On indexed Summability of a factored Fourier series through Local Property

34 International Journal of Research in Science & Technology

Volume: 2 | Issue: 2 | February 2015 | ISSN: 2349-0845 IJRST

I. INTRODUCTION

Let na be a given infinite series with sequence of partial

sums ns . Let np be a sequence of positive real constants

such that

n

iin ipPnaspP

0

(1.1))1,0(

The sequence- to - sequence transformation

)2.1(1

0

spP

tn

n

n

n

defines npN , - means of the sequence ns generated by

the sequence of coefficients np . The series na is said

to be summable knpN , , ,1k if

11

1

(1.3)n

k

nn

k

n

n ttp

P

For k=1, knpN , - summability is same as npN , -

summability.

For k=1, knpN , - summability is same as npN , -

summability.

On indexed Summability of a factored

Fourier series through Local Property

R.K. Jati1, S.K.Paikray

2 and U.K.Misra

3

1 Research Scholar, Department of Mathematics, Ravenshaw University , Cuttack, Odisha, India 2 Department of Mathematics , VSS University of Technology, Burla, Odisha, India

3 Department of Mathematics , NIST , Berhampur, Odisha, India

Article Info ABSTRACT

Article history:

Received on February 26th 2015

Accepted on March 2nd 2015

Published on 5th March 2015

In this paper we have established a theorem on the local Property

of , , ;n n kN p

summability of factored Fourier series.

Keyword:

knpN , - summability,

knnpN ,, - summability,

, ; ,n nk

N p - summability,

Fourier series,

Local Property

Copyright © 2015 International Journal of Research in Science & Technology

All rights reserved.

Corresponding Author:

S. K. Paikray Department of Mathematics,

VSS University of Technology, Burla, India

Email Id: [email protected]

mailto:[email protected]


On indexed Summability of a factored Fourier series through Local Property

When 1, for all and 1np n k ,knpN , - summability is

same as 1, C - summability.

Let n be any sequence of positive numbers. The series

na is said to be summable 1,,, kpNknn , if

k

ntnt

n

kn

11

1 (1.4)

Where nt is as defined in (1.2).The series na is said

to be , , ; , 1, 0n nk

N p k summable if

1

1

1

n

k

nn

kk

n tt

(1.5)

For 0 , the summability method

, , ; , 1, 0n nk

N p k , reduces to the summabilty

method , , , 1.n nk

N p k

A sequence n is said to be convex if 02 n for every

positive integer n.

Let tf be a periodic function with

period 2 and integrable in the sense of Lebesgue

over ),( . Without loss of generality we may assume

that the constant term in the Fourier series of tf is zero,

so that

11

)sincos(~n

n

n

nn tAntbntatf (1.6)

It is well known that the convergence of Fourier

series at t = x is a local property of f (t). (i.e., it depends only

on the behavior of f (t) in an arbitrarily small neighbourhood

of x) and hence the summability of the Fourier series at t = x

by any regular linear method is also a local property of f (t).

II. KNOWN THEOREMS

Dealing with the k

npN , - summability of an infinite series

Bor [1] proved the following theorem:

THEOREM 2.1:

Let 1k and let the sequences

andn np be such that

nOX n

1 (2.1)

1

11

n

k

n

k

nk

nn

X

(2.2)

and

1

)1(n

n

k

nX (2.3)

where .n

n

nnp

PX Then the summability

knpN , of the

factored Fourier series

1

)(n

nnn XtA at a point can be

ensured by the local property.

Subsequently Misra et al [2] proved the following

theorem on the local property of knnpN ,, summability

of factored Fourier series:

THEOREM 2.2:

Let 1k . Suppose n be a convex sequence

such that

nn 1 is convergent and np be a sequence

such that

nOX n

1

(2.2.1)

r

r

n

rn

n

rn

p

P

P

pO

P

P

1

11 (2.2.2)

r

r

n

rn

km

rn

nP

pO

P

p11

1

(2.2.3)

1

1

n

k

nk

nn

X

(2.2.4)

and

1

1

n

k

nk

nn

X

(2.2.5)

where .n

n

nnp


1,,, kpNknn of the factored Fourier series

1

)(n

nnn XtA at a point can be ensured by the local

property, where n is a sequence of positive numbers.



In what follows, in the present paper we

establish the following theorem on , , ,n n kN p -

summabilty of a factored Fourier series through its local

property.

III. MAIN THEOREM

Let 1k . Suppose n be a convex sequence such that

nn 1 is convergent and np be a sequence such that

nOX n

1 (3.1)

r

r

n

rn

n

rn

p

P

P

pO

P

P

1

11 (3.2)

11

1

k kmn r r

n

n r n r

p pO

P P

(3.3)

1

1

n

k

nk

nn

X

(3.4)

and

1

1

n

k

nk

nn

X

(3.5)

where .n

n

nnp


, , , , 1n n kN p k of the factored Fourier series

1

)(n

nnn XtA at a point can be ensured by the local

property, where n is a sequence of positive numbers .

In order to prove the above theorem we require the following

lemma:

IV. LEMMA

Let 1k . Suppose n be a convex sequence such that

nn 1 is convergent and np be a sequence such that

the conditions (3.1)-(3.5) are satisfied. If ns is bounded

then for the sequence of positive numbers n , the series

1n

nnn Xa is summable

, , , , 1, 0n n kN p k .

5. PROOF OF THE LEMMA:

Let nT denote the npN , -mean of the series

1n

nnn Xa .Then by definition we have

n

r

rrrn

n

n XapP

T0 0

1

n

r

n

n

r

rrr

n

pXaP

0

1

n

r

rrrnr

n

XPaP 0

1

Hence

1

1

1

11

1

11 n

r

rrrrn

n

n

r

rrrrn

n

nn XaPP

XaPP

TT

rrr

n

r n

rn

n

rn XaP

P

P

P

1 1

1

rrr

n

r

nrnnrn

nn

XaPPPPPP

1

11

1

1

rn

r

rrnrnnrn

nn

aXPPPPPP 1

1

1

11

1

1

1

1

1211

1

1

211

1

1

11

1

1

n

r

rrrnrnnrn

n

r

rrrnrnnrn

n

r

rrrnrnnrn

nn

sXPPPP

sXPPPP

sXPpPpPP

(By Abel’s transformation)



,1 ,2 ,3 ,4 ,5 ,6 (say)n n n n n nT T T T T T .

In order to complete the proof of the theorem by using

Minokowski’s inequality, it is sufficient to show that

1

,

1

1,2,3,4,5,6.kk k

n n i

n

T for i

Now, we have

1

1

,1

2

mkk k

n n

n

T

11 1

1

2 11

1k k

km n

n n r n r r r

n rn n

p P X sP P

1

11 1 1

2 1 1

1 1k k

km n n

k k k

n n r r r r n r

n r rn n

p s X pP P

11

1 1

(1)k km m

k k n rr r n

r n r n

pO X

P

r

rm

r

k

r

k

rP

pXO

1

)1( , by (3.3)

,)1(1

1

r

r

r

rm

r

k

r

k

rrp

P

P

pXO

as

rXO

k

rm

r

k

r

1

1)1(

masO )1( , by (3.4).

Next,

11

,2

2

k kmk

n n

n

T

11 1

1

2 11

1k k

km n

n n r n r r r

n rn n

p P X sP P

1

11 1 1

1 1

2 1 11 1

1 1k k

km n n

k k k

n n r r r r n r

n r rn n

p s X pP P

11

1

1 1 1

(1)k km m

k k n rr r n

r n r n

pO X

P

r

rm

r

k

r

k

rP

pXO

1

)1( , by (3.3)

,)1(1

1

r

r

r

rm

r

k

r

k

rrp

P

P

pXO

as

n

n

nnp

PX

rXO

k

rm

r

k

r

1

1)1(

masO )1( , by (3.4).

Further,

11

,3

2

k kmk

n n

n

T

11 1

1 1

2 11

1k k

km n

n n r n r r r

n rn n

P P X sP P

1

11 1 1

1 1

2 1 1

1 1k k

km n n

k k k

n n r r r r n r r

n r rn n

P s X PP P

11

1

1 1

(1)k km m

k k n rr r n

r n r n

PO X

P

1

1

1

1

1 )1(1 n

r

r

n

r

rrn

n

OPP

Since

r

rm

r

k

r

k

rP

pXO

1

)1( , by (3.3)

,)1(1

1

r

r

r

rm

r

k

r

k

rrp

P

P

pXO

as

n

n

nnp

PX

rXO

k

rm

r

k

r

1

1)1(

masO )1( , by (3.5).

Now,

11

,4

2

k kmk

n n

n

T

11 1

2

2 11

1k k

km n

n n r n r r r

n rn n

P P X sP P

1

11 1 1

2 2

2 1 11 1

1 1k k

km n n

k k k

n n r r r r n r r

n r rn n

P s X PP P

11

2

1 1 1

(1)k km m

k k n rr r n

r n r n

PO X

P

, (as above)



r

rm

r

k

r

k

rP

pXO

1

)1( , by (3.3)

,)1(1

1

r

r

r

rm

r

k

r

k

rrp

P

P

pXO

as

n

n

nnp

PX

rXO

k

rm

r

k

r

1

1)1(

masO )1( , by (3.5).

Again

11

,5

2

k kmk

n n

n

T

11 1

1 1 1

2 11

1k k

km n

n n r n r r r

n rn n

P P X sP P

11 1

11

2 1

k k

km n

n rn r r r

n r n

PX s

P

11 1

11

2 1 1

k k

km n

n r rn r r r

n r n r

p PX s

P p

, by (3.2)

11 1

11

2 1 1

1k k

km n

n r rn r r

n r n r

p Ps

P p r

, by (3.1)

11 1

11

2 1 1

k k

km n

n r r rn r r r

n r n r r

p P ps X

P p P

, as

n

n

nnp

PX

1

11 1 1

1 11

2 1 11 1

k k

km n n

k k kn r n rn r r r

n r rn n

p ps X

P P

11

11

1 1 1

(1)k km m

k k n rr r n

r n r n

pO X

P

,)1(1

1

1

r

r

r

rm

r

k

r

k

rrp

P

P

pXO

as

n

n

nnp

PX and

by (3.3)

,)1(1

11

m

r

k

r

k

rX

rO

masO )1( , by (3.4).

Finally,

11

,6

2

k kmk

n n

n

T

11 1

2 1

2 11

1k k

km n

n n r n r r r

n rn n

P P X sP P

11 1

21

2 1 1

k k

km n

n rn r r r

n r n

PX s

P

11 1

21

2 1 2

k k

km n

n r rn r r r

n r n r

p PX s

P p

, by (3.2)

11 1

21

2 1 2

1k k

km n

n r rn r r

n r n r

p Ps

P p r

, by (3.1)

11 1

21

2 1 2

k k

km n

n r r rn r r r

n r n r r

p P ps X

P p P

, as

n

n

nnp

PX

1

11 1 1

2 21

2 1 12 2

k k

km n n

k k kn r n rn r r r

n r rn n

p ps X

P P

11

21

1 1 2

(1)k km m

k k n rr r n

r n r n

pO X

P

,)1(1

1

1

r

r

r

rm

r

k

r

k

rrp

P

P

pXO

as

n

n

nnp

PX and

By (3.3)

masO )1( , by (3.4).

This completes the proof of the Lemma.

V. PROOF OF THE THEOREM

Since the behavior of the Fourier series, as far as convergence

is concerned, for a particular value of x depends on the

behavior of the function in the immediate neighborhood of

this point only, thus the truth of the theorem is necessarily the

consequence of the Lemma.

VI. CONCLUSION

The paper provides an analytic idea in the field of

summability theory. In future, the present work can be

extended to establish some theorems on different indexed

summability factors of Fourier series as well as conjugate

series of Fourier series under certain weaker conditions.

ACKNOWLEDGMENT

The authors are highly thankful to the anonymous referees

for their careful reading, constructive comments and

suggestions. The authors are also grateful to all the members



of editorial board of International Journal of Research in

Science and Technology.

REFERENCES

[1] H.Bor., 1992, On the Local Property of k

npN ,

Summability of

factored Fourier series ,journal of mathematical analysis and applications-

163(220-226)

[2] Misra, U.K., Misra, M., Padhy, B.P and Buxi, S.K., 2010. On the Local

Property of , ,n nk

N p

Summability of factored Fourier series,

International Journal of Research and Reviews in applied Sciences, Vol.5,

No. 2, pp 162 – 167.

R. K. Jati is Presently working as a research Scholar

in the Post Graduate Department of Mathematics,

Ravenshaw University, Cuttack, Odisha, India and a

faculty member in Dhaneswar Rath Institute of

Engineering and Technology, Cuttack, Odisha, India.

He has been published around 5 research papers in

various national and international journals of repute.

Mr. Jati has 8 years of teaching and 5 years of research

experience.

Dr. S. K. Paikray is Presently working as a

Reader in the Department of Mathematics, VSS

University of Technology, Burla, Sambalpur, Odisha,

India. Prior to this he had worked as a lecturer in the

Post Graduate Department of Mathematics,

Ravenshaw University, Cuttack, Odisha, India. Dr.

Paikray has 15 years of teaching experiences. Under

the guidance of him 5 Ph. D scholars are continuing their research in different

Universities. Dr. Paikray has been published around 21 research papers in

various national and international journals of repute. The research area of Prof.

Dr. Paikray is summability theory, Fourier series, Operations Research and

Inventory control.

Prof. U. K. Misra had been working as a faculty

member in the Post Graduate Department of

Mathematics, Berhampur University for last 30 years.

Presently he has been working as a professor of

Mathematics in National Institute of Science and

Technology, Palur Hills, Berhampur. Under the

guidance of him 18 Ph. D scholars and 2 D.Sc.

scholars have been awarded their degrees. Presently 8 scholars are working

under his supervision. Prof Misra has been published around 100 research

papers in various national and international journals of repute. The research

area of Prof. Misra is sequence space, summability theory, Fourier series,

Operations Research, Inventory control and mathematical modeling. He is a

reviewer of Mathematical Review published by American Mathematical

Society. Prof Misha has conducted several National and International seminar,

conferences and refresher courses sponsored by UGC.