Approximation of a Function f Belonging to Lip Class by (N, p, q)C1 Means of its Fourier Series

DOI : 10.17577/IJERTV2IS3279

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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

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.

1. Definitions

2Rn k0 n k 1

2. 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

3. 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

1. N, pn

C1 if qn

1 n.

n n

0 1

n1

2. N, qn

C1 if pn

1n.

=I1+I2 say. (6)

3. 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.

4. References

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

2. Binod Prasad Dhakal (2010) Approximation of functions belonging to Lip class by Matrix CesÃ ro summability method, International Mathematical forum, 5(35), 1729-1735.

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

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