Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 351016, 7 pages
doi:10.1155/2010/351016
Research Article
A Study on Degree of Approximation by
E, 1 Summability Means of
the Fourier-Laguerre Expansion
H. K. Nigam and Ajay Sharma
Department of Mathematics, Faculty of Engineering & Technology, Mody Institute of Technology and
Science, Deemed University, Laxmangarh - 332311, Sikar, Rajasthan, India
Correspondence should be addressed to Ajay Sharma, ajaymathematicsanand@gmail.com
Received 12 April 2010; Accepted 26 July 2010
Academic Editor: V. R. Khalilov
Copyright q 2010 H. K. Nigam and A. Sharma. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
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.
1. Introduction
Let
∞
n0
un be an infinite series with the sequence of its nth partial sums {sn }.
If
En1 n
1 n
Ck sn−k −→ s
2n k0
as n −→ ∞,
1.1
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 ∞
n0 un .
The Fourier-Laguerre expansion of a function fx ∈ L0, ∞ is given by
fx ∼
∞
n0
α
an Ln x,
1.2
2
International Journal of Mathematics and Mathematical Sciences
where
−1 ∞
α nα
e−y yα f y Ln y dy
an Γα 1
n
0
1.3
α
and Ln x denotes the nth Laguerre polynomial of order α > −1, defined by generating
function
−xω
,
1−ω
1.4
φ y {Γα 1}−1 e−y yα f y − f0 .
1.5
∞
α
Ln xωn 1 − ω−α−1 exp
n0
and existence of integral 1.3 is presumed.
We write
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
Ft t f y
0
y
∞
1p
1
dy o log
,
t
e
t −→ 0, −1 < p < ∞,
f y dy < ∞,
1.6
−y/2 3α−3k−1/3 y
1
then
p1
σnk 0 o log n
1.7
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,
p1
,
tn 0 − f0 o log n
1.8
provided that
δ 1p
φ y
1
dy
o
log
,
α1
t
y
t
t −→ 0, −1 < p < ∞,
1.9
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3
δ is a fixed positive constant,
n
1p ,
ey/2 y−2α3/4 φ y dy o n−2α1/4 log n
δ
∞
e
p1 φ y dy o log n
,
y/2 −1/3 y
1.10
n −→ ∞.
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 of α is increased to −1 < α < −1/2, which is more useful
for application.
In fact, we establish the following theorem.
Theorem 2.1. If
En1 n
1 n
Ck sk −→ ∞ as n −→ ∞,
2n k0
2.1
then the degree of approximation of Fourier-Laguerre expansion 1.2 at the point x 0 by E, 1
means En1 is given by
En1 0 − f0 o{ξn}
2.2
provided that
t
φ y dy o tα1 ξ 1
,
t
t −→ 0,
2.3
ey/2 y−2α3/4 φ y dy o n−2α1/4 ξn ,
2.4
Φt 0
δ is a fixed positive constant and −1 < α < −1/2,
n
δ
∞
ey/2 y−1/3 φ y dy o{ξn},
n −→ ∞,
n
where ξt is a positive monotonic increasing function of t such that ξn → ∞ as n → ∞.
2.5
4
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3. Lemmas
Lemma 3.1 see the study by Szegö, 1959, 3, page 175. Let α be arbitrary and real, let c and ∈
be fixed positive constants, and let n → ∞. Then
α
Ln x O x−2α1/4 n2α−1/4
c
≤ x ≤∈,
n
∈
if 0 ≤ x ≤ .
n
α
Ln x Onα if
3.1
3.2
4. Proof of the Main Theorem
Since
α
Ln 0 nα
,
α
4.1
therefore,
sn 0 n
k0
α
ak Lk 0
{Γα 1}−1
{Γα 1}
−1
∞
0
∞
0
n
α e−y yα f y
Lk y dy
k0
α1 e−y yα f y Ln
y dy.
4.2
Now,
En1 0 n
1 n
Ck sk 0
2n k0
∞
n
α1 1 −1
n
n
e−y yα f y Lk
y dy.
Ck {Γα 1}
2 k0
0
4.3
Using orthogonal property of Laguerre’s polynomial and 1.5, we have
∞
n
α1 1 n
φ y Lk
y dy
C
k
n
2 k0
0
1/n δ
n ∞ α1 1 n n
y dy
C k φ y Lk
n
2
0
1/n
δ
n
k0
En1 0 − f0 I1 I2 I3 I4
say .
4.4
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5
Using orthogonal property and condition 3.2 taking α 1 for α and δ for ∈ of Lemma 3.1,
we get
I1 n
1/n 1 α1
n
φ y dy
O
n
C
k
2n k0
0
n
1
1 α1
n
o
O
n
ξn
,
C
k
2n k0
nα1
n
1 n
I1 o
Ck ξn
2n k0
o{ξn}
since
n
n
4.5
n
Ck 2 .
k0
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
I2 n
δ 1 2α1/4
n
φ y y−2α3/4 dy.
O
n
Ck
2n k0
1/n
4.6
Now,
n
n
2α1/4
Ck n
k0
⎧
⎨n/2
⎩ k0
⎫
⎬
n
⎭
kn/21
n2α1/4
n/2
n
n
2α1/4
Ck n
Ck Cn/2 n
n
2α5/4
4.7
k0
≤ n2α1/4
n
n
2α5/4
Ck Cn/2 n
n
,
k0
n
n
2α1/4
Ck n
n2α1/4 2n n Cn/2 n2α5/4
k0
since
2n n
n
Ck
k0
n C0 n C1 n C2 · · · n Cn/2 n Cn/21 · · ·
≥ Cn/2 Cn/2 · · · Cn
n
n
n
≥ n Cn/2 n Cn/2 · · · n Cn/2 n n
1 n Cn/2 ≥ n Cn/2 .
2
2
4.8
6
International Journal of Mathematics and Mathematical Sciences
Therefore,
nn
n
Cn/2 ≤ 2 .
2
4.9
By 4.7 and 4.9, we have,
n
n
Ck
n2α1/4 ≤ n2α1/4 2n 22n n2α1/4
k0
4.10
O n2α1/4 2n .
Thus,
δ
2α1/4
y−2α3/4 φ y dy
I2 O n
1/n
δ
2α1/4
O n
y−2α3/4 Φ y
1/n
δ
1/n
2α 3 −2α7/4 y
Φ y dy
4
δ
!
1
1
2α1/4
−2α3/4
α1
−2α7/4
α1
dy
y
o y ξ
y
o y ξ
O n
y
y
1/n
δ
δ
!
1
1
2α1/4
2α1/4
2α−3/4
o y
dy
ξ
o
y
ξ
O n
y
y
1/n
1/n
!
δ
2α1/4
−2α1/4
2α−3/4
O1 o n
ξn ξno
y
dy
O n
1/n
o{ξn} o n2α1/4 ξn
δ
!
y2α−3/4 dy
1/n
o{ξn} o n2α1/4 ξn
o{ξn} o n2α1/4 ξn
y2α−3/41
{2α − 3/4} 1
y2α1/4
2α 1/4
o{ξn} o n2α1/4 ξn n−2α1/4
!δ
1/n
!δ
1/n
o{ξn} o{ξn},
I2 o{ξn}.
4.11
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7
Now, we consider
I3 1
2n
n
n
n
Ck
k0
e
−y/2 2α3/4 α1 φ y e
y
y dy
Ln
!
y/2 −2α3/4 y
δ
!
n
n
1
2α1/4
y/2 −2α3/4 n
φ y dy ,
e y
Ck O n
2n
δ
k0
I3 O n2α1/4 o n−2α1/4 ξn , using 2.4,
4.12
I3 o{ξn}.
Finally,
I4 1
2n
n
k0
n
∞
Ck
e
−y/2 3α5/6 α1 φ y e
y
y dy
Ln
!
y/2 −3α5/6 y
n
!
n
∞ ey/2 y−1/3 φy
1
α1/2
n
dy ,
Ck O k
2n
yα1/2
n
k0
1
n
α1/2 −α1/2
n
o{ξn}
, by 2.5
n
I4 O
2
2n
4.13
I4 o{ξn}.
Combining 4.4, 4.5, 4.11, 4.12, and 4.13, we get
En1 0 − f0 o{ξn}.
4.14
This completes the proof of the theorem.
References
1 G. H. Hardy, Divergent Series, University Press, Oxford, UK, 1st edition, 1949.
2 D. P. Gupta, “Degree of approximation by Cesàro means of Fourier-Laguerre expansions,” Acta
Scientiarum Mathematicarum, vol. 32, pp. 255–259, 1971.
3 G. Szegö, Orthogonal Polynomials, Colloquium Publications American Mathematical Society, New York,
NY, USA, 1959.
4 T. Singh, “Degree of approximation by harmonic means of Fourier-Laguerre expansions,” Publicationes
Mathematicae Debrecen, vol. 24, no. 1-2, pp. 53–57, 1977.