Cmjv05i02p0133

JOURNAL OF COMPUTER AND MATHEMATICAL SCIENCES An International Open Free Access, Peer Reviewed Research Journal www.compmath-journal.org

ISSN 0976-5727 (Print) ISSN 2319-8133 (Online) Abbr:J.Comp.&Math.Sci. 2014, Vol.5(2): Pg.133-139

On the Local Property of | , ࢔ , ࢔ ; |࢑ Summability of A Factored Fourier Series Chitaranjan Khadanga1 and Shubhra Sharma2 Professor, Department of Mathematics, R.C.E.T, Bhilai, C.G., INDIA. Assistant Professor, Department of Mathematics, R.C.E.T, Bhilai, C.G. INDIA. (Received on: January 31, 2014) ABSTRACT In this paper a theorem on local property of Index summability factored Fourier series has been established.

Keywords: Index summability, factored Fourier series, Infinite series.

coefficients {pn }. The series

1. INTRODUCTION Let

∑a

n

be a given infinite series

with sequence of partial sums {s n } .

n

Pn =

∑p

v

→ ∞ as n → ∞ (P−i = p −i = 0, i ≥ 1)

v =0

(1.1) The sequence-to-sequence transformation

tn =

1 Pn

n

∑

p n−v s v

(1.2)

is said

k −1

 Pn  k   t n − t n −1 < ∞ .

(1.3) ∑ n =1  p n  For k = 1, N , p n k -summability as N, p n -summability.

When

p n = 1 for all n and

k = 1, N , p n k -summability is same as C ,1 - summability.

v =0

Defines the N, p n -means of the sequence

{sn }

n

to be summable N , p n k , k ≥ 1 , if ∞

Let {pn } be a sequence of positive real constants such that

∑a

generated

by

the

sequence

of

Let

{α n }

be any sequence of

positive numbers. The series

∑a

Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)

n

is said

134

Chitaranjan Khadanga, et al., J. Comp. & Math. Sci. Vol.5 (2), 133-139 (2014)

to be summable N , p n , α n k , k ≥ 1 , if ∞

∑α

k −1 n

t n − t n −1

k

∞

∑X (1.4)

<∞

Clearly N , pn ,

n =1 ∞

∑ (X

n =1

λn

k −1 n

k

+ λn +1

∑α

)

k n

+ 1 ∆λ n < ∞

(2.3)

n =1

Pn pn

= N , pn

k

and N , pn , 1 1 = N , pn

k

Pn . Then the summability npn

Xn =

Where

∞

δk + k −1 n

Tn − Tn −1

k

< ∞,

(1.5)

n =1

then the series

N , pn , α n ; δ

k

∑a

n

is said to be

, k ≥ 1, δ ≥ 0 summable.

A sequence

{λn }

is said to be

convex if ∆2 λn ≥ 0 for every positive integer ‘ n ’.

Let f (t ) be a positive function with period 2π , integrable in the sense of Lebesgue over (− π , π ) . Without loss of generality we may assume that the constant term in the Fourier series of f (t ) is zero, so that ∞

f (t ) ~

(2.2)

<∞

n

Further, if ∞

k

∑ (a

n

cos nt + bn sin nt ) =

n =1

∞

∑ A (t ) n

n =1

(1.6) 2. Dealing with the N , p n -summability k

of an infinite series, Bor following theorem:

proved the

n

n

X n at a

n =1

the following result: Theorem 2.2 Let k ≥ 1 , Suppose convex sequence such that convergent and that

{p n }

{λn }

∑n

m +1

∑ (α )

k −1

n

∞

∑X

k −1 n

n =1

λn

−1

be a

λn is

be a sequence such

1 ∆X n = O  , n p Pn − r −1 P  = O n −r −1 r  , Pn  Pn −1 p r 

p  p n −r = O r  Pn  Pr 

(2.4) (2.5)

(2.6)

k

<∞

n

(2.7)

and ∞

(2.1)

∑ A (t ) λ

point can be ensured by the local property.

Extending the above result to N , p n , α n k -summability, Buxi established

n = r +1

Theorem- 2.1 Let k ≥ 1 and let the sequences {pn } and {λn } be such that

1 ∆X n = O  n

N , p n k of the series

∑X n =1

k −1 n

∆λ n n

k

<∞

Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)

(2.8)

Chitaranjan Khadanga, et al., J. Comp. & Math. Sci. Vol.5 (2), 133-139 (2014)

Where

Xn =

Pn . Then the summability npn

N , pn , α n k , k ≥ 1 of the factored Fourier ∞

series

∑ A (t ) λ n

n

X n at a point can be

n =1

ensured by the local property, where {a n } is a sequence of positive numbers.

∞

Factored Fourier Series

∑ A (t ) λ n

convex sequence such that convergent and that

{p n }

∑n

−1

be a

λn is

be a sequence such

X n at

a point can be ensured by the local property, where {a n } is a sequence of positive numbers.

In order to prove the above theorem we require the following lemma: LEMMA Let

{λn }

n

n =1

3. MAIN THEOREM: Let k ≥ 1 . Suppose

135

{λn }

k ≥ 1 . Suppose

∑n

convex sequence such that

−1

be a

λn is

convergent and {pn } be a sequence such that the conditions (3.1) to (3.5) are satisfied.

If {s n } is bounded, then the series ∞

∑a

1 ∆X n = O  n

Pn − r −1 Pn

k −1 ∑ (α n ) n = r +1

∑α

δk k

X nk −1

n =1

ሼߙܽ௡ ሽ is sequence of positive numbers.

(3.2)

p  p n −r = O r  Pn  Pr  λn

∑α

δk n

X

k −1 n

PROOF OF LEMMA Let {Tn } denote the N, p n -mean

(3.3)

∞

∑a

of the series

k

n ∆λ n

n

λn X n . Then by

n =1

(3.4)

<∞

and ∞

λn X n is summable

N , pn , α n ; δ k , k ≥ 1, δ ≥ 0 , where

p P  = O n −r −1 r   Pn −1 p r 

m +1

∞

(3.1)

n

n =1

definition we have

Tn =

k

<∞ (3.5) n P Where X n = n . Then the summability npn

1 Pn

n

n− v

v =0

n =1

N , p n , α n ; δ k , k ≥ 1, δ ≥ 0

of

the

v

∑ p ∑a = =

r

λr X r

r =0

1 Pn

∑ a r λr X r

∑p

r =0

v=r

1 Pn

∑a

n

n

r =0

n

n −v

n

r

λr X r

∑p v=r

Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)

n −v

136

Chitaranjan Khadanga, et al., J. Comp. & Math. Sci. Vol.5 (2), 133-139 (2014)

1 Pn

=

m

n

Hence

m

Tn − Tn−1 1 Pn−1

−

∑P

m

r =1

= O(1) ∑ a

∑ Pn−r −1 ar λr X r r =1

1   ∑ ∆{(Pn−r Pn−1 − Pn−r −1Pn ) λr X r } Pn Pn −1  r =1 n −1

r

,

∑a

v

n − r −1 Pn −1

− P n − r − 2 Pn ) ∆λr X r s r +

n −1

∑ (P

n − r −1 Pn −1

∑ α (α )

Tn , j

Tn ,1

m +1

k

=

n=2

m +1

k

< ∞ for i = 1,2,3,4,5,6 .

∑ α (a ) δk n

k −1

n

n=2

∑ α (α ) δk n

k −1

n

n=2

= O(1)

m

∑α r =1

1   ∑ p n − r −1 λr Pn  r =1

k −1

n

n=2

= O (1)

m

∑α

δk n

k

λr

1  n −1  ∑ p n−r λr Pn  r =1

δk n

λr

k

X rk

1 Pn Pn −1 k

sr

k

∑p

n− r

 X  

m +1

 1   Pn

k −1

∑ (α ) n

n = r +1

 p n − r  ∑ r =1  n −1

 pn −r   Pn

k −1

  

 1 X rk     Pn −1

k

k −1

n

k

k

δk n

X

r =1

n− r −1

  

k −1

 p n − r −1     Pn −1 

p r Pr P , as X n = n Pr rp r np n

X rk −1 k −1 r

n −1

∑p

pr Pr

X rk

k

λr r

r =1

= ܱሺ1ሻ ܽ‫∞ → ݉ ݏ‬

by

(3.4)

Further, m +1

∑α

k δk + k −1 Tn,3 n

m +1

=

n =2

∑a

∑α

δk n

(an )k −1

n =2

δk n

(an )k −1

n=2

Pn −1 λ r X r s r

r =1

n = r +1

m

≤

r =1

k r

sr

∑

pn − r −1Pn λr X r sr

∑ (α )

X rk

1 Pn

 n−1  pn − r −1 ∆λr   r =1

∑

k

n −1

k

k

n −1

m +1

r =1

m +1

δk + k −1 n

δk n

n −1

= O(1) ∑ α

Now, we have by using Holder’s inequality

≤

m +1

 − Pn − r − 2 Pn ) λr +1 ∆X r s r  

n =1

∑α

≤

n=2

= O (1) ∑ α nδk λr

To complete the lemma by using Minokowski’s inequality, it is sufficient to show that

m +1

n=2

1 Pn Pn −1

α nδk (α n )k −1

m

= Tn,1 + Tn, 2 + Tn ,3 + Tn, 4 + Tn ,5 + Tn ,6 (say)

δk + k −1 n

∑

=

(3.4)

r =1

1  n−1 ∑ ( pn−r Pn−1 − pn−r −1 Pn ) λr X r s r Pn Pn −1  r =1

(by Abel’s transformation)

∑α

∑

m +1

k

α nδk + k −1 Tn,2

= O(1) ∑ α nδk λr

r =1

∞

m +1

m

r =1

r

r =1

So

n −1

X

k

λr

k −1 r

= ܱሺ1ሻ ܽ‫ ∞ → ݉ ݏ‬by Again,

v =1

with p0 = 0

∑ (P

δk n

r =1

P  P = ∑  n −r − n− r −1  a r , λr X r Pn −1  r =1  Pn n 1 = ∑ (Pn−r Pn−1 − Pn−r −1 Pn ) ar λr X r Pn Pn−1 r =1

+

p r Pr P , as X n = n Pr rpr npn

X rk −1

r =1

ar λr X r

n− r

n −1

Tn − Tn−1 =

k

= O(1) ∑ a nδk λr

n

n

=

pr Pr

X rk

r =1

r =0

1 = Pn

k

= O(1) ∑ a nδk λr

∑ ar Pn−r λr X r

= O(1)

m

∑α

δk n

∆λr

k

r =1

 1  Since Pn  m

n −1

∑P

n − r −1

r =1

= O(1) ∑ α nδk ∆λr r =1

k

sr

k

k

n −1

1 Pn Pn −1

∑p

n −r −1 Pn −1

∆λr X r s r

r =1

 1 X rk     Pn 

n −1

∑p r =1

n − r −1

 ∆λr   

k −1

 p n −r −1    n = r +1  Pn  n −1  ∆λr ≤ ∑ ∆λr = O(1) r =1 

X rk

k

m +1

∑ (α )

X rk

k −1

n

pr Pr

Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)

Chitaranjan Khadanga, et al., J. Comp. & Math. Sci. Vol.5 (2), 133-139 (2014) m

= O(1) ∑ α nδk ∆λr

k

p r Pr P , as X n = n Pr rp r npn

X rk −1

r =1

m

= O(1) ∑ α

δk n

X

∆λ r

k −1 r

k

r

r =1

= O (1) as m → ∞

by

(3.5)

Now, m +1

∑α

δk + k −1 n

Tn , 4

m +1

k

= ∑ α nδk (α n )

n=2

≤

k −1

n =2

m +1

∑ α (α ) δk n

k −1

n

n =2

1 Pn Pn −1

k

n −1

∑p

n −r − 2

Pn ∆λr X r s r

r =1

1  n −1  ∑ p n − r − 2 ∆λr Pn −1  r =1 m

k

k

sr

 1 X     Pn

∆λ r

k

X rk

= O(1) ∑ α nδk ∆λr

k

X rk

= O(1)

∑α

δk n

r =1

m

r =1 m

= O(1) ∑ α nδk ∆λr

k

= O(1) ∑ α

δk n

X

k −1 r

X rk −1

r =1

= O (1) as m → ∞

∆λ r

n −1

pr Pr

r =1 m

k −1

 p n − r − 2 ∆λr  ∑ r =1  m +1   (α n )k −1  p n−r − 2  , (as above) ∑ n = r +1  Pn 

k r

P p r Pr , as X n = n Pr rp r npn

k

r by (3.5)

Again, m +1

∑α

δk + k −1 n

Tn ,5

m +1

k

=

n=2

∑ α (α ) δk n

k −1

n

n =2 m +1

=

k −1

∑ α (α ) δk n

n

n =2 m +1

=

k −1

m +1

=

k −1

n −1

∑ α (α ) ∑ n =2

δk n

n −1

∑ α (ε ) ∑ n= 2

m +1

n −1

n

k −1

n

r =1

∑p

P λr +1 ∆ X r s r

n − r −1 n −1

r =1

Pn − r −1 λr +1 ∆X r s r ∑ Pn r =1

r =1 δk n

k

n −1

n −1

∑ α (α ) ∑ δk n

n =2

=

1 Pn Pn −1

n

r =1

k

p n − r −1 Pr λr +1 ∆X r s r Pn −1 p r p n− r −1 Pr 1 λr +1 s r Pn −1 p r r

137

k

k

k

p n − r −1 Pr P p λr +1 s r X r r , as X n = n Pn −1 p r Pr np n

Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)

138

=

m +1

Chitaranjan Khadanga, et al., J. Comp. & Math. Sci. Vol.5 (2), 133-139 (2014)

∑ α (α ) δk n

k −1

n

n =2

 n −1 p n − r −1 λ r +! ∑  r =1 Pm−1

k

sr

  n−1 p  X  ∑ n − r −1    r =1 Pn −1 

k

m

= O(1) ∑ α nδk λr +1

k

X rk

r =1

∑α

= O(1)

m +1

∑ (α )

k −1

n

n = r +1

m

δk n

k −1

k r

k

λr +1

X rk −1

r =1

 p n −r −1     Pn−1 

P pr Pr , as X n = n Pr rpr npn ,

and by (3.3) m

= O(1)

∑α

k

λr +1

δk n

X rk −1

r = O (1) as m → ∞ r =1

by (3.4)

Finally, m +1

∑α

δk + k −1 n

Tn ,6

k

m +1

∑ α (α )

=

n=2

δk n

k −1

n

n =2 m +1

=

k −1

m +1

n

r =1 k −1

m +1

n

r =1 k −1

n

r =1

m +1

=

m +1

k −1

n −1

∑ α nδk (α n )

∑

n=2

r =1

∑ α (α ) δk n

n −1

∑ α (α ) ∑ δk n

n =2

=

n −1

∑ α (α ) ∑ δk n

n =2

=

n −1

∑ α (α ) ∑ δk n

n =2

=

1 Pn Pn −1

k −1

n

n =2

m

m

∑α r =1

δk n

λr +1

n −r −2

Pn −1λr +1 ∆ X r s r

r =1 k

p n − r − 2 Pr λr +1 ∆X r s r Pn − 2 p r p n − r − 2 Pr 1 λr +1 s r Pn − 2 p r r

k

k

k

p n − r − 2 Pr p λ r +1 s r X r r Pn − 2 p r Pr

k

, as X n =

  n −1 p  s r X  ∑ n− r − 2    r =1 Pn − 2  m +1   (α n )k −1  p n− r −2  α ∑ n = r +1  Pn − 2 

k

X rk

k

X rk −1

r =1

= O(1)

∑p

Pn − r − 2 λr +1 ∆X r s r p n −1

 n−1 p n − r − 2 λr +1 ∑  r =1 Pn − 2

= O(1) ∑ α nδk λr +1

k

n −1

k

k −1

k r

P pr Pr , as X n = n Pr rpr npn

and by (3.3) Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)

Pn np n

Chitaranjan Khadanga, et al., J. Comp. & Math. Sci. Vol.5 (2), 133-139 (2014) m

= O(1)

∑α

δk n

λr +1

r r =1 = O (1) as m → ∞

139

k

X rk −1 , by (3.4)

This completes the proof of the Lemma.

PROOF OF THE MAIN THEOREM Since the behaviour of the Fourier series, as far as convergence is concerned, for a particular value of x depends on the behaviour of the function in the immediate neighbourhood of the point only.

Hence the truth of the theorem is necessary consequence of the above result of the lemma.

REFERENCES 1. Hardy, G.H. Divergent series, Clarendow Press, Oxford,(1949).

2. Hardy, G.H and Riesz, M : The general

theory of Direchlet series, Cambridge, (1915).

3. Mahapatra, P. C. Absolute inclusion theorem for a method of Norlund summability, Ph.D. Thesis Sambalpur University,(1975).

4. Bor, H. On the local property of , ௡ |௞ summability of factored Fourier | series, Journal of Mathematical Analysis and Applications,163,220-226 (1992).

5. Buxi, S.K. On the local property of | , ௡ , ௡ |௞ summability of factored Fourier series, international Journal Review of Applied Sciences, Vol.5 (2), 162-167 November (2010).

Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)