Surveys in Mathematics and its Applications
ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 8 (2013), 125– 136
ON APPROXIMATION OF FUNCTIONS BY
PRODUCT OPERATORS
Hare Krishna Nigam
Abstract. In the present paper, two quite new reults on the degree of approximation of
a function f belonging to the class Lip(α, r), 1 ≤ r < ∞ and the weighted class W (Lr , ξ(t)),
1 ≤ r < ∞ by (C, 2)(E, 1) product operators have been obtained. The results obtained in the
present paper generalize various known results on single operators.
1
Let
Introduction
P∞
n=0 un
be a given infinite series with the sequence of its nth partial sum {sn }.
The (C,2) transform is defined as the nth partial sum of (C,2) summability and
is given by
n
tn =
X
1
(n − k + 1)sk → s as n → ∞
(n + 1)(n + 2)
k=0
then the series
P∞
n=0 un
is summable to s by(C,2) method.
If
(E, 1) =
En1
n 1 X n
sk → s as n → ∞
= n
k
2
(1.1)
k=0
then the series
to s [3].
P∞
n=0 un
with the nth partial sum sn is said to be summable (E,1)
The (C,2) transform of the (E,1) transform defines (C,2)(E,1) product transform
and we denote it by Cn2 En1 .
2010 Mathematics Subject Classification: Primary 42B05, 42B08.
Keywords: Degree of approximation; Function of class Lip(α, r); Function of class W (Lr , ξ(t));
(E,1) means; (C,1) means; (E,1)(C,1) product means; Fourier series; Lebesgue integral.
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126
H. K. Nigam
Thus if
n
Cn2 En1
X
2
=
(n − k + 1)Ek1 → s
(n + 1)(n + 2)
as n → ∞,
(1.2)
k=0
where En1 denotes P
the (E,1) transform of sn and Cn2 denotes the (C,2) transform of
sn , then the series ∞
n=0 un is said to be summable by (C,2)(E,1) means or summable
(C,2)(E,1) to s.
Let f (x) be periodic with period 2π and integrable in the sense of Lebesgue. The
Fourier series of f (x) is given by
∞
f (x) v
a0 X
+
(an cos nx + bn sin nx)
2
(1.3)
n=1
with nth partial sum sn (f ; x).
L∞ -norm of a function f : Rn → R is defined by
kf k∞ = sup{| f (x) |: x ∈ R}
(1.4)
Lr -norm is defined by
2π
Z
kf kr = (
1
| f (x) |r dx)r , r ≥ 1
(1.5)
0
The degree of approximation of a function f : Rn → R by a trigonometric polynomial
tn of order n under sup norm k k∞ is defined by
k tn − f k∞ = sup{| tn (x) − f (x) : x ∈ R} Zygmund [14]
(1.6)
and En (f ) of a function f belongs to Lr is given by
En (f ) = minktn − f kr
(1.7)
One says that a function f belongs to the class Lipα if
f (x + t) − f (x) = O(| t |α ) f or 0 < α ≤ 1
(1.8)
and that f belongs to the class Lip(α, r) if
Z
2π
r
| f (x + t) − f (x) | dx
r1
= O(| t |α ), 0 < α ≤ 1 and r ≥ 1
(1.9)
0
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Degree of approximation of functions...
(Mc Fadden [6], definition 5.38)
Given a positive increasing function ξ(t)and some r, 1 ≤ r < ∞, one says that a
function f belongs to the class Lip((ξ(t), r) if
Z
2π
r1
| f (x + t) − f (x) | dx)
= O(ξ(t))
r
(1.10)
0
and that f belongs to the class W (Lr , ξ(t)), 1 ≤ r < ∞ if
Z
2π
r1
| {f (x + t) − f (x)} sin x | dx)
= O(ξ(t)), β ≥ 0.
β
r
(1.11)
0
If β = 0, our newly defined class W (Lr , ξ(t)) reduces to the class Lip(ξ(t), r) and
if ξ(t) = tα then Lip(ξ(t), r) class reduces to the class Lip(α, r) and if r → ∞ then
Lip(α, r) class reduces to the class Lipα.
We use the following notations throughout this paper:
φ(t) = f (x + t) + f (x − t) − 2f (x)
n
X
1
Kn (t) =
π (n + 1)(n + 2)
k=0
2
(
n−k+1
2k
X
k ν=0
k
υ
sin(υ + 12 )t
sin 2t
)
Main Theorems
Several researchers ([1], [2], [4], [5], [7], [8], [9], [10], [11], [12]) initiated the studies
of error estimates of function belonging to the classes Lipα, Lip(α, r), Lip(ξ(t), r),
W (Lr , ξ(t)) using different linear operators. But till now nothing seems to have been
in the direction of present work. Therefore, in this paper, two quite new theorems
on degree of approximation of a function belonging to the class Lip(α, r) and the
weighted class W (Lr , ξ(t)) by (C, 2)(E, 1) summability means on Fourier series have
been established. In fact, we proved the following theorems.
Theorem 1. If f is a 2π-periodic function, Lebesgue integrable on [0, 2π] belonging
to the class Lip(α, r), 1 ≤ r < ∞, then its degree of approximation by (C, 2)(E, 1)
means on Fourier series (1.3) is given by
(
)
1
2 1
k Cn En − f kr = O
f or 0 < α ≤ 1,
(2.1)
1
(n + 1)α− r
where δ is an arbitrary number such that 0 < δ < 1r − α, where r−1 + s−1 = 1,
1 ≤ r < ∞ and Cn2 En1 is (C, 2)(E, 1) means of the series (1.3).
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128
H. K. Nigam
Theorem 2. If a positive increasing function ξ(t) satisfies the conditions
1
n+1
(Z
t | φ(t) |
ξ(t)
0
)1
r
r
βr
sin
t dt
=O
1
n+1
(2.2)
and
(Z
π
1
n+1
t−δ | φ(t) |
ξ(t)
)1
r
r
dt
n
o
= O (n + 1)δ
(2.3)
uniformaly in x, in which δ is an arbitrary number with s(1 − δ) − 1 > 0, where
r−1 + s−1 = 1, 1 ≤ r < ∞ and
ξ(t)
t
is non increasing in t,
(2.4)
then the degeree of approximation of a 2π-periodic function f belonging to the
weighted class W (Lr , ξ(t), 1 ≤ r < ∞, by (C, 2)(E, 1) means on Fourier series
(1.3) is given by
k
Cn2 En1
− f kr = O (n + 1)
β+ r1
ξ
1
n+1
,
(2.5)
where Cn2 En1 denote the sequence of (C,2)(E,1) means of the series (1.3).
3
Lemmas
Following leemas are required for the proof of our theorems.
Lemma 3.
| Kn (t) |= O(n + 1), for 0 5 t 5
1
n+1
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Degree of approximation of functions...
Proof. For 0 5 t 5
| Kn (t) | =
≤
≤
=
π
π
π
π
1
n+1 ,
sin nt 5 n sin t
n
X
1
(n + 1)(n + 2) k=0
n
X
1
(n + 1)(n + 2) k=0
n
1
X
(n + 1)(n + 2) k=0
n
X
1
(n + 1)(n + 2) k=0
n
X
"
"
n−k+1
2k
X
k υ=0
X
k k
υ
sin(υ + 12 )t
sin 2t
#
(2υ + 1) sin 2t
sin 2t
υ=0
"
#
k X
n−k+1
k
(2k + 1)
υ
2k
υ=0
(n − k + 1)(2k + 1)
n−k+1
2k
k
υ
#
n
X
1
k(2k + 1)
π (n + 1)(n + 2)
k=0
k=0
" n
#
n
n
X
X
X
1
1
2
=
(2k + 1) −
2
k +
k
π (n + 2)
π (n + 1)(n + 2)
k=0
k=0
k=0
n(n + 1)(2n + 1) n(n + 1)
(n + 1)2
1
=
−
+
π (n + 2) π (n + 1)(n + 2)
3
2
2
(n + 1)
n(2n + 1)
n
=
−
−
π (n + 2) 3π (n + 2) 2π (n + 2)
2n2 + 7n + 6
=
6π (n + 2)
= O(n + 1)
=
n+1
π (n + 1)(n + 2)
(2k + 1) −
Lemma 4.
1
1
| Kn (t) |= O
, for
5t5π
t
n+1
Proof. For
1
n+1
5 t 5 π, by applying Jordan’s lemma sin 2t ≥
t
π
and sin nt 5 1
| Kn (t) |
"
#
k n
X
sin(υ + 12 )t 1
n−k+1 X k
=
υ
π (n + 1)(n + 2) 2k
sin 2t
υ=0
k=0
n "
n "
#
#
k k X 1 X
X k X
1
n+1
1 1 k
k
≤
−
t
t k
k
υ
υ
π (n + 1)(n + 2) π (n + 1)(n + 2) 2
2
π
π
k=0
υ=0
k=0
υ=0
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130
H. K. Nigam
"
"
#
#
n
k n
k X
X
1
1 X k
1
k X k
=
−
k
k
υ
υ
t(n + 1)(n + 2) t(n + 2) 2
2
υ=0
k=0
k=0
υ=0
n
X
1
k
t(n + 1)(n + 2)
k=0
k=0
n+1
n(n + 1)
1
=
−
t(n + 2) t(n + 1)(n + 2)
2
n+1
n
=
−
t(n + 2) 2t(n + 2)
1
=
2t 1
=O
t
=
1
t(n + 2)
n
X
1−
Lemma 5. (Mc Fadden [6], lemma 5.40) If f (x) belongs to Lip(α, r) on [0, π] then
φ(t) belongs to Lip(α, r) on [0, π].
4
4.1
Proof of the main theorems
Proof of theorem 1
Following Titchmarsh [13] and using Riemann-Lebesgue theorem, sn (f ; x) of the
series (1.3) is given by
1
sn (f ; x) − f (x) =
2π
Z
0
π
sin(n + 12 )t
φ(t)
dt
sin 2t
Therefore using (1.3), the (E,1) transform En1 of sn (f ; x) is given by
En1
1
− f (x) =
2π 2n
Z
0
π
n X
sin(k + 12 )t
n
φ(t)
dt
t
k
sin
2
k=0
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Degree of approximation of functions...
Now denoting (C,2)(E,1) transform of sn (f ; x) by Cn2 En1 , we write
Cn2 En1 − f (x)
"
( k ) #
Z
n
X
X k (n − k + 1) π φ(t)
1
1
t dt
=
sin υ +
t
υ
π (n + 1)(n + 2)
2
2k
sin
0
2
υ=0
k=0
Z π
φ(t) Kn (t) dt
=
"0Z 1
#
Z
n+1
=
π
+
0
1
n+1
φ(t)Kn (t) dt
= I1.1 + I1.2 (say)
(4.1)
We consider,
Z
| I1.1 |5
1
n+1
| φ(t) || Kn (t) | dt
0
Using Hölder’s inequality and leema 3,
"Z 1 # r1 "Z 1 # 1s
n+1
n+1
t | φ(t) | r
| Kn (t) | s
dt
dt
| I1.1 |≤
tα
t1−α
0
0
"Z 1 # r1
n+1
1
| Kn (t) | s
=
dt
n+1
t1−α
0
"Z 1 s # 1s
n+1
1
=O
dt by lemma 1
t1−α
0
"Z 1
#1
n+1
=O
s
t
αs−s
dt
0
"
1
n+1
αs−s+1 #
"
1
n+1
α−1+ 1 #
"
1
n+1
α−(1− 1 ) #
"
1
n+1
α− 1 #
=O
=O
=O
I1.1 =O
s
s
s
r
(4.2)
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132
H. K. Nigam
Similarly as above, we have
"Z
| I1.2 |≤
π
1
n+1
"Z
π
=O
1
n+1
"Z
π
=O
1
n+1
"Z
π
=O
r # r1 "Z π # 1s
t−δ | φ(t) |
| Kn (t) | s
dt
dt
1
tα
t−δ−α
n+1
(
)r # 1 "Z
# 1s
1
r
π t−δ tα− r
| Kn (t) | s
dt
dt
1
tα
t−δ−α
n+1
# 1 "Z
s # 1s
r
n 1 or
π 1
t− r −δ dt
dt
1
t1−δ−α
n+1
# 1 "Z
#1
π
r
t−1−δr dt
1
n+1
1
n+1
I1.2
s
tsα+sδ−s dt
n
o1 δ
−sα−sδ+s−1 s
=O (n + 1) (n + 1)
h
i
1
=O (n + 1)δ (n + 1)−α−δ+1− s
#
"
1
=O
1
(n + 1)α− r
(4.3)
Combining (4.1), (4.2) and (4.3) we get
"
k Cn2 En1 − f kr = O
#
1
1
(n + 1)α− r
This completes the proof of theorem 1.
4.2
Proof of theorem 2
Following the proof of theorem 1
Cn2 En1 − f (x) =
"Z
0
1
n+1
Z
π
#
+
1
n+1
= I2.1 + I2.2 (say)
φ(t)Kn (t) dt
(4.4)
we have
| φ(x + t) − φ(x) | ≤ | f (u + x + t) − f (u + x) | + | f (u − x − t) − f (u − x) |
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Degree of approximation of functions...
Hence, by Minkowiski’s inequality,
Z
2π
β
r
r1
| {φ(x + t) − φ(x)} sin x | dx
Z
β
r
| {f (u + x + t) − f (u + x)} sin x | dx
5
0
2π
0
Z
2π
+
| {f (u − x − t) − f (u − x)} sinβ x |r dx
r1
= O{ξ(t)}.
0
Then f ∈ W (Lr , ξ(t)) ⇒ φ ∈ W (Lr , ξ(t)).
Using Hölder’s inequality and the fact that φ(t) ∈ W (Lr , ξ(t)) condition (2.2),
sin t ≥ 2t
π , lemma 1 and second mean value theorem for integrals, we have
#1
r # r1 "Z 1 s
n+1
t | φ(t) | sinβ t
ξ(t) | Kn (t) | s
dt
dt
| I2.1 | 5
ξ(t)
t sinβ t
0
0
#1
"Z 1 s
n+1
1
(n + 1) ξ(t) s
dt
=O
n+1
t1+β
0
#1
"Z 1
s
n+1
1
dt
1
=O ξ
for some 0 <∈<
(1+β)s
n+1
n+1
t
∈
1
1
I2.1 = O (n + 1)β+ r ξ
(4.5)
n+1
"Z
1
n+1
Now using Hölder’s inequality, | sin t |5 1, sin t = 2t
π , conditions (2.3) and (2.4),
lemma 2 and second mean value theorem for integrals, we have
#1
r # r1 "Z π s
t−δ | φ(t) | sinβ t
ξ(t) | Kn (t) | s
|5
dt
dt
1
1
ξ(t)
t−δ sinβ t
n+1
n+1
"Z
#1
s
π ξ(t) s
δ
= O{(n + 1) }
dt
1
tβ+1−δ
n+1
 s

 1s
Z n+1  ξ 1 
y
dy 
= O{(n + 1)δ } 
δ−1−β
1
y
 y2
"Z
| I2.2
π
π
δ
= O (n + 1) ξ
δ
= O (n + 1) ξ
1
n+1
Z
1
n+1
Z
n+1
1s
for some
y s(δ−1−β)+2
η
1
dy
n+1
dy
y s(δ−1−β)+2
1s
for some
1
<η <n+1
π
1
<1<n+1
π
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r1
134
H. K. Nigam
h
i
1
1
= O (n + 1) ξ
(n + 1)(1+β−δ)− s
n+1
1
1
β+ r
= O (n + 1)
ξ
n+1
I2.2
δ
(4.6)
Now combining (4.4), (4.5) and (4.6), we get
1
| Cn2 En1 − f (x) |r = O (n + 1)β+ r ξ
1
n+1
.
Hence,
k
Cn2 En1
Z
r1
2π
− f (x) kr =
|
Cn2 En1
r
− f (x) | dx
0
r
r1 #
1
dx
ξ
=O
(n + 1)
n+1
0
"Z 2π
r1 #
1
1
dx
= O (n + 1)β+ r ξ
n+1
0
1
β+ r1
ξ
= O (n + 1)
n+1
" Z
2π
β+ r1
This completes the proof of theorem 2.
5
Applications
Following corollaries can be derived from our main theorems:
Corollary 6. If ξ(t) = tα then the weighted class W (Lr , ξ(t)), 1 ≤ r < ∞ reduces
to the class Lip(α, r) and then the degree of approximation of a function f belonging
to the class Lip(α, r), r−1 ≤ α ≤ 1 is given by
(
)
1
k Cn2 En1 − f (x) kr = O
.
1
(n + 1)α− r
Proof. The result follows by setting β = 0 in (2.5)
Corollary 7. If r → ∞ in corollary 6, then the class f ∈ Lip(α, r) reduces to the
class Lipα and the degree of approximation of a function f belonging to the class
Lipα, 0 < α < 1 is given by
1
2 1
k Cn En − f (x) kr = O
(n + 1)α
Acknowledgment. The author is thankful to his parents for their encouragement
and support to this work.
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Degree of approximation of functions...
135
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Hare Krishna Nigam
Department of Mathematics,
Faculty of Engineering and Technology,
Mody Institute of Technology and Science (Deemed University)
Laxmangarh, Sikar-332311,
Rajasthan, India.
e-mail: harekrishnan@yahoo.com
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