MATEMATIQKI VESNIK
originalni nauqni rad
research paper
66, 2 (2014), 155–164
June 2014
APPROXIMATION OF FUNCTIONS BELONGING TO THE
GENERALIZED LIPSCHITZ CLASS BY C 1 · Np SUMMABILITY
METHOD OF CONJUGATE SERIES OF FOURIER SERIES
Vishnu Narayan Mishra, Kejal Khatri and Lakshmi Narayan Mishra
Abstract. In the present study, a new theorem on the degree of approximation of function f˜, conjugate to a periodic function f belonging to weighted W (Lr , ξ(t))-class using semimonotonicity on the generating sequence {pn } has been established.
1. Introduction
In 1941, Alexits [1] (later Zygmund [22] and Zamansky [20], too) proved a very
interesting result pertaining to the degree of approximation of conjugate functions.
The degree of approximation of functions belonging to Lip α, Lip(α, r), Lip(ξ(t), r)
and W (Lr , ξ(t))-classes, (r ≥ 1) by Nörlund (Np ) matrices and general summability
matrices has been proved by various investigators like Khan [6], Mohapatra and
Sahney [15,16], Qureshi [18], Mohapatra and Chandra [12–14], Holland et al. [5],
Das et al. [3], Mittal et al. [9-11], Chandra [2], Leindler [8], Rhoades et al. [19]
and Nigam and Sharma [17]. Recently, Lal [7] has proved a theorem on the degree
of approximation of function f belonging to weighted W (Lr , ξ(t))-class by C 1 · Np
summability method of its Fourier series of a 2π-periodic function f where ξ(t) is a
positive increasing function in t. Lal [7] has assumed monotonicity on the generating
sequence {pn }. The approximation of function f˜, conjugate to a periodic function
f ∈ W (Lr , ξ(t)) (r ≥ 1) using product C 1 · Np -summability has not been studied
so far. In this paper, we obtain a new theorem on the degree of approximation
of function f˜, conjugate to a periodic function f ∈ W (Lr , ξ(t))-class using semimonotonicity on the generating sequence {pn }.
P∞
Let n=0 an be a given infinite series with the sequence of nth partial sums
{sn }. Let {pn } be a non-negative sequence of constants, real or complex, and let
2010 Mathematics Subject Classification: 40G05, 42B05, 42B08.
Keywords and phrases: Generalized Lipschitz W (Lr , ξ(t))-class of functions; conjugate
Fourier series; degree of approximation; C 1 means; Np means; product summability C 1 · Np
transform.
155
156
V.N. Mishra, K. Khatri, L.N. Mishra
us write
Pn =
n
P
pk 6= 0 ∀ n ≥ 0, p−1 = 0 = P−1 and Pn → ∞ as n → ∞.
k=0
Pn pn−ν s̃n
The sequence to sequence transformation t̃N
defines the sequence
n =
ν=0
Pn
N
{t̃n } of Nörlund means of P
the sequence {s̃n }, generated by the sequence of co∞
efficients {pn }. The series n=0 an is said to be Np summable to the sum s if
limn→∞ t̃N
n exists and is equal to s. In the special case in which
¶
µ
n+α−1
Γ(n + α)
; (α > 0),
pn =
=
α−1
Γ(n + 1)Γ(α)
the Nörlund summability Np reduces to the familiar C α summability.
The product of C 1 summability with a Np summability defines C 1 ·Np summaPn
−1 Pk
1
bility. Thus the C 1 ·Np mean is given by t̃CN
n (f ) = n+1
k=0 Pk
ν=0 pk−ν s̃ν (f ).
P∞
CN
If t̃n (f ) → s as n → ∞, then the infinite series n=0 an or the sequence
{s̃n } is said to be C 1 · Np summable to the sum s.
−1
s̃n → s =⇒ Np (s̃n ) = t̃N
n = Pn
n
P
ν=0
pn−ν s̃n → s, as n → ∞, Np method is regular,
→ s, as n → ∞, C 1 method is regular,
=⇒ C 1 (Np (s̃n )) = t̃CN
n
=⇒ C 1 · Np method is regular.
Let f (x) be a 2π-periodic and Lebesgue integrable function. The Fourier series of
f (x) is given by
f (x) ∼
∞
∞
P
P
a0
+
(an cos nx + bn sin nx) ≡
An (f ; x)
2
n=1
n=0
(1.1)
with n-th partial sums sn (f ; x).
The conjugate series of Fourier series (1.1) is given by
∞
P
n=1
(bn cos nx − an sin nx) ≡
∞
P
n=1
Bn (f ; x).
A function f (x) ∈ Lip α if
|f (x + t) − f (x)| = O(|t| α ) for 0 ≤ α ≤ 1, t ≥ 0,
and f (x) ∈ Lip(α, r) for 0 ≤ x ≤ 2π, if
µ Z 2π
¶1/r
r
|f (x + t) − f (x)| dx
= O(|t|α ), 0 ≤ α ≤ 1, r ≥ 1, t ≥ 0,
0
f (x) ∈ Lip(ξ(t), r) if
µ Z 2π
¶1/r
|f (x + t) − f (x)|r dx
= O(ξ(t)), r ≥ 1, t ≥ 0,
0
(1.2)
Approximation of functions belonging to the generalized Lipschitz class
157
f (x) ∈ W (Lr , ξ(t)) [19] if
µ Z 2π
¶1/r
ωr (t; f ) =
= O(ξ(t)),
|(f (x + t) − f (x)) sinβ (x/2)|r dx
0
β ≥ 0, r ≥ 1, t ≥ 0, where ξ(t) is a positive increasing function of t.
If β = 0 then W (Lr , ξ(t)) reduces to the class Lip(ξ(t), r), if ξ(t) = tα , (0 ≤
α ≤ 1) then Lip(ξ(t), r) class coincides with the class Lip(α, r) and if r → ∞ then
Lip(α, r) reduces to the class Lip α.
L∞ -norm of a function f : R → R is defined by kf k∞ = sup{|f (x)| : x ∈ R}.
¡ R 2π
¢1/r
Lr -norm of f is defined by kf kr = 0 |f (x)|r dx
, r ≥ 1.
The degree of approximation of a function f : R → R by trigonometric polynomial tn of order n under sup norm k k∞ is defined by [21]: ktn − f k∞ =
sup{|tn − f (x)| : x ∈ R}, and En (f ) of a function f ∈ Lr is given by
En (f ) = min ktn (f ) − f (x)kr .
n
The conjugate function f˜(x) is defined for almost every x by
µ
¶
Z
Z
−1 π
−1 π
˜
f (x) =
ψ(t) cot(t/2) dt = lim
ψ(t) cot(t/2) dt .
h→0
2π 0
2π h
CN
are also trigonometric polynomials of degree (or order) n
We note that t̃N
n and t̃n
and the series, conjugate to a Fourier series, is not necessarily a Fourier series [21].
Hence a separate study of conjugate series is desirable and attracted the attention
of researchers.
Abel’s Transformation: The formula
n
n−1
P
P
uk vk =
Uk (vk − vk+1 ) − Um−1 vm + Un vn ,
(1.3)
k=m
k=m
where 0 ≤ m ≤ n, Uk = u0 + u1 + u2 + · · · + uk , if k ≥ 0, U−1 = 0, which can
be verified, is known as Abel’s transformation and will be used extensively in what
follows.
If vm , vm+1 , . . . , vn are non-negative and non-increasing, the left-hand side of
(1.3) does not exceed 2vm maxm−1≤k≤n |Uk | in absolute value. In fact,
¯ P
¯
o
n n−1
P
¯ n
¯
uk vk ¯ ≤ max |Uk |
(vk − vk+1 + vm + vn = 2vm max |Uk |.
(1.4)
¯
k=m
k=m
We write throughout
ψ(t) = f (x + t) − f (x − t),
˜ t) =
J(n,
Wn =
n
k
P
P
1
Pk−1
(ν + 1) |pν − pν−1 |,
2π(n + 1) k=0
ν=0
n
k
P
P
cos (k − ν + 1/2)t
1
Pk−1
pν
,
2π(n + 1) k=0
sin(t/2)
ν=0
(1.5)
τ = [1/t], where τ denotes the greatest integer not exceeding 1/t. Furthermore, C
denotes an absolute positive constant, not necessarily the same at each occurrence.
158
V.N. Mishra, K. Khatri, L.N. Mishra
2. Main theorem
In this section we state our main result.
Theorem 1. Let f˜ be the conjugate to a 2π-periodic function f belonging to
W (Lr , ξ(t))-class. Then its degree of approximation by C 1 · Np means of conjugate
series of Fourier series (1.2) is given by
³
³ 1 ´´
˜(x)kr = O (n + 1)β+1/r ξ
kt̃CN
(f
)
−
f
,
(2.1)
n
n+1
provided {pn } satisfies
Wn < C,
(2.2)
and ξ(t) satisfies the following conditions:
µZ
{ξ(t/t)} is non-increasing in t,
¶1/r
³ t |ψ(t)| ´r
sinβr (t/2) dt
= O((n + 1)−1 ) and
ξ(t)
µZ π
³ t−δ |ψ(t)| ´r ¶1/r
dt
= O((n + 1)δ ),
ξ(t)
π/(n+1)
(2.3)
π/(n+1)
0
(2.4)
(2.5)
where δ is an arbitrary number such that s(β − δ) − 1 ≥ 0, r−1 + s−1 = 1, 1 ≤ r ≤
∞; conditions (2.4) and (2.5) hold uniformly in x.
π
1
π
1
Remark 1. ξ( n+1
) ≤ π ξ( n+1
), for ( n+1
) ≥ ( n+1
).
Remark 2. Condition Wn < C implies (n + 1)pn < CPn [4].
Remark 3. The product transform C 1 · Np plays an important role in signal
theory as a double digital filter [11] and theory of machines in Mechanical Engineering.
Remark 4. The condition 1/ sinβ (t) = O(1/tβ ), 1/(n + 1) ≤ t ≤ π used by
Lal [7] is not valid since sin t → 0 as t → π.
Remark 5. There is a fatal error in the proof of Theorem 2 of Lal [7, p. 349].
In the calculation of |I1 | the author of [7] obtains
Z 1/(n+1)
h t1−βs−s i
dt
1
for some 0 < ² <
;
=
(1+β)s
1 − βs − s
n+1
t
²
¸
h
1
1
βs+s−1
note that −βs − s + 1 < 0. Therefore one has βs+s−1 ²βs+s−1 − (n + 1)
,
¡
¢
which need not be O (n + 1)βs+s−1 , since ² might be O(1/nγ ) for some γ > 1.
3. Lemmas
We need the following lemmas for the proof of our theorem.
˜ t)| = O(τ ) for 0 < t ≤ π/(n + 1).
Lemma 1. |J(n,
Approximation of functions belonging to the generalized Lipschitz class
159
Proof. For 0 < t ≤ π/(n + 1), sin(t/2) ≥ (t/π) and | cos nt| ≤ 1, and we have
¯
¯
n
k
¯
P
1
cos(k − ν + 1/2)t ¯¯
−1 P
˜ t)| = ¯
|J(n,
P
p
ν
k
¯ 2π(n + 1)
¯
sin(t/2)
ν=0
k=0
≤
n
k
P
P
| cos(k − ν + 1/2)t|
1
Pk−1
pν
2π(n + 1) k=0
| sin(t/2)|
ν=0
≤
n
k
n
P
P
P
1
1
Pk−1
pν =
P −1 Pk = O(τ ).
2t(n + 1) k=0
2t(n + 1) k=0 k
ν=0
This completes the proof of Lemma 1.
Lemma 2. Let {pn } be a non-negative sequence satisfying (2.2). Then
¶
³ τ 2 ´µ P
n
k−1
P
˜ t)| = O(τ ) + O
|J(n,
Pk−1
|∆pν | uniformly in 0 < t ≤ π.
(n + 1) k=τ
ν=0
(3.1)
Proof. We have
n
k
P
P
1
cos(k − ν + 1/2)t
Pk−1
pν
2π(n + 1) k=0
sin(t/2)
ν=0
¶
µτ −1
n
k
P
P
P
1
cos(k − ν + 1/2)t
=
+
Pk−1
pν
2π(n + 1) k=0 k=τ
sin(t/2)
ν=0
˜
˜
= J1 (n, t) + J2 (n, t), (say),
˜ t) =
J(n,
where
¯
¯
˜
|J1 (n, t)| = ¯¯
(3.2)
¯
k
τP
−1
cos(k − ν + 1/2)t ¯¯
1
−1 P
P
pν
¯
2π(n + 1) k=0 k ν=0
sin(t/2)
k
τP
−1
P
| cos(k − ν + 1/2)t|
1
Pk−1
pν
2π(n + 1) k=0
| sin(t/2)|
ν=0
³
´
k
τ
−1
P −1 P
1
τ2
≤
Pk
pν = O (n+1)
,
2t(n + 1) k=0
ν=0
≤
and using Abel’s transformation and sin(t/2) ≥ (t/π), for 0 < t ≤ π, we get
¯
¯
n
k
¯
P
1
cos(k − ν + 1/2)t ¯¯
−1 P
˜
¯
P
|J2 (n, t)| = ¯
pν
¯
2π(n + 1) k=τ k ν=0
sin(t/2)
¯
½
µ
¶¯
k−1
n
ν
¯ P
¯
P
1
−1 P
¯
Pk
|∆pν | ¯
≤
cos(k − γ + 1/2)t ¯¯
2t(n + 1) k=τ
ν=0
γ=0
¯µ k
¶¯ ¾
¯ P
¯
+ ¯¯
cos(k − γ + 1/2)t ¯¯ pk
γ=0
O(t−1 )
=
2t(n + 1)
µ
n
P
k=τ
Pk−1
k−1
P
ν=0
|∆pν | +
n
P
k=τ
¶
Pk−1 pk ,
(3.3)
160
V.N. Mishra, K. Khatri, L.N. Mishra
Pµ
by virtue of the fact that
|J˜2 (n, t)| = O
³
k=λ
τ2
(n + 1)
exp(−ikt) = O(t−1 ), 0 ≤ λ ≤ k ≤ µ. Hence,
¶
´µ P
n
k−1
n
P
(k+1)
−1 P
−1
Pk
|∆pν | +
Pk pk (k+1)
ν=0
k=τ
k=τ
µ n
¶
k−1
P
τ2 ´ P
(n + 1)
=O
Pk−1
|∆pν | +
,
(n + 1) k=τ
τ
ν=0
¶
³ τ 2 ´µ P
k−1
n
P
˜ t)| = O(τ ) + O
|∆pν | ,
Pk−1
|J(n,
(n + 1) k=τ
ν=0
³
(3.4)
in view of Remark 2. Combining (3.2)–(3.4) yields (3.1). This completes the proof
of Lemma 2.
4. Proof of Theorem 1
Let s̃n (f ; x) denote the partial sum of series (1.2). We have
Z π
1
cos(n + 1/2)t
s̃n (f ; x) − f˜(x) =
ψ(t)
dt.
2π 0
sin(t/2)
Denoting C 1 · Np means of s̃n (f ; x) by t̃CN
n (f ), we write
Z π
n
k
P
P
1
cos(k − ν + 1/2)t
˜(x) =
t̃CN
(f
)
−
f
ψ(t)
Pk−1
pν
dt
n
2π(n
+
1)
sin(t/2)
ν=0
k=0
0
Z π
˜ t) dt
=
ψ(t)J(n,
0
·Z
Z
π/(n+1)
=
π
¸
+
0
˜ t) dt
ψ(t)J(n,
π/(n+1)
= I1 + I2 say.
(4.1)
Clearly,
|ψ(x + t) − ψ(t)| ≤ |f (u + x + t) − f (u + x)| + |f (u − x − t) − f (u − x)|.
Hence, by Minkowski’s inequality, we have
µ Z 2π
¶ 1/r
|(ψ(x + t) − ψ(t)) sinβ (x/2)|r dx
0
µZ
≤
¶1/r
2π
β
r
|(f (u + x + t) − f (u + x)) sin (x/2)| dx
0
µZ
2π
+
|(f (u − x − t) − f (u − x)) sinβ (x/2)|r dx)1/r
0
= O(ξ(t)).
Then f ∈ W (Lr , ξ(t)) =⇒ ψ(t) ∈ W (Lr , ξ(t)).
Approximation of functions belonging to the generalized Lipschitz class
161
Using Hölder’s inequality, ψ(t) ∈ W (Lr , ξ(t)), condition (2.4), sin(t/2) ≥
(t/π), for 0 < t ≤ π, Lemma 1, Remark 2 and Second Mean Value Theorem
for integrals, we have
·Z π/(n+1) ³
¸1/r ·Z π/(n+1) ³
˜ t)| ´s ¸1/s
t|ψ(t) sinβ (t/2)| ´r
ξ(t)|J(n,
|I1 | ≤
dt
dt
ξ(t)
t sinβ (t/2)
0
0
³ 1 ´· Z π/(n+1) ³ ξ(t) ´s ¸1/s
dt
=O
n+1
t2+β
0
n³ 1 ´ ³ π ´o· Z π/(n+1) ³ 1 ´s ¸1/s
=O
ξ
dt
n+1
n+1
t2+β
0
³
³ 1 ´´
= O (n + 1)β+1/r ξ
, r−1 + s−1 = 1.
(4.2)
n+1
Using Lemma 2, we have
·Z π
¸
·Z π
µ n
¶ ¸
P −1 k−1
P
|ψ(t)|
|ψ(t)|
dt + O
τ
Pk
|I2 | = O
|∆pν | dt
t
ν=0
k=τ
π/(n+1)
π/(n+1) t(n + 1)
= O(I21 ) + O(I22 ).
Using Hölder’s inequality, conditions (2.3) and (2.5), | sin t| ≤ 1, sin(t/2) ≥ (t/π),
for 0 < t ≤ π, Remark 2 and Second Mean Value Theorem for integrals, we have
·Z π
³
´s ¸1/s
³ t−δ |ψ(t)| sinβ (t/2) ´r ¸1/r ·Z π
ξ(t)
dt
dt
|I21 | ≤
−δ+1 sinβ (t/2)
ξ(t)
π/(n+1) t
π/(n+1)
·Z π
³ ξ(t) ´s ¸1/s
¡
¢
= O (n + 1)δ
dt
−δ+1+β
π/(n+1) t
· Z (n+1)/π ³
¸1/s
¡
¢
ξ(1/y) ´s dy
= O (n + 1)δ
y δ−1−β y 2
1/π
´
³
π
¸1/s
³
´· Z (n+1)/π
ξ n+1
dy
δ
= O (n + 1)
π/(n + 1)
y (δ−β)s+2
1/π
³
³ 1 ´´³ (n + 1)(β−δ)s−1 − (π)(−β+δ)s+1 ´1/s
= O (n + 1)δ+1 ξ
n+1
(β − δ)s − 1
³
³ 1 ´´
= O (n + 1)β+1/r ξ
, r−1 + s−1 = 1.
(4.3)
n+1
Similarly as above, using conditions (2.2), (2.3) and (2.5), | sin t| ≤ 1, sin(t/2) ≥
(t/π), for 0 < t ≤ π, Remark 2 and Second Mean Value Theorem for integrals, we
have
·Z π
³ t−δ |ψ(t)| sinβ (t/2) ´r ¸1/r
|I22 | ≤
dt
ξ(t)
π/(n+1)
·Z π
µ
µ n
¶¶s
¤1/s
P −1 k−1
P
ξ(t)
1
×
τ
Pk
|∆pν |
dt
β
−δ+1
sin (t/2) n + 1
ν=0
k=τ
π/(n+1) t
162
V.N. Mishra, K. Khatri, L.N. Mishra
·Z
¡
¢
δ−1
= O (n + 1)
µ
π
π/(n+1)
·Z
¡
¢
δ−1
= O (n + 1)
π
π/(n+1)
·Z
¡
¢
δ−1
= O (n + 1)
π
t
µ
µ n
¶¶s ¸1/s
P −1 k−1
P
Pk
τ
|∆pν |
dt
−δ+1+β
ξ(t)
ξ(t)
t−δ+1+β
µ
ξ(t)
µ
k=τ
µ
ν=0
k=τ
n
P
n
P
Pk−1
Pk−1
k−1
P
ν=0
k
P
¶¶s ¸1/s
(ν + 1)|∆pν |
dt
¶¶s ¸1/s
(ν + 1)|∆pν |
dt
t−δ+1+β k=0
ν=0
µ
¶s ¸1/s
π
ξ(t)
Wn 2π (n + 1) dt
t−δ+1+β
π/(n+1)
·Z π
µ
¶s ¸1/s
¢
¡
ξ(t)
δ
= O (n + 1)
dt
t−δ+1+β
π/(n+1)
¶s ¸1/s
· Z (n+1)/π µ
¡
¢
ξ(1/y)
dy
δ
= O (n + 1)
δ−1−β
y
y2
1/π
³
³ 1 ´´
= O (n + 1)β+1/r ξ
, r−1 + s−1 = 1.
n+1
π/(n+1)
·Z
¡
¢
δ−1
= O (n + 1)
(4.4)
Collecting (4.1)–(4.4), we have
³
³
˜(x)| = O (n + 1)β+1/r ξ
(f
)
−
f
|t̃CN
n
1 ´´
.
n+1
(4.5)
Now, using the L r -norm of a function, we get
¸1/r
·Z 2 π
CN
˜(x)|r dx
˜(x)kr =
(f
)
−
f
(f
)
−
f
|
t̃
kt̃CN
n
n
0
·Z
¸1/r
1 ´´r
=O
(n + 1)
ξ
dx
n+1
0
µ
³ 1 ´µ Z 2π ¶1/r ¶
β+1/r
= O (n + 1)
ξ
dx
n+1
0
³
³ 1 ´´
= O (n + 1)β+1/r ξ
.
n+1
2π
³
β+1/r
³
This completes the proof of Theorem 1.
5. Applications
The following corollaries can be derived from Theorem 1.
Corollary 1. If ξ(t) = tα , 0 < α ≤ 1, then the class Lip(ξ(t), r), r ≥ 1,
reduces to the class Lip(α, r), 1r < α < 1 and the degree of approximation of a
function f˜(x), conjugate to a 2π-periodic function f belonging to the class Lip(α, r),
is given by
¢
¡
−α+1/r
˜
.
(5.1)
|t̃CN
n (f ) − f (x)| = O (n + 1)
Approximation of functions belonging to the generalized Lipschitz class
163
Proof. Putting β = 0 in Theorem 1, we have
·Z 2π
¸1/r
³
³ 1 ´´
CN
CN
r
˜
˜
kt̃n (f ) − f (x)kr =
|t̃n (f ) − f (x)| dx
= O (n + 1)1/r ξ
n+1
0
¡
¢
−α+1/r
= O (n + 1)
.
Thus we get
·Z
|t̃CN
n (f )
2π
− f˜(x)| ≤
0
|t̃CN
n (f )
− f˜(x)|r dx
¸1/r
¡
¢
= O (n + 1)−α+1/r ,
r ≥ 1. This completes the proof of Corollary 1.
Corollary 2. If ξ(t) = tα for 0 < α < 1, and r → ∞ in Corollary 1, then
f ∈ Lip α. In this case, using (5.1) we get that
¡
¢
−α
˜
kt̃CN
.
n (f ) − f (x)k∞ = O (n + 1)
Proof. For r → ∞, we get
˜
kt̃CN
n (f ) − f (x)k∞ =
¡
¢
−α
˜
sup |t̃CN
.
n (f ) − f (x)| = O (n + 1)
0≤x≤2π
This completes the proof of Corollary 2.
Acknowledgement. The authors are highly thankful to the anonymous
learned referees for their observations, careful reading, their critical remarks, valuable comments and several useful pertinent suggestions, which greatly helped us
for the overall improvements and the better presentation of this paper significantly.
The second author is thankful to the “Ministry of Human Resource and Development” of India for financial support to carry out the above work.
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(received 25.08.2012; in revised form 28.06.2013; available online 10.09.2013)
V.N. Mishra, K. Khatri, Department of Applied Mathematics and Humanities, Sardar Vallabhbhai
National Institute of Technology, Ichchhanath Mahadev Road, Surat-395 007 (Gujarat), India
E-mail: vishnunarayanmishra@gmail.com, kejal0909@gmail.com
L.N. Mishra, L. 1627 Awadh Puri Colony Beniganj, Phase - III, Opp. I.T.I. Ayodhya Main Road,
Faizabad - 224 001 (Uttar Pradesh), India
E-mail: lakshminarayanmishra04@gmail.com