 Open Access
 Total Downloads : 220
 Authors : Aditya Kumar Raghuvanshi, B. K. Singh, Ripendra Kumar
 Paper ID : IJERTV2IS70700
 Volume & Issue : Volume 02, Issue 07 (July 2013)
 Published (First Online): 23072013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Approximation of function belonging to W(L_{p,}, (t)) class by (E, q)(N, p_{n}) means of its conjugate Fourier series
Approximation of function belonging to W (Lp, (t)) class by (E, q)(NÂ¯ , pn) means of its conjugate Fourier series
Aditya Kumar Raghuvanshi, B.K. Singh & Ripendra Kumar,
Department of Mathematics
IFTM University, Moradabad – 244001, (U.P.) India
Abstract
In this paper a theorem on degree of approximation of a function f W (Lp, (t)) by product summability (E, q)(NÂ¯ , pn) of conjugate series of fourier series associated with f has been proved.
Keywords: Degree of approximation, W (Lp, (t)) class (E, q) mean (NÂ¯ , pn) mean, conjugate of Fourier series and Lebesgue integral.
Let an be a given infinite series with the sequence of partial sums {sn}. Let
{pn} be a sequence of positive real numbers such that
n
Pn = pv as n (p i = p i = 0, i 0) (1.1)
v=0
The sequence to sequence transformation
1 n
n
n
tn = pv sv p
v=0
(1.2)
defines the sequence {tn} of the (NÂ¯ , pn)mean of the sequence {sn} generated by the sequence of coefficient {pn} if
tn s as n (1.3)
Then the series an is said to be (NÂ¯ , pn) summable to s. It is clear that (NÂ¯ , pn) method is regular (Hardy [1]).
The sequence to sequence transformation, (Hardy [1])
Tn =
n
1
1
(1 + q)n
v=0
n
v
v
qnv Â· sv (1.4)
defines the sequences {Tn} of the (E, q) means of the sequence {sn} if
),
),
Tn s as n (1.5) Then the series an is said to be (E, q) summable to s. Clearly (E, q)
method is regular (Hardy [1]).
{ }
{ }
Further, the (E, q) transformation of the (NÂ¯ , pn) transform of sn is de fined by
n =
=
n
1
1
(1 + q)n
k=0
1
1
n
(1 + q)n
k=0
n
k
k
n
k
k
qnk Tk
( 1
( 1
k
qnk
k
k
P
v=0
pv Â· sv
(1.6)
If
),
),
n s as n (1.7)
then an is said to be (E, q) (NÂ¯ , pn)summable to s.
Let f (t) be a periodic function with period 2 and Lintegrable over ( , ).
The Fourier series associated with f at any point x is defined by
0
0
a
f (x) 2 +
n=1
(an
cos nx + bn
sin nx) An
n=0
(x) (1.8)
And the conjugate series of the Fourier series (1.8) is
(bn cos nx an sin nx) Bn(x) (1.9)
n=1
n=0
n=1
n=0
n=1
n=0
n=1
n=0
let sn(f : x) be the nth partial sum of (1.9)Lnorm of a function f :
R R is defined by
f  = sup{f (x) : x R} (1.10) and the Lv norm is defined by
r
r
2
f v = (
0
0
1
v
v
f (x) )v , v 1 (1.11)
0
0
The degree of approximation of a function f : R R by a trigonometric polynomial Pn(x) of degree n under norm  Â·  is defined by (Zygmund [4]).
Pn f  = sup{pn(x) f (x) : x R} (1.12)
and the degree of approximation En(f ) a function f Lv is given by
Pn
Pn
En(f ) = min Pn f v. (1.13)
A function f Lip() if
f (x + 1) f (x) = O(t ), 0 < 1, t > 0 (1.14) A function f (x) Lip(, r) if
r
r
(r 2
0
0
f (x + t) f (x) dx
\1/r
= O(t
), 0 < 1, r 1. (1.15)
0
0
(r
(r
A function f (x) Li(t, r) if
2
r
r
f (x + t) f (x) dx
\1/r
= O(t), r 1, t > 0 (1.16)
0
But f W (Lp, (t)) if
(r 2
0
0
1/p
\
\
p
[f (x + t) f (x)] sin
0
0
x dx
= O(t), 0 (1.17)
0
0
we use following Notation through out this paper
1
2
2
(t) = {f (x + t) f (x t)}
and
and
1
2
2
(t) = {f (x + t) f (x t) 2f (x)}
n
n
kÂ¯n(t) =
kÂ¯n(t) =
1
(1 + q)n
(1 + q)n
(n\
k
k
( 1 k
Pk
Pk
cos 1 cos(f + 1 )t
sin t/2
sin t/2
(1 + q)n
k=0
k
Pk
v=0
sin t/2
(1 + q)n
k=0
k
Pk
v=0
sin t/2
qnk
qnk
pv
pv
2
2
2
2
Further, the method (E, q)(NÂ¯ , pn) is assumed to be regular. Here we generalize the theorem of Mishra [2].

If f : R R is 2periodic, Lebesgue integrable [, ] and belonging to the class W (Lp, (t)), p 1 by n(x) on its conjugate Fourier series (1.9) is given
by
n f P = O ((n + 1)+ ( \\ (2.1)
p
p
1 1
n + 1
Provided (t) satisfies the following conditions
n+1
n+1
(r 1 ( t(t) P
1
1
1/p
sinP tdt
= O ( 1
(2.2)
0 (t)
n + 1
and
(r 1 (t(t) sin t p
1/p
1
1
dt
= O{(n + 1) } (2.3)
n+1
n+1
0 (t)
p
p
q
q
where is an arbitrary number such that q(1 ) 1 > 0, (2.2), (2.3) hold uniformly in x and where 1 + 1 = 1 such that 1 p .

In order to prove it, we shall required the following lemma (Misra [2]).
kÂ¯n(t) =
kÂ¯n(t) =
1
1
1
1
n+1
n+1
O
O
for
for
n+1 t
n+1 t
( Or(n)) for 0 t 1
t
t
t
t
Using RiemannLebesgue theorem, we have for the nth partial sum sÂ¯n(f : x) of the Conjugate Fourier series (1.9) of f (x) following (Titchmarch [3])
r
r
2
0
0
sÂ¯n(f : x) f (x) =
(t)kÂ¯ndt
the (NÂ¯ , pn) transform of sÂ¯n(f : x) using 1.2 is given by
n
n
tn f (x) =
tn f (x) =
(t)
(t)
pk
pk
2
2
2
2
dt
dt
2 r cos t sin(n + 1 )t
Pn
0
k=0
2 sin(t/2)
Pn
0
k=0
2 sin(t/2)
Pn
0
k=0
2 sin(t/2)
Pn
0
k=0
2 sin(t/2)
denoting the (E, q)(NÂ¯ , pn) transform of sÂ¯n(f : x) by n, where
r
r
1
0
0
(r
(r
n f = (1 + 1)n
(t)kÂ¯n(t)dt
r
r
1
n+1
=
0
0
1
1
+
1
n+1
(t)kÂ¯n(t)dt
Now
= I1 + I2 (4.1)
r
r
1
n+1
I1 = (t)kÂ¯n(t)dt
0
Applying HÂ¨older inequality
I1
I1
n+1
n+1
r 1 t(t) sin t p
0
(t)
0
(t)
1/p
dt
dt
r 1 (t)kÂ¯
0
0
(t) q
1/q
dt
dt
0
(t)
0
0
t)
0
Â·
Â·
n+1
n+1
n
t sin t
n
t sin t
n+1
n+1
( 1 \ r 1 n(t) q
1/q
= O
( \
( \
r (t)
r (t)
n + 1 0
t1+ dt
1
= O
( \
( \
n + 1
O(n)
1 q
n+1
r (t)
r (t)
0 t1+
1/q
dt
1
O n + 1
O(n + 1)
1 q
n+1
0 t1+
1/q
dt
= O (
( 1 \\ r 1
1/q
t(+1)qdt
n+1
n+1
n + 1 0
q
q
= O( ( 1 \ Â· (n + 1)+1 1
n + 1
p
p
= O( ( 1 \ Â· (n + 1)+ 1
n + 1
p
p
= O ((n + 1)+ 1 Â· ( 1 \\ . (4.2)
n + 1
And
Using HÂ¨older inequality.
I2 =
0
r
r
1
n+1
(t) kÂ¯n(t)dt
r
1
1
t(t) sin t p
1/p
r
Â·
Â·
kÂ¯n
(t) Â· (t) 1/q
1/q
I2
r
r
n+1
dt
(t)
1
n+1
t
sin t dt
= O(n + 1)
1
n+1
n+1
n+1
(t) q
t++1
1/q
dt
= O(n + 1)
r 1 (1/y) q dy 1/q
( \ I
( \ I
1/
yÂ¯(+1)+ y2
= O (n + 1)
1 (y+1)q1
n + 1
1/q n+1 1/
q
q
= O (n + 1) ( 1 \ (n + 1)+1 1
n + 1
p
p
= O ( ( 1 (n + 1)+ 1
n + 1
p
p
= O ((n + 1)+ 1 ( 1 (4.3)
n + 1
Now combining (4.1), (4.2), (4.3)
n f  = O ((n + 1)+ (
p
p
1 1
This completes the proof of main theorem.
n + 1
Following corollaries can be derived from our main results.
Corollary 5.1. If = 0 then the function f W (Lp, (t)) becomes f Lip((t), p) and degree of approximation is given by
Tn
f  = O ((n + 1) 1
1 , p > 0
(
(
n + 1
p
p
 / \
 / \
Corollary 5.2. If = 0 and (t) = t then f W (Lp, (t)) becomes f Lip((t), p) and degree of approximation is given by
Tn
f = O 1 , p > 0 (n + 1)p
1
1
Corollary 5.3. If = 0, (t) = t and p then f W (Lp, (t)) becomes
 (
 (
Lip and degree of approximation is given by
Tn
f = O 1 , 0 < < 1
(n + 1)

Hardy, G.H.; Divergent Series, First Ed. Oxford Uni. Press, 1949.

Mishra, U.K., Misra, M., Pandey, B.P. and Buxi, S.K.; On degree of Approximation by product means of conjugate Fourier series Malaya J. of Mathematics 2(1), 2013, 3742.

Titchmarch, E.C.; The theory of function, Oxford Uni. Press.

Zygmund, A.; Trigonometric Series, Second Edition, Cambridge Uni. Press Cambridge, 1959.