on the Euler’s Summability of a Differentiated Fourier Series

on the Euler’s Summability of a Differentiated Fourier Series

Pawan Saxena

Department of Applied Science (Mathematics) IIMT College of Engineering, Knowledge Park-III Plot No. A 20, Gr. Noida (G.B. Nagar) 201308 India

Abstract In this paper we have proved a theorem On the Euler Summability of a differentiated Fourier series which

1.2.7. P t = (t)

A sin 3 t

2

C , where C is constant K denotes an

generalizes known results however our theorem is as follows

Theorem: If (u) du = o(log 1) as t 0 and nq

is a

absolute constant not necessarily the same in each occurrence.

1. Introduction: The following theorem on Euler summability of derived Fourier series is due to Ray 7 .

   t u t

   monotonic convergence sequence

   Such that q x and

   n Theorem A: If g (t) is of bounded variation to the right of t=0 and g (t) = O (1) as t 0,then derived Fourier series at t = x,

   is summable E, q for q > 0,to the value s, where

   1x 2q (x)

   E n,t dt = o q x

   As x 1, then the series

   g(t)= t 4sin t

   1

   s and

   1x t3

   1x 2q (x) 2

   nBn (t), at t = x is summable E, q for q > 0 to the

   t = f x + t f x t .

   n=1

   value s.

   2000 Mathematics subjects classification:

   40D25, 40E25, 40F25, 40G25.

   Chandra 1 generalized above theorem for Euler summability of Fourier series in the following form:

   Theorem B: If t log 1 = o 1 , t 0, then, An x

   t n=0

   Keywords and Phrases;

   =1 (), at = is summable s E, q for q > 0.The object of this paper is to generalize

   =0

   1. Definition and Notations: Let be a given infinite series with partial sums . The sequence Eulers transform of

   Theorem A and B by establishing the following theorem.

2. We assert the following main theorem

   a sequence is defined by

   1.1 = + 1

   , where > 0, if

   Theorem:

   1. If (u) du = o(log 1) as t 0

      =0

      t u t

      As n , we say that sn or

      an are summable

      and nqn is a monotonic convex sequence such that

      n=0

      E , q q > 0 , to s or symbolically we write sn s

   2. q x

      ,

      1x 2 q (x)

      E , q for q > 0 , to s or symbolically we write sn

      and

      E n,t dt = o q x

      s E, q , for q > 0, Hardy 3 ,

      1x t3

      1x 2q (x)

      as x 1, then the series nBn (t), at t = x is summable

      It is evident that that , 0 is equivalent to convergence.

      1.2 Let () be a periodic function with period 2 and

      E, q for q > 0

      n=1

      to the value S.

      integrable in the Lebesgue sense over the interval , and

3. Proof of the theorem: note that

Br x = 1 t sin + 1 dt so that

let

1.2.1. f t ~ 1

2

a0 +

n=1

an cosnt + sinnt =

An (t) be

Sn x =

0

d

1

t

0 dt

n

n=1

1 + cos( + 1) dt 2

the Fourier series of f(t).Then the differentiated series of

1.2.1 at t = x, is

=1

3

= 1 t d sin n + 2 t

n=1

1.2.2.

n bn cosnx ansinnx =

nBn (x)

0 dt

3

2 sin 2 t

dt

n=1

Throughout we use the following notations for

0 < < 1.

1.2.3 t = f x + t f x t

3

1 2n sin 3/2 t cos n + 2 t sin n + 1 t

0

1. P q, t = 1 + q2 + 2q cost

   = t

   4sin2 3/2 t dt

2. Q q, t = tan1 sint

   q+cos t

3. E n, t = n nk sin k + 1 t

k=0 k q 2

3

1.5.5.

1 t sin(n + 1)

2n t

cos n + 2 t

2 +2 = 1 +1

=

0 4sin2

3

2 t

dt

0

3 dt

4 sin 2 t

=1

=1

= 1 3

= 1 +1 – 2 + 3

Also,

2

0 3

2

0 2

2 nq

n = n + 2 qn+2

2 n + 1 q

n+1 + nqn

= 0

+ 1

2 + 3 + 1

N + 1 2qn = n + 1 qn+2 2 n + 1 qn+1 + n + 1 qn

Therefore,

0 2

This implies that

2 nqn n + 1 2 qn = qn+2 qn

Or,

Q x = 2

P t

q xn sin n + 1 t dt

n + 1 2 q

= 2 nq

+ q q

q x 0 t

2

n=1 n

n

n

Hence,

n n n +2

q(x) P(t) nqn x

cos n + 3/2 t dt + o(1)

0 n=1

Or

Q x o(1) = 2

0

q x

2

P t

t E(n, t)dt

n + 1 2qn xn+2

n=1

= 2 nqn

n=1

xn+2

q(x) P t P(q, t) dt

=I I

0

, say

+ qn xn+2 qn+2xn+2

1 2

q xn+1

int

n=1

n=1

Now, E n, t = I

n=1 n e

n it

+1

3 1 x 3

= I n=1 qn =1 e x

= 1 x q x x q x

n=1

= I

qn

xeit (1xn +1eint )

it

1xe

1.5.6. and

n=1

= I

qn

xn +2ei n +1 t

1xeit

n + 1 q

n+2

xn+2 = x2 q x 1 x2 q x

=

n=1

qn

xn +2ei n +1 t

1xeit

n=1

So that

1

= ,

+2 + 2

1.5.7.

n q

xn +1

q x

+ 1 x2 q x

=1

2 +1 +2 + 2

n x2

n=1

Now using (1.5.1), (1.5.6) and (1.5.7)

=1

I 2

1

q x

1x P t

t dt +

P t E n, t dt

t

1.5.1.=

0

2 1x P t

=

dt +

1x

P t

dt 1

1

x

Q q,t

n=1

2qn xn+2sin n + 2 t + 1

q x 0 t

1x t

Q q, t

2

n=1

We have

1.5.2.

qn+1xn+2sin n + 2 t

2 +2 + 2

=1

+ 1 3 +1 +2 + 2

qn

xn+2 = 1 x Pn xn+1 = 1 x q x

2 1x P t

=1

P t 1

n=1

n=1

q x

t dt +

t dt Q q, t

1.5.3

0 1x

1 x 2

+ 1 + 2 +2 + 1 3

2 qn xn+2 = 1 x qn xn+1 =

x q x

n=1

1.5.4.

n=1

=1

+ + 1 +1

+2

nqn xn+2 = 1 x nqn xn+1

=1

n=1

n=1

= x 1 x 2q x

2qn xn+2sin n + 2 t 1

n=1

x3 qn+1xn+2sin n + 2 t

n=1

2 1x P t

P t 1

1

1

0

q x

dt +

t

1x

t dt Q x, t

3

1 0

+ n + 1 2qn xn+2 + 1 x3

+

2 +2

n=1

+ n + 1 qn+1xn+2

1

+ 1

2

=1

5

n=1

3 +1 + 2

2 1x 1

k 1 1x

=1

= q x P(t) dt + P(t) dt Q q, t

3 P(t) dt

0 1x

q(x) 1 x 0

P(t)

+ n + 1 1 x3 n qn xn+1

+ dt 2 nqn xn+2

k 1

n=1

1x P t dt +

n=1

1x t2

n=1

q x

1x 3 0

P t

+ 1 x3 nqn

xn+1

t2 dt 1 x 3q x x + 3

1x

n=1

1x

k 1

P t dt + P t dt

k x + 3 q (x)

(1x)

3 P(t)

q x

1 x 3 0

1x t2

= q(x) P(t) dt + 1 x

t2 dt

1 3 + 3

0 1x

k x+3 q (x)

1x

3 P(t)

= 2k x+3 q (x) 1x 3

E(n,t) dt + E(n,) by

= q (x) 0

k 1x 3q (x)

P t

E n,t

dt +

1 x

E n,

1x

t2 dt

q (x)

(1x) t3 2

= q (x) 1x

t3 dt +

2 , by integration by parts

integrating by parts

= o 1 as x 1 by using (1.4.2) and (1.4.3)

Proceeding as in E n, t , we easily get, 1.5.8

By hypothesis (1.4.2) and (1.4.3). This completes the proof of the theorems.

REFERENCES

P q, t = 1

Q(q, t)

2 nqn

n=1

xn+2

5

cos n + 2 t

1. P. Chandra, On Euler Summability of Fourier series, Ranchi University, Mathematical journal, Publisher, Location, Date, pp. 1-10.

1. On the E, q summability of a Fourier series, University Parma,

   + 1 3 nqn xn+1cos n +

   =1

   Therefore, Using (1.5.4), (1.5.5) and (1.5.8),

   2 1

   5 t 2

   Riv.Maths (4), 3(1977), 65-78.

2. H.G.Hardy, Divergent series, Oxford University Press, Oxford (1949).

3. K.Knopp, and G.G. Lorentz,: B. Kwee.

: The absolute Euler Summability series, Journal.Aust. Math. Soc., 13(1972), 129-140.

1 (6) R.Mohanty,ands. Mohapatra.: On the , Summability of Fouries

2 + ,

series and its Allied series, Jour. Indian Math.Soc.(New series),

0

2 +2

1

+ 5 +

2

32(1968).

1. B.K. Ray. : On , summability of derived Fourier series and a derived conzugate series Indian Jour. Math., 11(1969), 43-50.

2. G. Sonouchi. And T.Tsuchikura.: Absolute regularity for convergent

   1 3 +1 5

   integrals, To hoku Math. Jour. (2) 4(1952), 153-156.

   =1

   =1

   + 2

3. N. Tripathi.: On the absolute Hausdorff summability of Fourier series Jour. London Math. Soc.,44 (1969),15-25.