DOI : 10.17577/IJERTV2IS3279
- Open Access
- Total Downloads : 236
- Authors : Binod Prasad Dhakal
- Paper ID : IJERTV2IS3279
- Volume & Issue : Volume 02, Issue 03 (March 2013)
- Published (First Online): 20-03-2013
- ISSN (Online) : 2278-0181
- 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)C1 Means of its Fourier Series
International Journal of Engineering Research & Technology (IJERT)
ISSN: 2278-0181
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 Nörlund 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 Cesà 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 Nörlund 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 Nörlund 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 Cesà ro summability method, International Mathematical forum, 5(35), 1729-1735.
-
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.