Gen. Math. Notes, Vol. 6, No. 2, October 2011, pp. 24-32
ISSN 2219-7184; Copyright © ICSRS Publication, 2011
www.i-csrs.org
Available free online at http://www.geman.in
On Degree of Approximation
by Product Means (E, q )(N , p ) of Fourier
n
Series
Mahendra Misra1, U.K. Misra2, B.P. Padhy3 and M.K. Muduli4
1
Principal, Government Science College
Mlkangiri, Orissa
E-mail: mahendramisra@2007.gmail.com
2
Department of Mathematics, Berhampur University
Berhampur–760007, Orissa, India
E-mail: umakanta_misra@yahoo.com
3
Roland Institute of Technology, Golanthara-760008,
Odisha, India
E-mail: iraady@gmail.com
4
Roland Institute of Technology, Golanthara-760008,
Odisha, India
E-mail: malayamudulimath@gmail.com
(Received: 7-7-11/ Accepted: 14-10-11)
Abstract
In this paper a theorem on degree of Approximation of a function f ∈ Lip α
by product summability (E , q )(N , p n ) of Fourier series associated with f .
Keywords: Degree of Approximation, f ∈ Lip α class of function, (E, q )
mean, (N , p n ) mean, (E , q )(N , p n ) product mean, Fourier series, Lebesgue
integral
On Degree of Approximation…
1
Let
25
Introduction
∑a
n
be a given infinite series with the sequence of partial sums
{p n } be a sequence of positive real numbers such that
n
Pn = ∑ pυ → ∞
(1.1)
{s n } . Let
as n → ∞ , ( P−i = p −i = 0 , i ≥ 0 )
υ =0
The sequence –to-sequence transformation
1
tn =
Pn
(1.2)
n
∑p
υ =0
s
n −υ υ
{t n } of the (N , p n ) -mean of the sequence {s n } generated
by sequence of coefficient {p n }. If
defines the sequence
by the
t n → s , as n → ∞ ,
(1.3)
then the series
∑a
n
is said to be ( N , p n ) summable to s .
The conditions for regularity of Nörlund summability ( N , p n ) are easily seen to
p
be
(i) n → 0 as n → ∞
Pn
n
(ii)
∑p
k =0
k
= O ( Pn ) as n → ∞
The sequence –to-sequence transformation, [1]
Tn =
(1.4)
n
1
n
 q
(1 + q ) υ∑ υ 
n
=0
n −υ
sυ
.
defines the sequence {Tn } of the (E, q ) mean of the sequence {s n } .
If
(1.5)
Tn → s as n → ∞
then the series
∑a
n
is said to be (E, q ) summable to s .
Clearly (E, q ) method is regular.
Further, the (E, q ) transform of the (N , p n ) transform of {s n } is defined by
26
Mahendra Misra et al.
τn =
(1.6)
=
If
(1.7)
then
n
1
n
 q
(1 + q ) ∑  k 
n
n−k
k =0
 n  n−k
  q
∑
k =0  k 
n
1
(1 + q )n
Tk
1

 Pk
k
∑p
υ
=0

s 

k −υ υ
τ n → s as n → ∞
∑a
n
is said to be (E , q )(N , p n ) -summable to s .
Let f (t ) be a periodic function with period 2π, L-integrable over (-π,π), The
Fourier series associated with f at any point x is defined by
(1.8)
∞
∞
a0
f ( x) ~
+ ∑ (a n cos nx + bn sin nx ) ≡ ∑ An ( x)
2 n =1
n =0
Let s n ( f ; x ) be the n-th partial sum of (1.8).
The L∞ -norm of a function f : R → R is defined by
(1.9)
f
∞
= sup{ f ( x) : x ∈ R
}
and the Lυ -norm is defined by
1
(1.10)
f
υ
υ
 2π
υ 
=  ∫ f ( x ) 
0

, υ ≥1
f : R → R by a trigonometric
The degree of approximation of a function
polynomial Pn ( x) of degree n under norm
(1.11)
Pn − f
∞
.
∞
is defined by [4].
= sup{ p n ( x) − f ( x) : x ∈ R
}
and the degree of approximation E n ( f ) of a function f ∈ Lυ is given by
(1.12)
E n ( f ) = min Pn − f
Pn
υ
On Degree of Approximation…
27
This method of approximation is called Trigonometric Fourier approximation.
A function f ∈ Lipα if
(1.13)
( ) ,0 < α ≤ 1
f ( x + t ) − f ( x) = O t
α
We use the following notation throughout this paper :
(1.14)
φ (t ) = f ( x + t ) + f ( x − t ) − 2 f ( x),
and
(1.15)
K n (t ) =
1
n
2π (1 + q )
n
n
k =0
 
∑  k 

 1
q n −k 
 Pk

k
∑p
υ =0
k −υ
1 

sin υ +  t 
2 

.
t

sin

2
Further, the method (E , q )(N , p n ) is assumed to be regular and this case is
supposed through out the paper.
2
Known Theorem
Dealing with The degree of approximation by the product (E , q ) (c,1) -mean of
Fourier series, Nigam [2] proved the following theorem:
Theorem 2.1 If a function f ,2π - periodic, belonging to class Lipα , then its
degree of approximation by (E , q ) (c,1) summability mean on its Fourier series
∞
∑ An (t ) is given by E n cn − f
q 1
n =0
where Enq c1n represents the
3
∞
(E, q )
 1
= o
α
 (n + 1)

 ,0 < α < 1 ,


transform of (c,1) transform of s n ( f ; x ) .
Main Theorem
In this paper, we have proved a theorem on degree of approximation by the
product mean (E , q )(N , p n ) of Fourier series (1.8) .we prove
Theorem 3.1 If f is a 2π − Periodic function of class Lipα , then degree of
approximation by the product (E , q )(N , p n ) summability means on its Fourier
series (1.8) is given by
28
Mahendra Misra et al.
τn − f
4
∞
 1
= O
α
 (n + 1)

 ,0 < α < 1 where τ n on defined in (1.6) .


Required Lemmas
We require the following Lemmas to prove the theorem.
Lemma 4.1
K n (t ) = O(n)
1
n +1
,0 ≤ t ≤
Proof:
For 0 ≤ t ≤
1
, we have sinnt ≤ nsint then
n +1
K n (t ) =
≤
≤
n
 
∑
k =0  k 
n
1
n
2π (1 + q )

n
  n − k  1
  q 
∑
k =0  k 
 Pk

n
1
2π (1 + q )


n−k  1
q 
 Pk

n
2π (1 + q )
n
∑p
υ =0
=0
(
2n + 1) n  n  n − k
 q
≤
n ∑ 
2π (1 + q ) k =0  k 
=O(n).
This proves the lemma.
Lemma 4.2
1
1
K n (t ) = O  , for
≤t ≤π .
n +1
t 
1 

sin υ + t 
2 


t

sin
2

(2υ + 1)sin t 
k
∑p
υ
k −υ
k −υ
1
 n  n−k
  q (2k + 1)
∑
k =0  k 
 Pk
n
1
k
2
t
sin
2
k
p
∑
υ
=0
k −υ






On Degree of Approximation…
29
Proof:
For
Then
1
t t
≤ t ≤ π , we have by Jordan’s lemma, sin   ≥
, sin nt ≤ 1
n +1
2 π
K n (t ) =
≤

 n  n −k  1
  q 
∑
k =0  k 
 Pk

n
1
n
2π (1 + q )
 n  n−k  1
  q 
∑
k =0  k 
 Pk
n
1
2π (1 + q )
n
 n  n −k  1
  q 
=
∑
n
2 (1 + q ) t k =0  k 
 Pk
n
1
k
p
∑
υ
k
∑
υ
=0
π p k −υ 
t
=0
k
p
∑
υ
=0
k −υ
1 

sin υ + t 
2 


t

sin
2 
k −υ



.

 n  n−k
=
∑   q
n
2 (1 + q ) t k =0  k 
1
n
 1
= O  .
 t
This proves the lemma.
5
Proof of Theorem 3.1
Using Riemann –Lebesgue theorem, we have for the n-th partial sum s n ( f ; x ) of
the Fourier series (1.8) of f (x) ,
1

sin  n + t
1
2
s n ( f ; x ) − f ( x) =
φ (t ) 
dt ,
∫
2π 0
t
sin  
2
π
following Titechmarch [3],the (N , p n ) transform of s n ( f ; x ) using (1.2) is
given by
1

sin n +  t
π
n
1
2

t n − f ( x) =
φ (t )∑ p n − k
dt ,
∫
2π Pn 0
t
k =0
sin  
 2
30
Mahendra Misra et al.
Directing the
τn − f =
(E , q )(N , p n ) transform of
s n ( f ; x ) by τ n , we have

 n  n−k  1
  q 
∫0 φ (t )∑
k =0  k 
 Pk

π
1
n
2π (1 + q )
n
k −υ
1 

sin υ + t 
2 

 dt
t

sin
2

k −υ
1 

sin υ + t 
2 

 dt
t

sin
2

k
∑p
υ =0
π
=
∫ φ (t ) K
n
(t ) dt
0
 n1+1 π 


=  ∫ + ∫ φ (t ) K n (t ) dt
1 
0
n +1 

= I 1 + I 2 , say
(5.1)
Now
1
I1 =
1
2π (1 + q )
n
n +1
∫
n
k =0  k 
n
φ (t )∑  
0
≤ O (n)
1
n +1

 1
q n −k 
 Pk

φ ( t ) dt
∫
0
1
n +1
= O ( n) ∫
t α dt
0
α +1
1
n +1
t

= O ( n) 

α + 1 0


1
= O ( n) 
α +1  .
 (α + 1)(n + 1) 
k
∑p
υ =0
, using Lemma 4.1
On Degree of Approximation…
31


1
= O
α +1 
 (n + 1) 
(5.2)
Next
I2 ≤
π
∫
φ ( t ) K n ( t ) dt
1
n +1
π
∫
=
1
n +1
1
 
φ (t ) O   dt , using Lemma 4.2
t
π
∫
=
1
n +1
π
∫
=
t
1
t α O   dt
t
α −1
dt
1
n +1

(5.3)



(
)
n
+
1


1
= O
α
Then from (5.2) and (5.3) , we have


1

τ n − f ( x ) = O
α  , for 0 < α < 1
(
n
+
1
)



τ n − f (x ) ∞ = sup τ n − f (x ) = O
−π < x <π
1
α
 (n + 1)

 ,0 < α < 1


This completes the proof of the theorem.
6
Corollaries
The following corollaries can be derived from our main theorem.
Corollary 6.1 If p n = 1 , ∀ n ∈ N , theorem 2.1 follows from theorem 3.1.
.
32
Mahendra Misra et al.
Corollary 6.2 If p n = 1 , ∀ n and q = 1 then the theorem 3.1 reduces to degree
of approximation for (E,1) (C,1) method of Fourier series.
References
[1]
[2]
[3]
[4]
G.H. Hardy, Divergent Series (1st edition), Oxford University Press,
(1970).
H.K. Nigam and K. Sharma, The degree of approximation by product
means, Ultra Scientist, 22(3) (2010), 889-894.
E.C. Titechmalch, The Theory of Functions, Oxford University Press,
(1939).
A. Zggmund, Trigonometric Series (2nd Edition, Vol. I), Cambridge
University Press, Cambridge, (1959).