Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 22-30
ISSN 2219-7184; Copyright © ICSRS Publication, 2012
www.i-csrs.org
Available free online at http://www.geman.in
On Degree of Approximation of Fourier Series
by Product Means
S.K. Paikray1, R.K. Jati2, U.K. Misra3 and N.C. Sahoo4
1
P.G. Department of Mathematics
Ravenshaw University, Cuttack-753003
Odisha, India
E-mail: spaikray2001@yahoo.com
2
Department of Mathematics
DRIEMS, Tangi, Cuttack, Odisha, India
E-mail: rkjati1980@gmail.com
3
Department of Mathematics
National Institute of Science and Technology
Pallur Hills, Golanthara-761008, Odisha, India
E-mail: umakanta_misra@yahoo.com
4
Department of Mathematics
S.B. Women’s College (Auto)
Cuttack, Odisha, India
E-mail: kumarsahoo.chandan@gmail.com
(Received: 7-9-12 / Accepted: 12-11-12)
Abstract
(
)
In this paper a theorem on degree of approximation of a function
f ∈ Lip (α , r ) by product summability (E , q ) N , p n of Fourier series associated
with f has been established.
On Degree of Approximation of Fourier Series...
23
Keywords: Degree of Approximation, Lip (α , r ) class of function, (E, q )
mean, N, pn mean, (E , q ) N , p n product mean, Fourier series, Lebesgue
integral.
(
1
Let
(
)
)
Introduction
∑a
n
be a given infinite series with the sequence of partial sums {s n } .
Let {p n } be a sequence of positive real numbers such that
n
Pn = ∑ pυ → ∞
(1.1)
, n → ∞ , ( P−i = p −i = 0 , i ≥ 0 ).
υ =0
The sequence –to-sequence transformation
1
tn =
Pn
(1.2)
n
∑ pυ sυ ,
υ =0
(
)
defines the sequence {t n } of the N, pn -mean of the sequence
by the sequence of coefficient {p n }. If
t n → s ,as n → ∞ ,
(1.3)
then the series
{s n } generated
∑a
n
(
)
is said to be N, pn summable to s .
The conditions for regularity of
(N, p )- summability are easily seen to be
n
(i ) Pn → ∞ , as n → ∞,

n

(
ii
)
pi ≤ C Pn , as n → ∞.
∑

i =0

(1.4)
The sequence –to-sequence transformation, [1]
Tn =
(1.5)
1
(1 + q )n
 n  n −υ
∑   q sυ ,
υ = 0 υ 
n
defines the sequence {Tn } of the (E, q ) mean of the sequence {s n } .
If
(1.6)
Tn → s , as n → ∞ ,
then the series
∑a
n
is said to be (E, q ) summable to s .
24
S.K. Paikray et al.
(
Clearly (E, q ) method is regular. Further, the (E, q ) transform of the N, pn
transform of {s n } is defined by
τn =
(1.7)
=
If
(1.8)
then
n
1
n
 q
(1 + q ) ∑  k 
n
n−k
(1 + q )
n
tk
k =0
 n  n−k
  q
∑
k =0  k 
n
1
)
1

 Pk
k

=0

∑ pυ sυ 
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 and integrable in the sense of
Lebesgue over (-π,π) . Then the Fourier series associated with f at any point x is
defined by
∞
a0 ∞
f ( x) ~
+ ∑ (a n cos nx + bn sin nx ) ≡ ∑ An ( x ) .
(1.9)
2 n =1
n =0
Let s n ( f ; x ) be the n-th partial sum of (1.9). The L∞ -norm of a function
f : R → R is defined by
f
(1.10)
∞
= sup{ f ( x) : x ∈ R
}
and the Lυ -norm is defined by
1
2π
(1.11)
f
υ

υ 
=  ∫ f ( x ) 
0

υ
, υ ≥ 1.
The degree of approximation of a function f : R → R by a trigonometric
polynomial Pn ( x) of degree n under norm
(1.12)
Pn − f
∞
.
is defined by [5]
∞
= sup{ p n ( x) − f ( x) : x ∈ R
}
and the degree of approximation E n ( f ) of a function f ∈ Lυ is given by
(1.13)
E n ( f ) = min Pn − f
Pn
υ
.
This method of approximation is called Trigonometric Fourier approximation.
A function f ( x ) ∈ Lip α if
On Degree of Approximation of Fourier Series...
and f ( x ) ∈ Lip (α , r ) , for 0 ≤ x ≤ 2π , if
1
(1.15)
( ) , 0 < α ≤ 1, t > 0
f ( x + t ) − f ( x) = O t
(1.14)
25
α
 2π
r
r
α
 ∫ f ( x + t ) − f ( x) dx  = O t
0

( ) , 0 < α ≤ 1, r ≥1, t > 0 .
We use the following notation throughout this paper:
φ (t ) = f ( x + t ) + f ( x − t ) − 2 f ( x),
(1.16)
and
1
n
2π (1 + q )
K n (t ) =
(1.17)
(
n
n
k =0
 
∑  k 

 1
q n −k 
 Pk

1 

sin υ +  t 
2 

pυ
.
∑
t
υ =0

sin

2
k
)
Further, the method (E , q ) N , p n is assumed to be regular through out the paper.
2
Known Theorems
Dealing with The degree of approximation by the product (E , q ) (C ,1) -mean of
Fourier series, Nigam et al [3] proved the following theorem:
Theorem 2.1: If a function f ,2π - periodic, belonging to class Lip α , then its
degree of approximation by (E , q ) (C ,1) summability mean on its Fourier series
∞
∑ An (t ) is given by En Cn − f
q
1
n =0
where E nq C n1 represents the
∞
(E, q )
 1
= O
α
 (n + 1)

 ,0 < α < 1 ,


transform of (C ,1) transform of s n ( f ; x ) .
Recently, Misra et al [2] proved the following theorem using
of Fourier series:
(E , q )(N , p n ) mean
Theorem 2.2: If f is a 2π − periodic function of class Lip α , then degree of
(
)
approximation by the product (E , q ) N , p n summability means of its Fourier
series (1.9) is given by τ n − f
defined in (1.7) .
∞
 1
= O
α
 (n + 1)

 ,0 < α < 1


where τ n is as
26
S.K. Paikray et al.
3
Main Theorem
In this paper, we have proved a theorem on degree of approximation by the
product mean (E , q ) N , p n
of the Fourier series of a function of class
Lip (α , r ) . We prove:
(
)
Theorem 3.1: If f is a 2π − Periodic function of class Lip α , then degree of
(
)
approximation by the product (E , q ) N , p n summability means on its Fourier
series (1.9) is given by τ n − f
∞

1
= O
1

α−
r
n
1
+
(
)


 , 0 < α < 1, r ≥ 1
, where τ n is


as defined in (1.7).
4
Required Lemmas
We require the following Lemma for the proof the theorem.
Lemma 4.1:
1

 O(n) ,0 ≤ t ≤ n + 1
K n (t ) = 
1
1
O  ,
≤t ≤π
  t  n + 1
.
Proof:
For 0 ≤ t ≤
1
, we have sin nt ≤ n sin t , then
n +1
K n (t ) =
1
n
2π (1 + q )
1
≤
n
2π (1 + q )

n
 n  n −k  1
  q 
∑
k =0  k 
 Pk


 n  n−k  1
  q 
∑
k =0  k 
 Pk

1
≤
n
2π (1 + q )
n
1 

υ
sin
+

t
k
2  


∑ pυ
t
υ =0

sin
2 
k
∑ pυ
υ
=0
(2υ + 1)sin t 
2
t
sin
2
1
 n  n−k
  q (2k + 1)
∑
k =0  k 
 Pk
n



k

=0

pυ 
∑
υ
On Degree of Approximation of Fourier Series...
27
(
2n + 1) n  n  n − k
 q
≤
n ∑ 
2π (1 + q ) k =0  k 
= O(n)
1
t t
≤ t ≤ π , by Jordan’s lemma we have, sin   ≥
, sin nt ≤ 1 .
n +1
2 π
Then
For
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  P
n
2 (1 + q ) t k =0  k 
 k
n
1
=
1 

sinυ + t 
2 

pυ

∑
t
υ =0

sin

2
k
k
∑
υ
=0
π pυ 
t



p
∑ υ .
υ =0

k
 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 (1.9) of f (x) ,
1

sin  n + t
1
2
s n ( f ; x ) − f ( x) =
φ (t ) 
dt .
∫
2π 0
t
sin  
2
π
28
S.K. Paikray et al.
(
)
Following Titchmarch [4], 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 )∑ pk
dt ,
∫
2π Pn 0
t
k =0
sin  
2
(
)
Writing the (E , q ) N , p n transform of s n ( f ; x ) by τ n , we have
τn − f =
1
n
2π (1 + q )

 n  n −k  1
  q 
∫0 φ (t )∑
k =0  k 
 Pk

π
n
1 

sin υ + t 
2 

pυ
 dt
∑
t
υ =0

sin
2 
k
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
n
2π (1 + q )
≤
1
n +1
∫

n
 n  n− k  1
φ (t )∑   q 
k =0  k 
 Pk

n +1
∫
0
1 

sin
υ
+

t
k
2  

 dt
∑ pυ
t
υ =0

sin
2 
φ (t ) K ( t ) dt
0
1
1
 n1+1
 r  n1+1
s
r
s

 

=  ∫ (φ ( t ) ) dt   ∫ ( K n ( t ) ) dt  , using Holder’s inequality
 0
  0


 

On Degree of Approximation of Fourier Series...


= O(
)
α
( n + 1) 

1
1
n +1
∫
0


n s dt 


29
1
s
1
 ns  s
1
= O(
)

α 
n
+
1
( )  n +1

n
= O
1

α+
s
+
1
n
(
)



 .


1
= O
1

α +1−
+
1
n
) r
(
(5.2)




Next
I2


≤ 


π
∫ (φ ( t ) )
1
n +1
r


dt 


1
r





π
∫ ( K (t ))
n
1
n +1
 π

s
 1 
= O(
)
  dt
α
( n + 1)  ∫1  t  
 n +1

1
 − s +1 π  s
t  1 
= O(
)
α 
( n + 1)    n+1 
 1 
= O(
)

α 
( n + 1)  n + 1 
1

1

O
=
1

α −1+
s
n
+
1
(
)

1− s
s




s


d t  ,using Holder’s inequality


1
s
1
1
1
s
, using Lemma 4.1
30
S.K. Paikray et al.

1
= O
 (n + 1)α − 1r





Then from (5.2) and (5.3), we have

τ n − f ( x ) = O

1
α−
 (n + 1) r
1

 , for 0 < α < 1, r ≥1 .


Hence,
This completes the proof of the theorem.
References
[1]
[2]
[3]
[4]
[5]
G.H. Hardy, Divergent Series (First Edition), Oxford University Press,
(1970).
U.K. Misra, M. Misra, B.P. Padhy and S.K. Buxi, On degree of
(E , q ) N , p n of Fourier series,
approximation by product mean
International Jour. of Math. Sciences, Technology and Humanities,
22(2012), 213-220.
H.K. Nigam and A. Sharma, On degree of approximation by product
means, Ultra Scientist of Physical Sciences, 22(3) (M) (2010), 889-894.
E.C. Titchmarch, The Theory of Functions, Oxford University Press,
(1939).
A. Zygmund, Trigonometric Series (Second Edition) (Vol. I), Cambridge
University Press, Cambridge, (1959).
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)