# On(N,p,q)(E,1) Summability of Fourier Series

Authors: H. K. Nigam; Ajay Sharma
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

A new theorem on(N,p,q)(E,1) summability of Fourier series has been established.


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## Body

## 1. Introduction

Let{pn} and {qn} be  the  sequences of constants, real or complex, such that(1.1)Pn=p0+p1+p2+⋯+pn=∑ν=0npν→∞,asn→∞,Qn=q0+q1+q2+⋯+qn=∑ν=0nqν→∞,asn→∞,Rn=p0qn+p1qn-1+⋯+pnq0=∑ν=0npνqn-ν→∞,asn→∞.
Given two sequences{pn} and {qn} convolution (p*q) is defined as
(1.2)Rn=(p*q)n=∑k=0npn-kqk.
Let  ∑n=0∞un be an infinite  series with the sequence of its nth partial sums {sn}.We write(1.3)tnp,q=1Rn∑k=0npn-kqksk.IfRn≠0,  for   all  n,  the  generalized    Nörlund   transform  of   the   sequence  {sn}   is  the sequence {tnp,q}.Iftnp,q→S,asn→∞,   then  the  series   ∑n=0∞un  or   sequence  {sn}   is   summable   to  S  by generalized Nörlund method (Borwein [1]) and is denoted by(1.4)Sn→S(N,p,q).The necessary  and sufficient conditions for(N,p,q) method to be regular are
(1.5)∑k=0n|pn-kqk|=O(|Rn|),andpn-k=o(|Rn|),asn→∞ for every fixed k≥0, for which qk≠0.Now(1.6)En1=12n∑k=0n(nk)sk.IfEn1→s,asn→∞,   then  the   series ∑n=0∞un is  said  to  be (E,1)  summable  to   s  (Hardy [2]):
(1.7)tnp,q,E=1Rn∑k=0npn-kqkEk1=1Rn∑k=0npn-kqk·12k∑ν=0k(kν)sν.Iftnp,q,E→∞,asn→∞, then we say that the series  ∑n=0∞un or the sequence {sn} is summable to S by (N,p,q)(E,1) summability method.Particular Cases
(1)
(N,p,q)(E,1) mean reduces to (N,pn)(E,1) summability mean if qn=1,∀n.(2)
(N,p,q)(E,1) mean reduces to (N,1/(n+1))(E,1) mean if pn=1/(n+1)andqn=1,∀n.(3)
(N,p,q)(E,1) method reduces to (N¯,qn)(E,1)  if  pn=1,∀n.(4)
(N,p,q)(E,1) method reduces  to(C,α)(E,1) if pn=(n+α-1α-1),α>0,andqn=1,∀n.Let f(t) be a  periodic function with period 2π and integrable in the sense of  Lebesgue over the interval (-π,π).
Let its Fourier series be given by
(1.8)f(t)~12a0+∑n=1∞(ancosnt+bnsinnt).
We use the following notations:
(1.9)ϕ(t)=f(x+t)-f(x-t)-2f(x),Φ(t)=∫0t|ϕ(u)|du,τ=[1t]=theintegralpartof1t,R(1t)=Rτ,Rn=R(n),Kn(t)=12πRn∑k=0npn-kqkcosk(t/2)cos(k+1)(t/2)sin(t/2).


## 2. Theorem

A quite good amount of work is known for Fourier series by ordinary summability method. The purpose of this paper is to study Fourier series by(N,p,q)(E,1) summability method in the following form.Theorem 2.1.
Let{pn}and{qn} be positive monotonic, nonincreasing sequences of real numbers such that
(2.1)Rn=∑k=0npkqn-k→∞,asn→∞.
Let α(t) be a positive, nondecreasing function of t.  If
(2.2)Φ(t)=∫0t|ϕ(u)|du=o(tα(1/t)),ast→+0,(2.3)α(n)→∞,asn→∞,
then a  sufficient  condition  that  the  Fourier  Series  (1.8)  be   summable   (N,p,q)(E,1)   to f(x)  at the point t=x  is
(2.4)∫1nR(u)uα(u)du=O(Rn),asn→∞.


## 3. Lemmas

Proof of the theorem needs some lemmas.Lemma 3.1.
For0≤t≤1/n,(3.1)|Kn(t)|=O(n).Proof.
(3.2)|Kn(t)|=12πRn|∑k=0npn-kqkcosk(t/2)sin(k+1)(t/2)sin(t/2)|≤12πRn∑k=0npn-kqk(k+1)|sin(t/2)||sin(t/2)|=O(n)1Rn∑k=0npn-kqk=O(n).Lemma 3.2.
If{pn} and {qn} are nonnegative and nonincreasing, then for  0≤a≤b<∞, 0≤t≤π, and any  n we have
(3.3)12πRn|∑k=abpn-kqkcosk(t/2)sin(k+1)(t/2)sin(t/2)|=O(RτtRn).Proof.
(3.4)12πRn|∑k=abpn-kqkcosk(t/2)sin(k+1)(t/2)sin(t/2)|≤1tπRn|∑k=abpn-kqkcosk(t2)sin(k+1)t2|=1tπRn|Im{∑k=abpn-kqkcosk(t2)ei(k+1)(t/2)}|≤1tπRn|∑k=abpn-kqkcosk(t2)eikt/2||eit/2|≤1tπRn|∑k=abpn-kqkcosk(t2)eikt/2|≤1tπRn{|∑k=aτ-1pn-kqkcosk(t2)eikt2|+|∑k=τbpn-kqkcosk(t2)eik(t2)|}.
Now considering first term of (3.4), we have
(3.5)1tπRn|∑k=aτ-1pn-kqkcosk(t2)eik(t/2)|≤1tπRn∑k=aτ-1pn-kqk|eik(t/2)|≤1tπRn∑k=aτ-1pn-kqk≤1tπRn∑k=aτ-1pτ-kqk≤1tπRn(Rτ)=O(RτtRn).
Now considering second term of (3.4) and using Abel’s lemma, we have
(3.6)1tπRn|∑k=τbpn-kqkcosk(t2)eik(t/2)|≤1tπRn|∑k=τbpn-kqkeik(t/2)|≤2pn-bqτtπRnmaxτ+1≤k≤b|1-ei(k+1)(t/2)1-eit/2|≤4pn-bqτtπRn|e-it/4eit/4-e-it/4|≤2qτtπRn(pn-bPτ)Pτ|1sin(t/4)|(wherePτ=∑k=0τpτ-k)≤8qτtπRn(pn-bPτ)Pτ|1t|≤8qτPτtπRn≤8RτtπRn(sinceRτ=∑k=0τpτ-kqk≥Pτqτ)=O(RτtRn).
Using (3.5) and (3.6), we get the required result of Lemma 3.2.


## 4. Proof  of Theorem

Following Zygmund [3], the nth partial sum sn(x) of the series (1.8) at t=x  is given by
(4.1)sn(x)=f(x)+12π∫0πϕx(t)sin(n+1/2)tsin(t/2)dt.So the(E,1) mean of the series (1.8) att=x is given by
(4.2)En1(x)=12n∑k=0n(nk)sk(x)=f(x)+12n+1π∫0πϕx(t)sin(t/2){∑k=0n(nk)sin(k+12)t}dt=f(x)+12n+1π∫0πϕx(t)sin(t/2)Im{eit/2(1+eit)n}dt=f(x)+12n+1π∫0πϕx(t)sin(t/2)Im{eit/2(1+cost+isint)n}dt=f(x)+12n+1π∫0πϕx(t)sin(t/2)Im{eit/22ncosn(t2)(cost2+isint2)n}dt=f(x)+12n+1π∫0πϕx(t)sin(t/2)Im{eit/22ncosn(t2){(cosnt2+isinnt2)}}dt=f(x)+12π∫0πϕx(t)cosn(t/2)sin(n+1)(t/2)sin(t/2)dt.Therefore(N,p,q) transform of {En1(x)}  is given by
(4.3)tnp,q,E(x)=1Rn∑k=0npn-kqkEk1(x)=f(x)+12π∫0π1Rn∑k=0npn-kqkϕx(t)cosk(t)sin(k+1)(t/2)sin(t/2)=f(x)+∫0πKn(t)ϕx(t)dt,tnp,q,E(x)-f(x)=[∫01/n+∫1/nδ+∫δπ‍]Kn(t)ϕx(t)dt=I1+I2+I3(say).
We have
(4.4)|I1|≤∫01/n|Kn(t)||ϕx(t)|dt=O(n)∫01/n|ϕx(t)|dt(usingLemma3.1)=O(n)o(1nα(n))(by(2.2))=o(1α(n))=o(1)asn→∞(by(2.3)).Now we consider(4.5)|I2|≤∫1/nδ|Kn(t)||ϕx(t)|dt(where0<δ<1)=∫1/nδO(R(1/t)tRn)|ϕx(t)|dt(usingLemma3.2)=O(1Rn)∫1/nδ(R(1/t)t)|ϕx(t)|dt=O(1Rn)[{(R(1/t)t)ϕx(t)}1/nδ-∫1/nδd(R(1/t)t)ϕx(t)]=O(1R(n))[{o(R(1/t)α(1/t))}1/nδ-∫1/nδϕx(t)d(R(1/t)t)](by(2.1))=o(1R(n))+o(1α(n))+o(1R(n))[∫1/nδϕx(t){d(R(1/t)α(1/t)tα(1/t))}]=o(1R(n))+o(1α(n))+o(1R(n))×[∫1/nδo(tα(1/t)){dα(1/t)(R(1/t)tα(1/t))}+α(1t)d(R(1/t)tα(1/t))]=o(1R(n))+o(1α(n))+o(1)[∫1/nδdα(1/t){α(1/t)}2+o(1R(n))∫1/nδtd(R(1/t)tα(1/t))]=o(1R(n))+o(1α(n))+o(1){1α(1/t)}1/nδ+o(1R(n))[{tR(1/t)tα(1/t)}1/nδ-∫1/nδ(R(1/t)tα(1/t))dt]=o(1R(n))+o(1α(n))+o(1α(n))+o(1R(n))∫1/n1(R(1/t)tα(1/t))dt=o(1R(n))+o(1α(n))+o(1R(n))∫1n(R(u)uα(u))du{∵1t=u}=o(1R(n))+o(1α(n))+o(1R(n))O(Rn)(by(2.4))=o(1R(n))+o(1α(n))+o(1)=o(1),asn→∞(byvirtueof(2.1)and(2.2)).Now by Riemann-Lebesgue theorem and by regularity of the method of summability we have(4.6)I3=∫δπ|kn(t)||ϕx(t)|dt=o(1),asn→∞.This completes the proof of the theorem.


## 5. Corollaries

Following corollaries can be derived from our main theorem.Corollary 5.1.
If(5.1)Φ(t)=o[tlog(1/t)],ast→+0,
then the Fourier series (1.8) is (C,1)(E,1) summable to f(x) at the  point t=x.Corollary 5.2.
If(5.2)Φ(t)=o(t),ast→+0,
then the Fourier series (1.8) is (N,pn)(E,1) summable to f(x) at the point t=x, provided that {pn} be a positive, monotonic, and nonincreasing sequence of real numbers such that
(5.3)pn=p0+p1+⋯+pn→∞,asn→∞.



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*Source: 989865-2009-06-08.xml*