This document presents a theorem about the almost Norlund summability of conjugate Fourier series. It generalizes previous results by Pati (1961) and Singh and Singh (1993). The main theorem states that if the conjugate partial sums of a Fourier series satisfy certain conditions, including being bounded by a function that approaches 0 as n approaches infinity, then the conjugate Fourier series is almost Norlund summable to the integral of the function at every point where the integral exists. The proof utilizes lemmas about the behavior of the conjugate partial sums and applies mean value theorems to show the necessary conditions are met. References to previous related works are also provided.

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International Journal of Emerging Technologies in Computational
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IJETCAS 14- 605; © 2014, IJETCAS All Rights Reserved Page 266
ISSN (Print): 2279-0047
ISSN (Online): 2279-0055
ALMOST NORLUND SUMMABILITY OF CONJUGATE SERIES OF A FOURIER SERIES
V. S. Chaubey
Department of Mathematics, B R D P G College,
Deoria (274001), U.P., India
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Abstract: In this paper a more general result than those of Pati, T (1961), U.N Singh and V.S Singh (1993) has been obtained so that their results come out as particular cases. U.N Singh and V.N Singh has been given some interesting result on this in 1993.
Keywords: Almost Norlund Summability, Fourier Series
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I. INTRODUCTION
Let n be an infinite series with {sn} as the sequence of its n-th partial sums. Lorentz (1948) has given the following definition:
Definition: A bounded sequence { } is said to be almost convergent to a limit s if
v = s (1.1)
uniformly with respect to m.
Let { Pn} be a sequence of non – zero real constants and
Pn= p0 + p1 + p2 …………….pn , pn-1= Pn-1 =0 (1.2)
We define that the series an or sequence {sn} is said to be almost (N,Pn) summable to s if
tn,m = n –v ,Sv,m tends to s (1.3)
As n , uniformly with respect to m
Sv,m k (1.4)
Let the fourier series of a 2 -periodic and Lebesgue Integrable function f in ( – ) be given by
F (t) a0 + cosnt + bn sin nt)= n (t) (1.5)
And then the conjugate series of (1.5) is
( bn cos nt – an sin nt ) = n (t) (1.6)
Let (x) denote the n- th partial sum of the series Bn (x) . Then we write
v,m= k (1.7)
Let us write (t) = f(x + t) + f(x-t) – 2f(x ) (1.8)
(t) = f(x +t ) – f(x – t) (1.9)
(t) = (u) | du (1.10)
(t) = (u) | du (1.11)
= n-v (1.12)
= n-v (1.13)
T = [ ] = the integral part of
II. MAIN THEOREM
Pati (1961) has established the following theorem on the norlund summability of a Fourier series:
Theorem: If (N, Pn ) be a regular norlund method defined by real , non – negative monotonic non –increasing sequence of co- efficient {po} such that pn as n
Pn = v as n and log n = O (Pn) , as n (2.1)
Then if (t) = (u) | du = O [ ] as t to (2.2)
The series (1.5) is summable (N,Pn) to f(x) at the point t = x

2. V. S. Chaubey, International Journal of Emerging Technologies in Computational and Applied Sciences, 9(3), June-August, 2014, pp. 266-
268
IJETCAS 14- 605; © 2014, IJETCAS All Rights Reserved Page 267
Theorem: Let {pn} be a real non – negative monotonic , non increasing sequence of coefficient such that pn , h
if ( t ) = (u)| du = O [ ] as t 0. and du = O (1), as n
where o uniformly with respect to m, then the series (1.6 ) is almost (N, pn) summable to t dt at every point where this integral exist is provided (n) log n = O (pn) , as n .
III. LEMMA
The following lemma is essential for the proof of our theorem.
Lemma: if (t) is given by (1.13 ) Then (t) = { O (n+ m) , for 0
= { O ( ), for
Proof of the lemma:-
We have (t) = n-v
= n-v (
=O (n+m), for
Similarly, on expanding sine, cosine in powers of t,
we get (t) = o( ) = for
IV. PROOF OF MAIN THEOREM
Let (x) denote the n- th partial sum of the series n (x) .Then we have
(x) = – )t dt
(x) - t dt = dt
So that
= – f(x) }
= - dt
= (t) dt
= dt
Now by (1.12) we have
-
= n-v sin dt
= - ( t) dt =0(1) = R( say)
In order to prove the theorem we have to show that under our assumptions
(t) dt = o (1) as n
For o
(t) dt = o [ (t) (t) dt
=R1 + R2 + R3 Say (4.1)
Let us first consider R1
Now |R1| = o[ (t) dt]
=O(n+m) [ (lemma)
=O (n+m) [ ] as t .
=0(1) as n Next considering R2 we have
|R2| =0 [ ]=0 (1) as n by (1.2)
Lastly we have

3. V. S. Chaubey, International Journal of Emerging Technologies in Computational and Applied Sciences, 9(3), June-August, 2014, pp. 266-
268
IJETCAS 14- 605; © 2014, IJETCAS All Rights Reserved Page 268
R3 = n-v dt
= I3.1 – I3.2 ( say )
Now using second mean value theorm
I3.1 n-v
Where
= O (1) as n (4.2)
uniformly with respect to m.
Similarly I3.2 = n-v
= O(1). As n (4.3)
uniformly with respect to m.
Collecting from (1.4) to (1.5) we get the required result
This completes the proof of our main theorem.
REFERENCES
[1]. Lorentz G.G (1948) . Acta Mathematics 80,167.
[2]. Pati , T(1961) : Indian journal of Mathematics , 3, 85.
[3]. Singh V.N.and Singh V.S. (1993) : Bull .cal Math. Soc 87,57- 62.