# A Study on Degree of Approximation by(E,1) Summability Means of the Fourier-Laguerre Expansion

Authors: H. K. Nigam; Ajay Sharma
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

A very new theorem on the degree of approximation of the generating function by(E,1) means of its Fourier-Laguerre series at the frontier point x=0 is obtained.


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## Body

## 1. Introduction

Let∑n=0∞un be an infinite series with the sequence of its nth partial sums {sn}.If(1.1)En1=12n∑k=0nCnksn-k→sasn→∞,
then we say that {sn} is summable by (E,1) means (see the study by Hardy [1]), and it is written as sn→s(E,1), where {sn} is the sequence of nth partial sums of the series ∑n=0∞un.The Fourier-Laguerre expansion of a functionf(x)∈L(0,∞) is given by
(1.2)f(x)~∑n=0∞anLn(α)(x),
where
(1.3)an={Γ(α+1)(n+αn)}-1∫0∞e-yyαf(y)Ln(α)(y)dy
and Ln(α)(x) denotes the nth Laguerre polynomial of order α>-1, defined by generating function
(1.4)∑n=0∞Ln(α)(x)ωn=(1-ω)-α-1exp(-xω1-ω),
and existence of integral (1.3) is presumed.We write(1.5)ϕ(y)={Γ(α+1)}-1e-yyα{f(y)-f(0)}.
Gupta [2] estimated the order of the function by Cesàro means of series (1.2) at the point x=0 after replacing the continuity condition in Szegö’s theorem [3] by a much lighter condition. He established the following theorem.Theorem 1.1.
If(1.6)F(t)=∫0t|f(y)|ydy=o{log(1t)}1+p,t→0,-1<p<∞,∫1∞e-y/2y(3α-3k-1)/3|f(y)|dy<∞,
then
(1.7)σnk(0)=o(logn)p+1
provided that k>α+1/2,α>-1, with σnk(0)  being the nth Cesàro mean of order k.Denoting the harmonic means by{tn}, Singh [4] estimated the order of function by harmonic means of series (1.2) at point x=0 by weaker conditions than those of Theorem 1.1. He proved the following theorem.Theorem 1.2.
For-5/6<α<-1/2,(1.8)tn(0)-f(0)=o(logn)p+1,
provided that
(1.9)∫tδ|ϕ(y)|yα+1dy=o{log(1t)}1+p,t→0,-1<p<∞,δ is a fixed positive constant,
(1.10)∫δney/2y-(2α+3)/4|ϕ(y)|dy=o{n-(2α+1)/4(logn)1+p},∫n∞ey/2y-1/3|ϕ(y)|dy=o{(logn)p+1},n→∞.


## 2. Main Theorem

The objects of present paper are as follows:(1)
We prove our theorem for(E,1) means which is entirely different from (C,k) and  harmonic means.(2)
We employ a condition which is weaker than condition (1.9) of Theorem 1.2.(3)
In our theorem the range ofα is increased to  -1<α<-1/2, which is more useful for application.In fact, we establish the following theorem.Theorem 2.1.
If(2.1)En1=12n∑k=0nCnksk→∞asn→∞,
then the degree of approximation of  Fourier-Laguerre expansion (1.2) at the point x=0 by (E,1) means En1  is given by
(2.2)En1(0)-f(0)=o{ξ(n)}
provided that
(2.3)Φ(t)=∫0t|ϕ(y)|dy=o{tα+1ξ(1t)},t→0,δ is a fixed positive constant and -1<α<-1/2,
(2.4)∫δney/2y-(2α+3)/4|ϕ(y)|dy=o{n-(2α+1)/4ξ(n)},(2.5)∫n∞ey/2y-1/3|ϕ(y)|dy=o{ξ(n)},n→∞,
where ξ(t) is a positive monotonic increasing function of t such that ξ(n)→∞ as n→∞.


## 3. Lemmas

Lemma 3.1 (see the study by Szegö, 1959, [3, page 175]).
Letα be arbitrary and real, let c and ∈ be fixed positive constants, and let n→∞. Then
(3.1)Ln(α)(x)=O(x-(2α+1)/4n(2α-1)/4)ifcn≤x≤∈,(3.2)Ln(α)(x)=O(nα)if0≤x≤∈n.


## 4. Proof of the Main Theorem

Since(4.1)Ln(α)(0)=(n+αα),
therefore,
(4.2)sn(0)=∑k=0nakLk(α)(0)={Γ(α+1)}-1∫0∞e-yyαf(y)∑k=0nLk(α)(y)dy={Γ(α+1)}-1∫0∞e-yyαf(y)Ln(α+1)(y)dy.
Now,
(4.3)En1(0)=12n∑k=0nCnksk(0)=12n∑k=0nCnk{Γ(α+1)}-1∫0∞e-yyαf(y)Lk(α+1)(y)dy.
Using orthogonal property of Laguerre’s polynomial and (1.5), we have
(4.4)En1(0)-f(0)=12n∑k=0nCnk∫0∞ϕ(y)Lk(α+1)(y)dy=(∫01/n+∫1/nδ+∫δn+∫n∞)12n∑k=0nCnkϕ(y)Lk(α+1)(y)dy=I1+I2+I3+I4(say).
Using orthogonal property and condition (3.2) (taking α+1 for α and δ for ∈) of Lemma 3.1, we get
(4.5)I1=12n∑k=0nCnkO{nα+1}∫01/n|ϕ(y)|dy=12n∑k=0nCnkO{nα+1}o{1nα+1ξ(n)},I1=o(12n∑k=0nCnkξ(n))=o{ξ(n)}since∑k=0nCnk=2n.
Further, using orthogonal property and condition (3.1) (taking α+1 for α, 1 for c, and δ for ∈) of Lemma 3.1, we get
(4.6)I2=12n∑k=0nCnkO{n(2α+1)/4}∫1/nδ|ϕ(y)|y-(2α+3)/4dy.
Now,
(4.7)∑k=0nCnkn(2α+1)/4={∑k=0[n/2]+∑k=[n/2]+1n}Cnkn(2α+1)/4=(n)(2α+1)/4∑k=0[n/2]Cnk+Cn[n/2](n)(2α+5)/4≤(n)(2α+1)/4∑k=0nCnk+Cn[n/2](n)(2α+5)/4,∑k=0nCnkn(2α+1)/4=(n)(2α+1)/42n+Cn[n/2](n)(2α+5)/4
since
(4.8)2n=∑k=0nCnk=Cn0+Cn1+Cn2+⋯+Cn[n/2]+Cn[n/2]+1+⋯≥Cn[n/2]+Cn[n/2]+⋯+Cnn≥Cn[n/2]+Cn[n/2]+⋯+Cn[n/2]={[n2]+1}Cn[n/2]≥n2Cn[n/2].
Therefore,
(4.9)n2Cn[n/2]≤2n.
By (4.7) and (4.9), we have,
(4.10)∑k=0nCnk{n(2α+1)/4}≤(n)(2α+1)/42n+2(2n)(n)(2α+1)/4=O{(n)(2α+1)/42n}.
Thus,
(4.11)I2=O{(n)(2α+1)/4}∫1/nδy-(2α+3)/4|ϕ(y)|dy=O{(n)(2α+1)/4}[{y-(2α+3)/4Φ(y)}1/nδ+∫1/nδ(2α+3)4y-(2α+7)/4Φ(y)dy]=O{(n)(2α+1)/4}[y-(2α+3)/4o{yα+1ξ(1y)}]+{∫1/nδy-(2α+7)/4o{yα+1ξ(1y)}dy}=O{(n)(2α+1)/4}[o{y(2α+1)/4ξ(1y)}]1/nδ+o{∫1/nδy(2α-3)/4ξ(1y)dy}=O{(n)(2α+1)/4}[O(1)+o{n-(2α+1)/4ξ(n)}]+ξ(n)o{∫1/nδy(2α-3)/4dy}=o{ξ(n)}+o{(n)(2α+1)/4ξ(n)}{∫1/nδy(2α-3)/4dy}=o{ξ(n)}+o{n(2α+1)/4ξ(n)}{y(2α-3)/4+1{(2α-3)/4}+1}1/nδ=o{ξ(n)}+o{n(2α+1)/4ξ(n)}{y(2α+1)/4(2α+1)/4}1/nδ=o{ξ(n)}+o{n(2α+1)/4ξ(n)}{n-(2α+1)/4}=o{ξ(n)}+o{ξ(n)},I2=o{ξ(n)}.
Now, we consider
(4.12)I3=[(12n){∑k=0nCnk∫δney/2y-(2α+3)/4|ϕ(y)|e-y/2y(2α+3)/4|Ln(α+1)(y)|dy}]=(12n){∑k=0nCnkO{n(2α+1)/4}∫δney/2y-(2α+3)/4|ϕ(y)|dy},I3=O{(n)(2α+1)/4}o{(n)-(2α+1)/4ξ(n)},using(2.4),I3=o{ξ(n)}.
Finally,
(4.13)I4=[(12n){∑k=0nCnk∫n∞ey/2y-(3α+5)/6|ϕ(y)|e-y/2y(3α+5)/6|Ln(α+1)(y)|dy}]=(12n){∑k=0nCnkO{k(α+1)/2}∫n∞ey/2y-1/3|ϕ(y)|y(α+1)/2dy},I4=O[(12n)(2n){n(α+1)/2n-(α+1)/2}o{ξ(n)}],by(2.5)I4=o{ξ(n)}.
Combining (4.4), (4.5), (4.11), (4.12), and (4.13), we get
(4.14)En1(0)-f(0)=o{ξ(n)}.
This completes the proof of the theorem.



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*Source: 351016-2010-08-18.xml*