Approximation of function belonging to W(L_p,, ξ(t)) class by (E, q)(N, p_n) means of its conjugate Fourier series

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)(N¯ , 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)(N¯ , pn) of conjugate series of fourier series associated with f has been proved.

Keywords: Degree of approximation, W (Lp, (t)) class (E, q) mean (N¯ , 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 (N¯ , 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 (N¯ , pn) summable to s. It is clear that (N¯ , 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 · 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 (N¯ , 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 · sv

(1.6)

If

),

),

n s as n (1.7)

then an is said to be (E, q) (N¯ , pn)-summable to s.

Let f (t) be a periodic function with period 2 and L-integrable 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 n-th partial sum of (1.9)L-norm 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 || · || 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

k¯n(t) =

k¯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)(N¯ , pn) is assumed to be regular. Here we generalize the theorem of Mishra [2].

1. If f : R R is 2-periodic, 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 .

2. In order to prove it, we shall required the following lemma (Misra [2]).

   |k¯n(t)| =

   |k¯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 Riemann-Lebesgue theorem, we have for the n-th partial sum s¯n(f : x) of the Conjugate Fourier series (1.9) of f (x) following (Titchmarch [3])

r

r

2

0

0

s¯n(f : x) f (x) =

(t)k¯ndt

the (N¯ , pn) transform of s¯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)(N¯ , pn) transform of s¯n(f : x) by n, where

r

r

1

0

0

(r

(r

||n f|| = (1 + 1)n

(t)k¯n(t)dt

r

r

1

n+1

=

0

0

1

1

+

1

n+1

(t)k¯n(t)dt

Now

= I1 + I2 (4.1)

r

r

1

n+1

I1 = (t)k¯n(t)dt

0

Applying H¨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)|k¯

0

0

(t)| q

1/q

dt

dt

0

(t)

0

0

t)

0

·

·

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 \ · (n + 1)+1 1

n + 1

p

p

= O( ( 1 \ · (n + 1)+ 1

n + 1

p

p

= O ((n + 1)+ 1 · ( 1 \\ . (4.2)

n + 1

And

Using H¨older inequality.

I2 =

0

r

r

1

n+1

(t) k¯n(t)dt

r

1

1

t|(t)| sin t p

1/p

r

·

·

k¯n

(t) · (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/

y¯(+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)

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

2. 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, 37-42.

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

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