International Journal of Analysis and Applications
ISSN 2291-8639
Volume 1, Number 1 (2013), 33-39
http://www.etamaths.com

ON THE DEGREE OF APPROXIMATION OF A FUNCTION BY

(C, 1)(E,q) MEANS OF ITS FOURIER-LAGUERRE SERIES

XHEVAT Z. KRASNIQI

Abstract. In this note a theorem on the degree of approximation of a function

by (C, 1)(E, q) means of its Fourier-Laguerre series at the frontier point x = 0

is proved.

1. Introduction

Let us consider the infinite series
∑∞
n=0 un with the sequence of its n-th partial

sums s := {sn}.
If for q > 0

(1.1) Eqn(s) =
1

(1 + q)n

n∑
k=0

(
n

k

)
qksk → s1 as n →∞,

then it is said that s := {sn} is summable by (E,q) means (see Hardy [3]), and we
write sn → s1(E,q).

The Fourier-Laguerre expansion of a function f(x) ∈ L(0,∞) is given by

(1.2) f(x) ∼
∞∑
n=0

anL
(α)
n (x),

where

(1.3) an =
1

Γ(α + 1)
(
n+α
n

) ∫ ∞
0

e−yyαL(α)n (y)dy,

L
(α)
n (x) denotes the n-th Laguerre polynomial of order α > −1, defined by gener-

ating function

(1.4)

∞∑
n=0

L(α)n (x)ω
n =

e
xω
ω−1

(1 −ω)α+1
,

and it is assumed that the integral (1.3) exists.
In 1971, D. P. Gupta [2] estimated the order of the function by Cesáro means of

series (1.2) at the point x = 0, after replacing the continuity condition in Szegö’s
theorem [6] by a much lighter condition. He proved the following theorem.

2010 Mathematics Subject Classification. 42C10, 40G05, 41A25.
Key words and phrases. (C, 1)(E, q) summability, Fourier-Laguerre series, Degree of

approximation.

c©2013 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

33



34 KRASNIQI

Theorem 1.1 ([2]). If

F(t) =

∫ t
0

|f(y)|
y

dy = o

(
log

(
1

t

))1+p
, t → 0,−1 < p < ∞,

and ∫ ∞
1

e−y/2y(3α−3k−1)/3|f(y)|dy < ∞,

are fulfilled, then
σkn(0) = o (log n)

p+1

provided that k > α + 1/2, α > −1, with σkn(0) being the n-th Cesàro mean of order
k.

Further, we use the notation

(1.5) φ(y) =
e−yyα[f(y) −f(0)]

Γ(α + 1)
,

and denote by tn harmonic means of the series (1.2). T. Singh [5] estimated the
deviation tn(x) −f(x) at the point x = 0 by some weaker conditions than those of
Theorem 1.1. Namely, he verified the following theorem.

Theorem 1.2 ([5]). For α ∈ (−5/6,−1/2)

tn(0) −f(0) = o (log n)
p+1

provided that ∫ δ
t

|φ(y)|
yα+1

dy = o

(
log

(
1

t

))1+p
, t → 0,−1 < p < ∞,

(1.6)

∫ n
δ

ey/2y−(2α+3)/4|φ(y)|dy = o
(
n−(2α+1)/4 (log n)

p+1
)
,

and ∫ ∞
n

ey/2y−1/3|φ(y)|dy = o
(

(log n)
p+1
)
, n →∞,

where δ is a fixed positive constant.

Very recently, Nigam and Sharma [4] proved a theorem of such type using (E, 1)
means which is entirely different from (C,k) and harmonic means of the series (1.2),
they employed a condition which is weaker than condition (1.6), and increased the
range of α to (−1,−1/2) which is more appropriate for applications. In their paper
they established the following statement.

Theorem 1.3 ([4]). If

(1.7) E1n =
1

2n

n∑
k=0

(
n

k

)
sk →∞ as n →∞,

then the degree of approximation of Fourier-Laguerre expansion (1.2) at the point
x = 0 by (E, 1) means E1n is given by

(1.8) E1n(0) −f(0) = o (ξ(n))
provided that

(1.9) Φ(t) =

∫ t
0

|φ(y)|dy = o
(
tα+1ξ

(
1

t

))
, t → 0,



ON THE DEGREE OF APPROXIMATION 35

(1.10)

∫ n
δ

ey/2y−(2α+3)/4|φ(y)|dy = o
(
n−(2α+1)/4ξ (n)

)
,

and

(1.11)

∫ ∞
n

ey/2y−1/3|φ(y)|dy = o (ξ (n)) , n →∞,

where δ is a fixed positive constant, α ∈ (−1,−1/2), and ξ(t) is a positive monotonic
increasing function of t such that ξ(n) →∞ as n →∞.

As is pointed out in [1] the infinite series

1 − 4
∞∑
n=1

(−3)n−1

is not (E, 1) summable nor (C, 1) summable. However, it is proved that the above
series is (C, 1)(E, 1) summable. Therefore the product summability (C, 1)(E, 1) is
more powerful than the individual methods (C, 1) and (E, 1). Thus, (C, 1)(E, 1)
mean gives an approximation for a wider class of Fourier-Laguerre series than the
individual methods (C, 1) and (E, 1). The main aim of this paper is to prove
the counterpart of the Theorem 1.3 using the product mean (C, 1)(E,q), which
obviously, based on what we discussed above, will give more general results. To
achieve this aim we need an auxiliary result (see [6], page 175).

Lemma 1.1. Let α be arbitrary and real, c and d be fixed positive constants, and
let n →∞. Then
(1.12) L(α)n (x) = O (n

α) , if 0 ≤ x ≤
c

n

and

(1.13) L(α)n (x) = O
(
x−(2α+1)/4n(2α−1)/4

)
if

c

n
≤ x ≤ d.

2. Main Result

We prove the following theorem.

Theorem 2.1. Te degree of approximation of Fourier-Laguerre expansion (1.2) at
the point x = 0 by (C, 1)(E,q), q ≥ 1 means [(C, 1)(E,q)]n is given by

[(C, 1)(E,q)]n(0) −f(0) = o (ξ(n))
provided that

(2.1) Φ(t) =

∫ t
0

|φ(y)|dy = o
(
tα+1ξ

(
1

t

))
, t → 0,

(2.2)

∫ n
δ

ey/2y−(2α+3)/4|φ(y)|dy = o
(
n−(2α+1)/4ξ (n)

)
,

and

(2.3)

∫ ∞
n

ey/2y−1/3|φ(y)|dy = o (ξ (n)) , n →∞,

where δ is a fixed positive constant, α ∈ (−1,−1/2), and ξ(t) is a positive monotonic
increasing function of t such that ξ(n) →∞ as n →∞.



36 KRASNIQI

Proof. Based on the equality

(2.4) L(α)n (0) =

(
n + α

α

)
,

we obtain

sn(0) =

n∑
k=0

akL
(α)
n (0)

=
1

Γ(α + 1)

∫ ∞
0

e−yyαf(y)

n∑
k=0

L
(α)
k (y)dy

=
1

Γ(α + 1)

∫ ∞
0

e−yyαf(y)L(α+1)n (y)dy.(2.5)

Thus,

[(E,q)]n(0) =
1

(1 + q)n

n∑
k=0

(
n

k

)
qksk(0)

=
1

(1 + q)n

n∑
k=0

(
n

k

)
qk

Γ(α + 1)

∫ ∞
0

e−yyαf(y)L
(α+1)
k (y)dy,

and

[(C, 1)(E,q)]n(0) =
1

n + 1

n∑
v=0

1

(1 + q)v

v∑
k=0

(
v

k

)
qksk(0)

=
1

n + 1

n∑
v=0

1

(1 + q)v

v∑
k=0

(
v

k

)
qk

Γ(α + 1)

∫ ∞
0

e−yyαf(y)L
(α+1)
k (y)dy.(2.6)

Therefore, using (1.5) we have

(C, 1)(Eqn)(0) −f(0) =

=
1

n + 1

n∑
v=0

1

(1 + q)v

v∑
k=0

(
v

k

)
qk
∫ ∞
0

φ(y)L
(α+1)
k (y)dy

=

(∫ 1/n
0

+

∫ δ
1/n

+

∫ n
δ

+

∫ ∞
n

)
1

n + 1

n∑
v=0

1

(1 + q)v

v∑
k=0

(
v

k

)
qkφ(y)L

(α+1)
k (y)dy

:=

4∑
m=0

rm.(2.7)



ON THE DEGREE OF APPROXIMATION 37

Using the property of the orthogonality, condition (2.1) and Lemma 1.1, we
obtain

r1 =
1

n + 1

n∑
v=0

1

(1 + q)v

v∑
k=0

(
v

k

)
qkO

(
kα+1

)∫ 1/n
0

|φ(y)|dy

=
1

n + 1

n∑
v=0

1

(1 + q)v

v∑
k=0

(
v

k

)
qkO

(
nα+1

)
o

(
ξ (n)

nα+1

)

= o

(
1

n + 1

n∑
v=0

1

(1 + q)v

v∑
k=0

(
v

k

)
qkξ (n)

)
= o (ξ (n)) ,(2.8)

since
n∑
v=0

1

(1 + q)v

v∑
k=0

(
v

k

)
qk = n + 1.

Again, using the property of the orthogonality and Lemma 1.1, we have

r2 =
1

n + 1

n∑
v=0

1

(1 + q)v

v∑
k=0

(
v

k

)
qkO

(
k(2α+1)/4

)∫ δ
1/n

y(2α+3)/4|φ(y)|dy.

Since

v∑
k=0

(
v

k

)
qkk(2α+1)/4 =

[ v2 ]∑
k=0

(
v

k

)
qkk(2α+1)/4 +

v∑
k=[ v2 ]+1

(
v

k

)
qkk(2α+1)/4

≤
v∑
k=0

(
v

k

)
qkk(2α+1)/4 +

(
v[
v
2

]) v∑
k=[ v2 ]+1

qkk(2α+1)/4

≤ (1 + q)v v(2α+1)/4 +
(
v[
v
2

])v(2α+5)/4qv
= (1 + q)

v
v(2α+1)/4 +

(
v[
v
2

])v(2α+1)/4vqv q ≥ 1.
and

(1 + q)v =

v∑
k=0

(
v

k

)
qk

=

(
v

0

)
q0 +

(
v

1

)
q1 + · · · +

(
v[
v
2

])q[ v2 ] + ( v[v
2

]
+ 1

)
q[

v
2 ]+1 + · · · +

(
v

v

)
qv

≥
(
v[
v
2

])q[ v2 ] + ( v[v
2

]
+ 1

)
q[

v
2 ]+1 + · · · +

(
v

v

)
qv

≥
[(

v[
v
2

]) + ( v[v
2

]) + · · · + ( v[v
2

])]q[ v2 ]
≥ K

([v
2

]
+ 1
)( v[

v
2

])qv ≥ K
2
v

(
v[
v
2

])qv, (for K ≤ 1/q),



38 KRASNIQI

then

1

(1 + q)v

v∑
k=0

(
v

k

)
qkk(2α+1)/4 ≤

(
1 +

2

K

)
v(2α+1)/4.

and moreover,

1

n + 1

n∑
v=0

1

(1 + q)v

v∑
k=0

(
v

k

)
qkk(2α+1)/4 = O

(
n(2α+1)/4

)
.

Using latter estimation, and doing the same reasoning as in [4] page 6, we obtain

(2.9) r2 = O
(
n(2α+1)/4

)∫ δ
1/n

y(2α+3)/4|φ(y)|dy = O (ξ(n)) .

Further we estimate r3:

r3 ≤
1

n + 1

n∑
v=0

∑v
k=0

(
v
k

)
qk

(1 + q)v

∫ n
δ

ey/2y−(2α+3)/4|φ(y)|e−y/2y(2α+3)/4|L(α+1)k (y)|dy

=
1

n + 1

n∑
v=0

∑v
k=0

(
v
k

)
qk

(1 + q)v
O
(
k(2α+1)/4

∫ n
δ

ey/2y−(2α+3)/4|φ(y)|dy
)

=
1

n + 1

n∑
v=0

∑v
k=0

(
v
k

)
qk

(1 + q)v
O
(
k(2α+1)/4o

(
n−(2α+1)/4ξ(n)

))
= o (ξ(n)) .(2.10)

Finally, we have

r4 ≤
1

n + 1

n∑
v=0

∑v
k=0

(
v
k

)
qk

(1 + q)v

∫ ∞
n

ey/2y−(3α+5)/6|φ(y)|e−y/2y(3α+5)/6|L(α+1)k (y)|dy

=
1

n + 1

n∑
v=0

∑v
k=0

(
v
k

)
qk

(1 + q)v
O
(
k(α+1)/4

∫ ∞
n

ey/2|φ(y)|
y(α+1)/2+1/3

dy

)

=
1

n + 1

n∑
v=0

∑v
k=0

(
v
k

)
qk

(1 + q)v
O
(
k(α+1)/2k−(α+1)/2o (ξ(n))

)
= o (ξ(n)) .(2.11)

Now, putting estimations (2.8)-(2.11) into (2.7) we obtain

[(C, 1)(E,q)]n(0) −f(0) = o (ξ(n)) .
The proof of the theorem is completed. �

References

[1] V. N. Mishra et al.: Approximation of signals by product summability transform, Asian
Journal of Mathematics and Statistics, 6(1): 12–22, 2013.

[2] D. P. Gupta: Degree of approximation by Cesàro means of Fourier-Laguerre expansions,
Acta Sci. Math. (Szeged), vol. 32, pp. 255–259, 1971.

[3] G. H. Hardy: Divergent Series, Oxford University Press, Oxford, UK, 1st edition, 1949.
[4] H. K. Nigam, A. Sharma: A study on degree of approximation by (N, p, q)(E, 1) summa-

bility means of the Fourier-Laguerre expansion summability of Fourier series, Int. J. Math.
Math. Sci. Volume 2010, Article ID 351016, 7 pages doi:10.1155/2010/351016.



ON THE DEGREE OF APPROXIMATION 39

[5] T. Singh: Degree of approximation by harmonic means of Fourier-Laguerre expansions,
Publ. Math. Debrecen, vol. 24, no. 1–2, pp. 53–57, 1977.

[6] G. Szegö: Orthogonal Polynomials. Colloquium Publications American Mathematical So-
ciety, New York, NY, USA, 1959.

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of Mathematics and Informatics, Avenue ”Mother Theresa” 5, 10000 Prishtina, Kosova