Copyright © ICSRS Publication, 2014
www.i-csrs.org
Available free online at http://www.geman.in
Best One-Sided Approximation of Entire
Functions in Lp,w Spaces
S.K. Jassim1 and Alaa A. Auad2
1
Department of Mathematics, College of Science
University of Al-Mustansiry
E-mail: dr.sahebmath@yahoo.com
2
Department of Mathematics
College of Education of Pure Science
University of Al-Anbar
E-mail: alaaadnan2000@yahoo.com
(Received: 28-12-13 / Accepted: 29-1-14)
Abstract
In this paper, we estimate the degree of best one-sided approximation of an
entire unbounded function by using some discrete operator in Lp,w weighted space
( 1 ≤ p ˂ ∞ ). We construct new theorems regarding by Jackson polynomials and
Valee-Poussin operator of these functions.
Keywords: Entire unbounded function, discrete operator and weight space.
1
Introduction
Let X = [0, 1], we denoted by L∞(X) the set of all bounded measurable functions
with usual norm [5]:
‖ ‖∞ = sup | ( )| ∶
(1.1)
96
S.K. Jassim et al.
For 1 ≤ p ˂ ∞, let Lp = {f: f is bounded measurable function for which
‖ ‖ = (
| ( )|
) <∞
(1.2)
Further for δ > 0 and (1 ≤ p ˂ ∞) the locally global norm of f is defined
‖ ‖
,
(sup | ( )| ∶
= (
∈
− , + !)
)
(1.3)
Now, let W be the set of all weight functions on X. Consider Bw the set of all
functions f on X such that | ( )| ≤ #$( ) integrable, that is
→ *; | ( )| ≤ #$( ),- ‖ ‖
%& = ' :
,&
=(
.&(0).
/(0)
) < ∞1
for δ > 0 and (1 ≤ p ˂ ∞ ), the weighted locally global norm of f ∈Bw is defined by
‖ ‖
, ,&
= (
(sup .&(2). ∶
/(2)
∈
− , + !)
) < ∞.
(1.4)
The kth average modulus of f∈ Lp with respect to algebraic polynomials from Lp
space is given by [7]:
34 ( , 5) = ‖6( , . , 5)‖
, 5>0
(1.5)
Where
64 ( , , 5) = :;< =∆4? | (@)|: @ ∈
− , − ! , ℎ ≤ 5B . 5 > 0
(1.6)
B ,
(1.7)
?
?
The kthLp,w modulus of smoothness for f ∈ Bw is define by
64 ( , 5)
,&
= :;< =C∆4? C
,&
I
∆4? (@) = ∑4GL (−1)4FG H K H +
J
where
4?
K
Now, we shall introduce the average modulus of smoothness for f ∈ Bw. The kth
average modulus of smoothness for f ∈ Bw is define by
34 ( , 5)
,&
= ‖64 ( , . , 5)‖
,&
,
(1.8)
where
64 ( , 5)
,&
= :;< =M∆4? (@)/$(@)M: @ ∈
−
4?
, −
4?
! , ℎ ≤ 5B .
(1.9)
Best One-Sided Approximation of Entire…
97
The degree of best weighted approximation to a given function f ∈ Bw with
trigonometric polynomials or algebraic polynomials on the interval X is given:
OP ( )
,&
= infT‖ − U‖
,&
∶ U ∈ VP W ,
(1.10)
Where Pn denote the set of all trigonometric polynomials or algebraic polynomials
of degree ≤ n.
Further, the degree of best one-sided weighted approximation of f ∈ Bw with
respect to trigonometric polynomials or algebraic polynomials of degree n on X
given by [1]:
ẼP ( )
,&
= minY∈Z T‖< − U‖
, <, U ∈ VP ,- U( ) ≤ ( ) ≤ <( )W
,&
(1.11)
Now, let us consider the Dirichlet kernel of degree n [7]
[P (;) =
Let aP (;) =
+ ∑PGL cos(J;)
Pb
, ; ∈ * , - = 1,2,3, …
[[ (;) + [ (;) + … + [P (;)]
(1.12)
(1.13)
be Fejer kernel of degree least than or equal n.
Let eP ( , ) =
4,P
where
∑P4L
f
4,P gaP (
−
4,P )
(1.14)
= Pb , k = 0,1,…,n be the so called Jackson polynomial of function
4h
f ∈Bw, and i P (@) =
j
Pb
Pb
h
[[P (@) + [Pb (@) + … + [ P (@)], Let
= lPbk , j = 0,1,2,…,3n. Then
i P,lP ( , ) =
lPb
∑lP
jL
f j gi P ( −
j)
(1.15)
be the Valee-poussin operator.
The unique linear trigonometric polynomial which is interpolating a given
function f ∈ Bw at the point xi is denoted by In(t) which has the following
representation:
mP ( , ) =
Pb
∑GLP
( G )[P ( −
G ).
(1.16)
98
S.K. Jassim et al.
Now, let B̿ w be the set of all entire functions, since the derivative of polynomial
exists everywhere , then we obtain every polynomial is an entire function, so we
consider that f ∈B̿ w , Jn(f)∈B̿ w , V2n,3n(f)∈B̿ w and In(f)∈B̿ w .
Further in (2002) Jiasong Deng [1] develop an analytic solution for the best one
sided approximation of polynomial under L1 norm, also (2005) Friedrich Littman
[4] estimate the error of one sided approximation of function f: R→R by entire
functions of exponential type δ > 0 in Lp space, (1 ≤ p ˂ ∞) and in (2011) S.K.
Jassim and N.J. Mohamed [3] evaluate the degree of the approximation of entire
function by some discrete operators in locally global quasi-norms ( Lδ,p-space ).
2
Auxiliary Theorems
In this section we introduce the results that we make use of in our lemmas.
Lemma (1) [3] If f ∈ 2n-periodic bounded measurable function, we have for
(0˂p ≤ 1) ‖ − Jp ( )‖q ≤ c(p)τ ( , p)q .
(2.1)
(0 ˂ p ≤ 1) C − i P,lP ( )C ≤ r(<, I, s)34 ( , P)
(2.2)
Lemma (2) [3] If f ∈ 2n-periodic bounded measurable function, we have for
where n = 1,2,… , p,k and l are constant depends on p.
Lemma (3) [3] If f ∈ 2n-periodic bounded measurable function, we have then for
(0˂ p ≤ 1) ‖ − Ip ( )‖q ≤ c(p, k, l)τw ( , p)q
where n = 1,2,… ,
(2.3)
p,k and l are constant depends on p.
Lemma (4) [6] If f is bounded measurable function on [a, b], then we have
z
{
(x)dx =
zF{
p
where x| = a +
∑p|L
(x| )
(2.4)
(zF{)( |F )
p
and 1 ≤ i ≤ n.
Lemma (5) [2] If f is bounded measurable function, then for (0 ˂ p ˂ 1) and δ > 0,
we have
‖ ‖
,
= r(<)(1 + -5)
F
(-:) ‖ ‖ .
(2.5)
Best One-Sided Approximation of Entire…
3
99
Main Results:
In this section we introduce our main results
Theorem (1): Let f ∈B̿ w, then for (1 ≤ p ˂ ∞). Then
τw (f, )q,~ ≤ τw (f, )δ,q,~
p
p
(3.1)
Theorem (2): Let f ∈B̿ w, then for (1 ≤ p ˂ ∞). Then
‖ − Jp ( )‖δ,q,~ ≤ c(p)τw H , K
p δ,q,~
(3.2)
Theorem (3): Let f ∈B̿ w, for (1 ≤ p ˂ ∞). Then
C −V
p,lp (
)Cδ,q~ ≤ c(p, k, l)τw ( , p)δ,q,~
(3.3)
Theorem (4): Let f ∈B̿ w, for (1 ≤ p ˂ ∞). Then
‖ − Ip ( )‖δ,q,~ ≤ c(p, k, l)τw ( , )δ,q,~
p
(3.4)
To prove our results we need the following lemmas:
Lemma (A): Let f ∈B̿ w, then for (1 ≤ p ˂ ∞) and δ > 0. Then
‖ ‖
Proof:
‖ ‖
We have
,&
,&
≤‖ ‖
=H
, ,&
.&(0).
/(0)
≤ r(<)(1 + -5) ‖ ‖
K
≤ r(<) €• ‚
= €• sup ‚
= ( • (sup | ( )/$( )| ∶
=‖ ‖
, ,& .
We need to prove that ‖ ‖
, ,&
( )
‚
$( )
( )
‚
$( )
,& .
ƒ
ƒ
5
5
∈„ − , + … )
2
2
≤ r(<)(1 + -5) ‖ ‖
,&
)
100
S.K. Jassim et al.
We have
‖ ‖
, ,&
5
5
∈„ − , + … )
2
2
= ( • (sup | ( )/$( )| ∶
)
By using (2.4) we get
P
1
= ( † sup '| ( )/$( )| ,
G
Suppose that
,
G
+ ]
GL
∈ [ − , + ] such that
Thus
‖ ‖
, ,&
P
,&
‖ ‖
We get
/(0‡ )
&(0‡ )
P
P
GL
GL
/(2‡ )
B
&(2‡ )
= :;< =
)
and
GL
= r(<) €• ‚
≤ r(<)(1 + -5) ‖ ‖
, ,&
,&
≤ r(<)(1 + -5) ‖ ‖
( )
‚
$( )
,&
∈[
G
−
ƒ
, for δ > 0 and 1≤ p ˂ ∞
,&
.
Lemma (B): Let f ∈B̿ w, then for (1 ≤ p ˂ ∞) and δ > 0. Then
‖ − eP ( )‖
G
( G)
( G)
1
1
≤ ( †|
| ) ≤ ( † sup |
| )
$( G )
$( G )
( G)
r(<)
=(
†|
| )
$( G )
-
= r(<)‖ ‖
5
5
∈ [ − , − ]1
2
2
≤ r(<)3 H , PK
,&
■
, where c is constant depends on p.
Proof:
From (2.1), we have for
Now, for f ∈B̿ w
‖ − eP ( )‖
,&
= ‰& −
/
∈ ˆ ‖ − eP ( )‖ ≤ r(<)3 ( , P)
Š(/)
&
‰ where $ ∈ ‹
From definition of f ∈B̿ w we get
obtain & − eP (&) is integrable.
/
/
/
&
is integrabl and also eP (&) integrable, we
/
Best One-Sided Approximation of Entire…
e( )
Œ −
Œ
$
$
Thus
Hence ‖ − eP ( )‖
,&
101
1
1
≤ r(<)3 • , Ž = r(<)3 ( , )
$ -
≤ r(<)3 H , K
P
,&
,&
Lemma (C): Let f ∈B̿ w, then for (1 ≤ p ˂ ∞) and δ > 0, we have
■
1
‖ − Ip ( )‖q,~ ≤ c(p, k, l)τw ( , )q,~
n
Prove of this lemma the same to proof of lemma (B).
We shall prove the theorems (3.1), (3.2), (3,3) and (3,4) to find the degree of
weighted approximation of unbounded entire function by some discrete operators
in weighted spaces Lp,w, (1 ≤ p ˂ ∞).
4
The Proof of Theorem (1)
We have
34 ( , 5)
,&
= •:;< •‚∆4?
4
= ‖64 ( , . , 5)‖
,&
(@)
Iℎ
Iℎ
‚ : @, @ + Iℎ ∈ „ −
, − … , ℎ ≤ 5‘•
$(@)
2
2
,&
I
I
I (@ + Jℎ)
= ’• sup “”†(−1)Gb4 H K
” , @, @ + Iℎ ∈ [ −
, − ]• @–
J $(@ + Jℎ)
22GL
4
I (@ + Jℎ)
≤ ’• sup “sup “”†(−1)Gb4 H K
” , @, @ + Iℎ
J $(@ + Jℎ)
GL
∈„ −
= •:;< •‚∆4?
I
I
, − …—• @–
22-
(@)
Iℎ
Iℎ
‚ : @, @ + Iℎ ∈ „ −
, − … , ℎ ≤ 5‘•
$(@)
2
2
= ‖64 ( , . , 5)‖
, ,&
= 34 ( , 5)
, ,&
, ,&
We obtain (3.1) hold.
■
102
5
S.K. Jassim et al.
The Proof of Theorem (2)
From lemma (A) we obtain
‖ − eP ( )‖
, ,&
Since δ > 0, then
≤ r(<)(1 + -5) ‖ − eP ( )‖
‖ − eP ( )‖
, ,&
,&
,
≤ r (<)‖ − eP ( )‖
,&
Now, by using lemma (B) and (3.1) respectively, we obtain
r (<)‖ − eP ( )‖
We obtain (3.2) hold.
,&
1
≤ r (<)34 ( , )
-
,&
1
≤ r (<)34 ( , )
-
, ,&
■
6
The Proof of Theorem (3.3)
From lemma (A), we obtain
C − i P,lP ( )C
, &
≤ r(<)(1 + -5) C − i P,lP ( )C
Since δ > 0, then
C − i P,lP ( )C
, &
≤ r (<)C − i P,lP ( )C
Also from theorem (2.2), we obtain
C − i P,lP ( )C
, &
So from theorem (3.1), we have
1
r (<, I, s)34 • , Ž
-
,&
1
≤ r (<, I, s)34 ( , )
1
≤ r(<, I, s)34 • , Ž
-
,&
,&
, ,&
We obtain (3.3) hold.
7
The Proof of Theorem (3.4)
From lemma (A), we obtain
‖ − mP ( )‖
, ,&
≤ r(<)(1 + -5) ‖ − mP ( )‖
,&
,&
Best One-Sided Approximation of Entire…
Since δ > 0, then
‖ − mP ( )‖
, ,&
103
≤ r (<)‖ − mP ( )‖
From (2.3) and (3.1) respectively, we obtain
‖ − mP ( )‖
, ,&
1
≤ r(<, I, s)34 • , Ž
-
,&
,&
1
≤ r(<, I, s)34 • , Ž
-
, ,&
We obtain (3.4) hold.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
J. Deng, Y. Feng and Falaichen, Best one-sided approximation of
polynomials under L1 norm, Journal of Computational and Applied Math.,
144(2002), 161-174.
D. Dranov, Eqi converge and Eqi approximation for entire functions,
constructive theory of functions 91, International Conference, Varna, May
28- June 3, (1991).
S.K. Jassim and N.J. Mohmamed, Approximation of entire function in
locally global norm, J. Ibn Al-Hathiam for Pure and Applied Science,
24(3) (2011), 238-243.
F. Littmann, One-sided approximation by entire functions, Journal of
Approximation Theory, 141(2006), 1-8.
G.G. Lorentz, Approximation of Functions, Printed in U.S.A., (1966).
B. Sendov and V.A. Popov, Average Modulus of Smoothness, Sofa,
(1983).
A. Zygmand, Trigonometric Series, Cambridge, UK, (1988).