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Ijetcas14 605
1.
International Association of
Scientific Innovation and Research (IASIR) (An Association Unifying the Sciences, Engineering, and Applied Research) (An Association Unifying the Sciences, Engineering, and Applied Research) International Journal of Emerging Technologies in Computational and Applied Sciences (IJETCAS) www.iasir.net 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 __________________________________________________________________________________________ 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 ______________________________________________________________________________________ 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.
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