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

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.

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 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

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 Cesàro 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

   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 Cesàro 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.