 Open Access
 Total Downloads : 236
 Authors : Binod Prasad Dhakal
 Paper ID : IJERTV2IS3279
 Volume & Issue : Volume 02, Issue 03 (March 2013)
 Published (First Online): 20032013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Approximation of a Function f Belonging to Lip Class by (N, p, q)C_{1} Means of its Fourier Series
International Journal of Engineering Research & Technology (IJERT)
ISSN: 22780181
Vol. 2 Issue 3, March – 2013
Binod Prasad Dhakal
Central Department of Education (Mathematics), Tribhuvan University, Nepal
Abstract
(t) f (x t) f(x t) 2f(x)
(2)
An estimate for the degree of approximation of
1 n p q
sin 2 n k 1 t
(3)
2
2
NC n (t)
k nk 2
function f Lip class by (N, p, q) C1 means of Fourier series has been established.

Definitions
2Rn k0 n k 1

Theorem
sin 2 t
The Fourier series of 2 periodic Lebesgue
Theorem. If
f : R R is 2 periodic, Lebesgue
integrable f (t) over [, ] is given by integrable over [,] and Lip class function,
f (t) 1 ao
2
n1
an
cosnt bn
sin nt
(1)
then the degree of approximation of function f by (N, p, q) C1 summability means,
p,q,c 1 n
The degree of approximation En(f) of a function f:
tn 1
pk qnk nk of the Fourier series (1) is
R
R
n k0
R R by a trigonometric polynomial tn of degree n is defined by (Zygmund [1])
given by, for n = 0, 1, 2 ,
p,q,c
logn 1e
En (f) tn f
sup. tn (x) f (x) 😡 . tn
1 f
O
n 1
for0 1,
(4)
A function f Lip if,
f (x t) f (x) O( t ) for 0 1. (Dhakal [2])
provide {pn}and {qn}are two sequences of positive real constants of regular generalized NÃ¶rlund method (N, p, q) such that
Let
be an infinite series such that whose
n pk qnk O Rn V n 0.
(5)
un
n k 1 n 1
m0
1
1
n th partial sum s n u . Write n S is
k0
n 1
n 1
n k k0
n k
k0

Proof of the Theorem
(C,1) means of the sequence {Sn}. If
n S, as n then the sequence {Sn} is said
to be summable by CesÃ ro method (C,1) to S.
Following Titchmarsh [4], nth partial sum Sn(x) of the Fourier series (1) at t = x [, ] is given by
1
sin n 1 t
The generalized NÃ¶rlund transform (N, p, q) of the
Sn (x) f (x)
(t) 2 dt .
sequence {S } is the sequence p,q where
2 0
sin t
n
n
tp,q 1 n
R
n
pk q
nk
Snk . If
t n
n
n
tp,q S as n then the
The (C,1) transform i.e. n
2
of Sn
is given by
n k0
sequence {Sn} is said to be summable by
1 n
1 (t) n
1
n 1
n 1
generalized NÃ¶rlund method (N, p, q) to S (Borwein [3]).
Sk (x)
k0
f (x)
2 (n 1) sin t sin k
2 t dt
n (x) f (x) 1
0 2 k1
0 2 k1
sin 2 (n 1) t
2
2
2 (t) dt .
The (N, p, q) transform of the (C,1) transform
2(n 1) 0
sin 2 t
n
n
defines the (N, p, q)C1 transform { t p,q,c1 } of the
partial sum {Sn} of the series
p,q,c 1 n
un .Thus,
n0
Denoting (N, p, q) transform of n i.e. (N, p, q)C1 transform of S by tp,q,c1 , we have
n n
n n
2 t
tn 1
pkqnk nk S, as n then the
1 n p q
(x) f (x)
1 n
pk qnk
sin (n k 1) 2 (t)dt
Rn k0
R k nk nk
2R n k 1 sin 2 t
sequence {Sn} is said to be summable by (N, p, q)C1 method to S..
Some important particular cases of (N, p, q)C1
n k0
n
n
tp,q,c1 (x) f (x)
0
1
0 n k0 2
NCn (t) (t)
means are:
n1
NC
(t) (t) dt
NC
(t) (t) dt

N, pn
C1 if qn
1 n.
n n
0 1
n1

N, qn
C1 if pn
1n.
=I1+I2 say. (6)

C,C1 if pn n 1, 0 andqn 1n. For I1 and 0 t 1
1
n 1
We shall use the following notations:
1 n p q
sin 2 (n k 1) t 1
NCn (t)
k nk 2
2
For I2 and
n 1 t
2
2
2Rn k0 n k 1
sin t
1 n p q
sin 2 n k 1 t
1 n p q
sin 2 t
NCn t
k nk 2
k nk n k 12 2
2Rn k0
n k 1
sin 2 t
2
2
2 Rn k0
n k 1
sin 2 t
1 n pk qnk 2
2
, by Jordans Lemma
1
2R
n k0
n k 1 t2
Since sinn nsin n for 0
n
pk qnk
n
2 R t2
n k 1
1
2 R
n
pk
n k0
qnk
n k 1
n
2 R t 2
k0
O Rn , by the hypothesis of the theorem
n 1
n 1 n p q
n
2 Rn
k0
k nk
O 1 .
n 1t 2
(10)
n 1 2
Using (8) and (10), we have
On 1.
(7)
I2
1
n1
NCn (t)
(t) dt
Since, f (x t) f (x) O( t
) for 0 1 ,
1
O
O(t
2
2
) dt
if fLip.
1 n1
n 1t
1
O t
n 1
2 dt
We have, (t) f (x t) f (x t) 2f(x)
f (x t) f (x) f (x t) f (x)
1
1 n1
t1
O
, for 0 1
O(t) O(t)
n 1 1 1
n1
n1
1 log t ,
for 1
O 1
O(t ) . (8)
n 1
n1
O
1
1 1
1 ,
for 0 1
Now, using (7) and (8) and the fact that
n 1 1
n 11
1
1
(t)Lip , we have
O log log
,
for
1
n 1
n 1
1 n1
1
1 1
I1
NCn (t)
(t) dt
O 1 n 1
,
for 0 1
0 n 1
1 log n 1,
for
1
O
1 n1
0
0
O(n 1) O t dt
n 1
1 1 1
O 1 ,
for 0 1
1
n 1
O(n 1) n1 t dt
logn 1
O
, for
1
0
n 1
1
,
,
1 O
for 0 1
t1 n1
n 1
O(n 1)
(11)
1 0
O log n 1 ,
for
1.
n 1
O(n 1) 1
1 n 11
Collecting (6), (9), (11); we have
1
,
,
O
for01
1 p,q,c n 1
O .
n 1
(9) tn
1 (x) f (x)
1
log n 1
O n 1 O
n 1
,
for 1
1
,
,
O
for01
n 1
logn 1e
O
n 1
,
for 1
logn 1e
O
,
for01
n 1
O log n 1 e ,
for 1
n 1
O log n 1e ,
for 0 1
n 1
Hence,
n
n
tp,q,c1 f
n
n
sup tp,q,c1 (x) f (x)
😡 R
log n 1e
,
,
O
n 1
0 1.
Thus, the theorem is completely established.


References

A. Zygmund (1959) Trigonometric series,Cambridge University Press.

Binod Prasad Dhakal (2010) Approximation of functions belonging to Lip class by Matrix CesÃ ro summability method, International Mathematical forum, 5(35), 17291735.

D. Borwein (1958) On products of sequences, J. London Math. Soc., 33, 352 357.

E. C. Titchmarsh (1939) The Theory of functions, Second Edition, Oxford University Press.