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ISSN 2229 – 5046
A NOTE ON THE USE OF PRODUCT MEAN (E, q)B
IN THE TRIGONOMETRIC FOURIER APPROXIMATION
PANKAJINI TRIPATHY
1AND BIRUPAKHYA PRASAD PADHY
2†1,2
Department of Mathematics,
School of Applied Sciences, Kalinga Institute of Industrial Technology,
Deemed to be University, Bhubaneswar, Odisha, India.
(Received On: 12-04-18; Revised & Accepted On: 18-05-18)
ABSTRACT
A. Zygmund introduced trigonometric Fourier approximation where as L. Mcfadden [2] introduced Lipchitz class. Dealing with degree of approximation of conjugate series of a Fourier series Padhy et al. have established some theorems. In this paper, we have extended their result and established a theorem on the use of product mean
(
E,q)
Bin the degree of Approximation of the conjugate series of Fourier series of a function of weighted LipchitzclassW(Lp,
ξ
(u)).Keywords: Degree of Approximation, W(Lp,
ξ
(u)) class of function,(
E,q)
- mean, B - mean,(
E,q)
B-product mean,Conjugate Fourier series, Lebesgue integral.
AMS subject classification: 42B05, 42B08.
1. INTRODUCTION
The sequence
{ }
u
n
of theB
-mean of the sequence{ }
sn is given by
,
2
,
1
,
0
=
=
=
∑
n
n
s
m
b
n
u
λ
λ
λ
(1.1)is the sequence –to-sequence transformation, where
B
=
(
b
mn
)
∞
×
∞
be a ∞×∞matrix and∑
b
n
be a given infinite series with the sequence of partial sums{ }
sn .The series
∑
b
n
is said to beB
- summable tos
ifs
n
u
→
asn
→
∞
, (1.2) The regularity conditions forB
-summability are:(i)
L
n
mn
b
m
<
∞
=
∑
0
sup
whereL
is an absolute constant. (ii)lim
=
0
∞
→
b
mn
m
(iii)
1
0
lim
=
∞
=
∞
→
n
∑
b
mn
m
Corresponding Author: Birupakhya Prasad Padhy
2†The sequence
{ }
v
n
[1] of the( )
E
,
q
mean of the sequence{ }
s
n
is given by(
)
λ
λ
q
n
λ
s
λ
n
n
n
q
n
v
−
=
+
=
∑
0
1
1
. (1.3)
The series
∑
b
n
is said to be( )
E
,
q
summable tos
ifs
n
v
→
asn
→
∞
(1.4) Clearly( )
E
,
q
method is regular [7].Further, let
w
n
be the(
E
,
q
)
transform of theB
-transform of{ }
s
n
defined by=
n
w
(
)
q
n
k
u
k
n
k
k
n
n
q
−
=
+
∑
0
1
1
=
(
)
=
−
=
+
∑
λ
∑
λ
s
λ
k
k
b
k
n
q
n
k
k
n
n
q
0
0
1
1
(1.5)
The series
∑
b
n
is said to be( )
E
,
q
B
-summable tos
ifs
n
w
→
asn
→
∞
(1.6) Letg
(
u
)
be a Lebesgue integrable function with period 2π in (-π,π), The Fourier series ofg
at any point ‘x’ is given by(
)
∞
∑
=
≡
∑
∞
=
+
+
0
)
(
1
sin
cos
2
0
~
)
(
n
x
n
G
n
nx
n
d
nx
n
c
c
x
g
(1.7)where
c
0
,c
n
andd
n
are the Fourier coefficients and the conjugate series of the Fourier series (1.7) is(
) ∑
∑
∞
=
≡
∞
=
−
1
)
(
1
sin
cos
n
x
n
H
n
nx
n
d
nx
n
c
(1.8)Let
s
n
( )
g
;
x
be the n-th partial sum of (1.8).For a function
g
:
R
→
R
, theL
∞
-norm of is definedby{
g
x
x
R
}
g
∞
=
sup
(
)
:
∈
(1.9)and the
L
υ
-norm is defined by( )
ν
π
ν
ν
1
2
0
=
∫
g
x
g
,ν
≥
1
. (1.10)and the degree of approximation
E
n
(
g
)
of a functiong
∈
L
ν
is given by [6]ν
g
n
Q
n
Q
g
n
E
(
)
=
min
−
(1.12)This method of approximation is called Trigonometric Fourier approximation. A function
g
∈
Lip
α
if [2]1
0
,
)
(
)
(
<
≤
=
−
+
u
g
x
O
u
α
α
x
g
(1.13)and
g
(
x
)
∈
Lip
(
α
,
r
)
, for0
≤ ≤
x
2
π
,if[2].
0
,
1
,
1
0
,
1
2
0
)
(
)
(
<
≤
≥
>
=
−
+
∫
g
x
u
g
x
r
dx
r
O
u
α
α
r
u
π
(1.14)
For a given positive increasing function
ξ
( )
u
, the functiong
( )
x
∈
Lip
(
ξ
( )
u
,
r
)
, if[2](
(
)
)
,
1
,
0
1
2
0
)
(
)
(
=
≥
>
−
+
∫
g
x
u
g
x
r
dx
r
O
ζ
u
r
u
π
(1.15)
For a given positive increasing function
ξ
( )
u
and an integer p>1 the functiong
( )
x
belongs to
L
p
,
(
u
)
W
ζ
,if [2]
(
(
)
)
,
0
.
1
2
0
)
(sin
)
(
)
(
=
≥
−
+
∫
π
g
x
u
g
x
p
x
p
β
dx
p
O
ζ
u
β
(1.16)We use the following notation throughout this paper:
{
(
)
(
)
}
2
1
)
(
u
=
g
x
+
u
−
g
x
−
u
ψ
and
(
)
+
−
=
−
=
+
=
∑
∑
2
sin
2
1
cos
2
cos
0
0
1
1
)
(
u
u
u
k
k
b
k
n
q
n
k
k
n
n
q
u
n
K
λ
λ
λ
π
(1.17)
Further, the method
( )
E
,
q
B
is assumed to be regular and this case is supposed through out the paper.2. KNOWN THEOREM
Theorem 2.1: If
g
is a2
π
−
Periodic function belonging to classLip
α
, then its degree of approximation by( ) ( )
E
,
q
c
,
1
summability mean on its Fourier series∑
∞
=
0
)
(
n
x
n
G
is given by(
1
)
,
0
1
1
1
<
<
+
=
∞
−
α
α
n
O
g
n
c
q
n
E
,where
E
n
q
c
1
n
represents the( )
E
,
q
transform of( )
C
,
1
transform ofs
n
( )
g
;
x
.Padhy et al. [5] proved the following theorem using
( )
E
,
q
B
mean of the conjugate series of the Fourier series.Theorem 2.2: If
g
is a2
π
−
Periodic function of classLip
α
, then degree of approximation by the product( )
E
,
q
B
summability means on its conjugate series of Fourier series (1.8) is given by(
1
)
,
0
1
1
<
<
+
=
∞
−
α
α
n
O
g
n
w
, wherew
n
as defined in (1.5) .Recently, Padhy et al [4] proved the following theorem using
( )
E
,
s
(
N
,
p
n
,
q
n
)
mean of the conjugate series of the Fourier series of a function of classLip
(
α
,
r
)
in the following form:Theorem 2.3: If
g
is a2
π
−
Periodic function of classLip
(
α
,
r
)
, then degree of approximation by the product( )
E
,
s
(
N
,
p
n
,
q
n
)
summability means on its conjugate series of Fourier series (1.8) is given by(
)
1
,
1
0
,
1
1
1
<
<
≥
−
+
=
∞
−
r
r
n
O
g
n
w
α
α
,wherew
n
as defined in (1.5) .3. MAIN THEOREM
In this paper, we have proved a theorem on degree of approximation by the product mean
( )
E
,
q
B
of Fourier series (1.8).We proveTheorem 3.1: Let
ξ
( )
u
be a positive increasing function andg
be a2
π
−
Periodic function of the class0
,
1
,
)
(
,
>
>
L
p
u
p
u
W
ζ
. Then degree of approximation by the product( )
E
,
q
B
means of the conjugateseries of the Fourier series (1.8) is given by
(
)
1
,
1
1
1
1
≥
+
+
+
=
−
f
r
O
n
r
n
r
n
w
β
ξ
(3.1)provided
+
=
+
∫
1
1
1
1
1
0
(
)
sin
)
(
n
O
r
n
du
r
u
u
u
u
ξ
β
ψ
(3.2)
and
+
=
+
−
∫
δ
π
ξ
ψ
δ
)
1
(
1
1
1
(
)
)
(
n
O
r
n
du
r
u
u
u
(3.3)
hold uniformly in
x
with1
1
1
r
+ =
s
, whereδ
is an arbitrary number such thats
(
1
−
δ
)
− >
1 0
andw
n
is asdefined in (1.5).
4. Lemma: We require the following Lemma to prove the theorem.
Lemma -4.1[5]:
,
1
1
,
1
1
1
0
,
)
(
)
(
≤
≤
+
+
≤
≤
=
π
u
n
u
O
n
u
n
O
u
n
K
Where
K
n
(
u
)
is as defined in (1.17). 5. PROOF OF THEOREM- 3.1:Using Riemann –Lebesgue theorem, we have the n-th partial sum
s
n
( )
g
;
x
of the Fourier series (1.8) ofg
(
x
)
,0
1
cos
sin
2
2
2
( ; )
( )
( )
,
2 sin
2
n
u
n
u
s
g x
g x
u
d u
u
π
ψ
π
−
+
−
=
∫
The
B
−
transform ofs
n
( )
g
;
x
[2] using (1.1) is given by0 0
1
cos
sin
2
2
2
( )
( )
,
2 sin
2
n
n nk
k
u
n
u
u
g x
u
b
du
u
π
ψ
π
=
−
+
−
=
∑
∫
If
w
n
be the( )
E
,
q
B
transform ofs
n
( )
g
;
x
, then we have∫
∑
∑
=
=
+
−
−
+
=
−
π
ν
ψ
π
0
0
0
2
sin
2
2
1
sin
2
cos
)
(
)
1
(
2
n
k
du
k
u
u
n
u
nk
b
k
n
q
k
n
u
n
q
g
n
w
∫
=
π
ψ
0
)
(
)
K
n
u
du
n
n
)
(
1
1
1
1
0
+
+
+
=
∫
π
∫
=
I
1+
I
2,
say
(5.1) Now∫
∑
∑
+
=
=
+
−
−
+
=
1
1
0
0
0
2
sin
2
2
1
sin
2
cos
)
(
)
1
(
2
1
n
n
k
du
k
u
u
n
u
nk
b
k
n
q
k
n
u
n
q
I
ν
ψ
π
∫
+
=
1
1
0
)
(
)
(
n
du
u
n
K
u
ψ
s
n
du
s
u
u
u
n
K
u
r
n
du
r
u
u
u
u
1
1
1
0
sin
)
(
)
(
1
1
1
0
(
)
sin
)
(
+
+
≤
∫
∫
β
ξ
ξ
β
ψ
, where1
1
1
r
+ =
s
,(using Hölder’s inequality)
s
n
du
s
u
u
1
1
1
0
1
)
(
+
+
=
∫
ξ
β
(Using lemma-4.1 and (3.2))(
)
s
n
s
u
du
n
O
1
1
1
0
1
1
1
+
+
+
=
ξ
∫
β
(
)
1 11
1
1
s
O
O
n
n
β
ξ
− + +
=
+
+
(
)
11
1
1
rO
n
n
βξ
+
=
+
+
(5.2)Next
∫
∑
∑
=
=
+
−
−
+
=
π
ν
ψ
π
1
0
0
2
sin
∫
+
=
π
ψ
1
1
)
(
)
(
n
du
u
n
K
u
s
n
du
s
u
u
u
n
K
u
r
n
du
r
u
u
u
u
1
1
1
sin
)
(
)
(
1
1
1
(
)
sin
)
(
+
−
+
−
≤
π
∫
π
∫
ξ
δ
β
ξ
β
ψ
δ
, (Where1
1
1
=
+
s
r
, using Hölder’s inequality)(
)
( )
s
n
du
s
u
u
n
O
1
1
1
1
1
+
−
+
+
=
δ
π
∫
ξ
β
δ
, using Lemma-4.1 and (3.3)(
)
s
n
z
dz
s
z
z
n
O
1
1
1
(
1
)
2
1
1
+
+
−
+
=
∫
π
β
δ
ξ
δ
,(
z
z
1
/
1
ξ
is a positive increasing function asξ
( )
u
is a positive increasing function )(
)
n
s
s
z
dy
n
n
O
1
1
2
)
1
(
1
1
1
1
+
+
−
−
+
+
+
=
∫
ε
δ
β
ξ
δ
, For some1
≤
≤
+
1
.
n
ε
π
(Using second mean value theorem)
=
(
)
(
)
1
1
1
11
1
1
s
O
n
O
n
n
δ β δ
ξ
+ + − −
+
+
+
(
)
11
1
1
rO
n
n
βξ
+
=
+
+
(5.3)Then from (5.2) and (5.3), we have
(
)
1
,
1
1
1
1
≥
+
+
+
=
−
g
O
n
r
n
r
n
w
β
ξ
(
)
,
1
1
2
0
1
1
1
1
≥
+
+
+
=
−
∫
r
(
)
1 2 1
0
1
1
1
r r
O
n
dx
n
π β
ξ
+
=
+
+
∫
(
)
11
1
1
r
O
n
n
β
ξ
+
=
+
+
.This completes the proof of the theorem.
Remark: If we put
β
=
0
andξ
( )
u
=
u
α
in the main theorem then the degree of approximation of a functiong
belonging to the classLip
( )
α
,
r
, 0
< ≤
α
1,
r
≥
1
is given by(
)
.
1
1
−
+
+
=
−
g
r
O
n
r
n
w
α
and if we take
r
→ ∞
then the degree of approximation of a functiong
belonging to the class( )
, 0
1
Lip
α
< ≤
α
is given by(
1
)
.
+
−
=
−
g
r
O
n
α
n
w
REFERENCES
1. Hardy,G.H., Divergent series, First edition, Oxford University press (1970). 2. Mcfadden, L., Absolute Norlund summability, Duke Math. J., 9, (1942) 168-207.
3. Nigam, H. K. and Sharma, K., The degree of Approximation by product means, Ultra scientist, vol22 (3) M, (2010) 889-894.
4. Padhy, B.P., Das, P.K., Misra, M., Samanta, P. and Misra, U.K.,Trigonometric Fourier approximation of the conjugate series of a function of generalized Lipchitz class by product means, Computational Intelligence in data mining, vol.1, Advances in intelligent system and computing 410, DOI 10.1007/978-81-322-2734-2_19, (2016) 181-190.
5. Padhy, B.P., Mallik, B., Misra, U.K. and Misra, M., On degree of Approximation of Conjugate series of Fourier series by product means
( )
E
,
q
A
, International Archive of applied sciences and technology, vol. 2(2), (2011) 31-37.6. Titechmarch, E.C., The theory of functions, oxford university press, (1939).
7. Zygmund, A., Trigonometric Series, Cambridge University press, Cambridge, second Edition, Vol. I, (1959).
Source of support: Nil, Conflict of interest: None Declared.
