International Journal on Advanced Computer Theory and Engineering (IJACTE)
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On Degree of approximation by Product Means of the Fourier Series
of a function of Lipchitz class
1
R. K. Jati, 2S. K. Paikray, 3U.K.Misra
Department of Mathematics DRIEMS Tangi, Cuttack, Odisha, India
Department of MathematicsVSSUT,Burla-768018, Odisha, India
Department of Mathematics, NIST, Golanthara-761008, Odisha, India
Email: spaikray2001@yahoo.com, umakanta_misra@yahoo.com
Abstract: This paper provides a new approach to a
theorem on degree of approximation of Fourier series of
f Lip  , r 
a function
 E, s  N , pn , qn 
by
product summability.
Keywords: Degree of Approximation,
class of function,
E, q
mean,
 E, s  N , pn , qn  product
(i)
p n  q
 0 for each integer   0 as n  
rn
and
(4)
n
Lip  , r 
p
(ii)
 N , pn , qn  mean,
mean, Fourier series,
 0
2010-Mathematics subject classification: 42B05, 42B08.
I. INTRODUCTION:
Let
a
n
sn  . Let pn and qn  be
n
Let
tn 
1 n
 pn q s
rn  0
n 
t n is called the sequence of ( N , pn , qn ) mean
of the sequence s n  . If t n  s , as n  
(3)
n
is said to be

1  q 
of the
n
(6)
 E, q 
mean of the
(7)
is said to be
1  q 
If
n
n
k 0
 
 k  q
n
nk
E, q summable to
n
 n  nk  1

 q 
k 0  k 
 rk
n  s
a
n
, as
the
 N , pn , qn 
tk
(8)
n
1
then
( N , pn , qn )
summable to s .
The necessary and sufficient conditions for regularity
of
a
1
,
Then
a
Tn 
 E, q  transform of
transform of s n  is defined by
(2)
rn  p0 qn  p1qn1    pn q0 ( 0)
then the series
s
 
0
Further, the
where
p1  q1  r1  0 .
n 
s .Clearly E, q  method is regular[5].
(1)
 0
 0
   q

sn  . If
then the series
n
Qn   q
and
n
Tn  s , as n   ,
sequences
of positive real numbers such that
Pn   p
1  q 
sequence
n
n
1
defines the sequence
be a given infinite series with sequence of
partial sums
q  H rn where H is a positive
number independent of n .
(5)
The sequence –to-sequence transformation [5],
Tn 
Lebesgueintegral .
n 
k
p

0
n ,
is said to be

qs 

k   
(9)
 E, q  N , pn , qn  -
summable to s .
Let
f (t ) be a periodic function with period 2 , L-
integrable over (-,), The Fourier series associated
( N , pn , qn ) method are[3 ]:
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International Journal on Advanced Computer Theory and Engineering (IJACTE)
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f
with
at
any
point
x is
defined
a0 
  an cos nx  bn sin nx
2 n 1
f ( x) ~
by
(10)

  An ( x)
n 0
Let
s n  f ; x  be the n-th partial sum of (1.10).The
L -norm of a function f : R  R is defined by
f   sup  f ( x) : x  R 
and
L
the
-norm
(11)
defined
is
by
direction Sahney and Rao[11], and Khan[6] have
established results on the degree of approximation of
the function belonging to the class Lip and
Lip  , r  by
 N , pn 
and
 N , pn , qn 
respectively. However, dealing with product
summability Nigam et al [9] proved the following
theorem on the degree of approximation by the product
E, q C,1 -mean of Fourier series.

 
Theorem 2.1:
If a function f is 2 - periodic and of class Lip ,
then its degree of approximation by
E, q C,1
1
f
The



 2
 
   f x  
0

,
summability mean on its Fourier series
 1 .(12)
Pn  f


is defined by [11]
 sup  Pn ( x)  f ( x) : x  R
and the degree of approximation
Pn
q
 (13)
En ( f ) of a function
.

1
r


r

, 0    1, r 1, t  0
  f ( x  t )  f ( x) dx   O t
0

We use the following notation throughout this paper:
 
 (t )  f ( x  t )  f ( x  t )  2 f ( x),
(17)
and

 1 
sin    t 
k

 n  nk  1
1
2 .
K n (t ) 
s   pk  q 

n  
t
2 1  s  k 0  k 
 rk  0

sin
2 

n
Further, the method
 E, q  N , pn , qn 
transform of
E, q N , pn  of the Fourier series:
2  Periodic function of class Lip  , r  ,
If f is a
then degree
of approximation by the
E, q N , pn  summability
(defined
n  f
where

n
means on its Fourier
above)

1
 O
1


r
n

1



product
is
given
by

 , 0    1, r 1


,
as defined in (8) .
Extending the above result Paikray et al [14]
established a theorem on degree of approximation by
the product mean
E, qN , pn 
of the conjugate
series of Fourier series of a function of
Lip  , r  . The result is as below:
class
Theorem -2.3:
is assumed
to be regular and this case is supposed throughout the
paper.
II. KNOWN THEOREMS:
Bernestein[2], Alexits[1], Sahney and Goel[10],
Chandra[4] and several others have determined the
degree of approximation of the Fourier series of the
function f  Lip by
E, q
Subsequently Misra et al [8] have established the
following theorem on degree of approximation by the
series
f Lip  , r  , for 0  x  2 , if [7]
2
represents the

 ,0    1


Theorem 2.2:
(14)
  ,0    1.(15)
and
1
product mean


 1
 O

 n  1
C,1 transform of s n  f ; x  .
This method of approximation is called Trigonometric
Fourier approximation. A function f  Lip if[7]
f ( x  t )  f ( x)  O t
Enq Cn1  f
,where E n C n
f  L is given by[12]
En ( f )  min Pn  f
n
given by
of approximation of a function
f : R  R by a trigonometric polynomial Pn (x) of
.
 A (t ) is
n 0
degree
degree n under norm
means
 C,1 , C,   ,  N , pn 
If f is a
2  Periodic function of class Lip  , r 
, then degree of approximation by the product
E, qN , pn summability
means of on the
conjugate series (10) of the Fourier series (9) is given
by
and
 N , p  means. Subsequently, working on the same
n
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International Journal on Advanced Computer Theory and Engineering (IJACTE)
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n  f


1
 O
1


r
n

1




 , 0    1, r 1 ,


where  n is as defined in (7).
Denoting
III. MAIN THEOREM:
In this paper, we have studied a theorem on degree of
approximation
by
the
product
mean
 E, s  N , pn , qn 
function of class
of the Fourier series of a
Lip , r  . We prove:
f is a
2 
Periodic function of the class
Lip  , l  , then degree of approximation by the
product
its
 E, s  N , pn , qn  summability
Fourier
n  f

series
(1.9)
is

 1 
sin    t 
k

n  1
1
2 
 (t )   s nk   pk  q 
dt
n 
t 
2 1  s  0 k 0  k   rk  0
sin
2 

   (t ) K n (t ) dt
0
 n11  


      (t ) K n (t ) dt
1 
0
n 1 

 I1  I 2 , say
We require the following Lemma for the proof the
theorem.
,0  t 
1
.
n 1
1
1
K n (t )  O  , for
t  .
n 1
t 
1

sin  n  t
1
2
s n  f ; x   f ( x) 
 (t ) 
dt

2 0
t
sin  
2

 N , pn , qn  transform of

0

 1 
sin    t 
 n  n  k  1 k
2 
 (t )   s   pk  q 
 dt
t
k
r
k 0  
 k  0

sin
2 

n
1
n 1
   t  K  t  dt
n
1
f (x) and following Titchmarch [12], we have
Using (2), the
n
n 1
0
5. Proof of Theorem 3.1:
Using Riemann –Lebesgue theorem, for the nth partial sum s n  f ; x  of the Fourier series (10) of
.
1
1

Lemma-4.2:[14]
(18)
I1 
2 1  s 
Lemma -4.1:[14]
K n (t )  O(n)
n



1

 , 0    1, l  1.
O
1

 

  n  1 l 
IV. REQUIRED LEMMAS:
of
n  f 
by
, where  n is as defined in (8).
transform
s n  f ; x  by  n , we have
means on
given
n
 E, q  N , p, q 
the

Theorem -3.1:
If
1

sin  n   t
1
2

tn  f ( x ) 
 (t ) pn k q
dt

2 rn 0
t
k 0
sin  
2

1
 n11
 l  n11
m
l  
m
1 1

     t     K n  t   , where   1
l m
 0
  0


 

, using Holder’s inequality


 O(
)

 n  1 

1
1
n 1

0


n m dt 


1
m
s n  f ; x  is
given by
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International Journal on Advanced Computer Theory and Engineering (IJACTE)
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
1
 O
1

1
m
n

1



REFERENCES
1
 1
 O
  n  1

  nm  m

  n  1 




1


 O
1 


  n  1  l 



(19)
Next
1
I2
[1]
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function durch die CesarochenMittel in hrer
Fourierreihe, Math. Annal 100(1928), 264-277
[2]
S.Bernstein :Sur l’ order de la Melleure
approximation des function continue par des
polynomes de degree’ donne’e, Memories
Acad. Roy-Belyique 4(1912), 1-104.
[3]
D.Borwein :On product of sequences, Journal of
London Mathematical Society, 33(1958), 352357.
[4]
P.Chandra :On degree of approximation of
functions belonging to Lipchitz class,Nanta
Math. 80(1970), 88-89.
[5]
G.H. Hardy: Divergent series, First edition,
Oxford University press 70(19).
[6]
Huzoor H. Khan :On degree of approximation
of function belonging to the class , Indian
Journal of pure and applied Mathematics,
13(1982), 132-136.
[7]
L. McFadden :Absolute N orlundsummabilty,
Duke Maths. Journal, 9(1942),168-207.
[8]
U.K.Misra, M. Misra, B.P. Padhy and M.K.
Muduli: On degree of approximation by product
1
 
l  
m
l
m

 

     t  dt    K n  t  dt 
 1
  1

 n 1
  n 1

using Holder’s inequality, as above.
1
m
 1
 O
  n  1



    1 m 
     dt 
 1 t 


 n 1

 1
 O
  n  1

   m 1   m
 1 
  t

n 1 

using Lemma 4.2
 1  1 
 O

  n  1   n  1 




1


O
1

 1 

  n  1 m 


1


O
1

 

  n  1 l 
1
1 m
m
mean
.


1

 , 0    1, l  1 .
 n  f  x  O
1

 

  n  1 l 

of Fourierseries, Gen.
Math. Notes ISSN 2219 – 7184, Vol.6, No.2
(2011),
[9]
H.K. Nigam and Ajay Sharma: On degree of
Approximation by product means, Ultra
Scientist of
Physical Sciences, Vol.22 (3)
M, 889-894, (2010).
[10]
B.N. Sahney and D.S.Goel :On degree of
approximation of continuous functions, Ranchi
University Mathematical Journal, 4(1973), 5053.
[11]
B .N. Sahney and G.Rao : Errors bound in the
approximation function, Bulletin of Australian
Mathematical Society, 6(1972)
[12]
E.C. Titchmarch: The theory of functions,
oxford university press, p.p402-403(1939).
[13]
A .Zygmund: Trigonometric Series, second
Edition ,Vol.I , Cambridge University press,
Cambridge , (1959).
[14]
Paikray, S.K., Misra, U.K., Jati R.K and Sahoo,
N. C,”On degree of Approximation of Fourier
series by product means”, Buletin of Society
for Mathematical Services and Standards vol.1,
No4(2012), pp. 12-20,
Then from (18) and (19) , we have
n  f  x
 E, q  N , pn 


1
 , 0    1, l 1.
 sup  n  f  x   O 
1

 
  x 
l 
n

1




This completes the proof of the theorem.
V. CONCLUSION
The present idea can be extended to establish the
theorems on indexed summability factors of Fourier as
well as conjugate series. Also other summability
methods with order of Lip condition may be taken into
consideration.

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