Khayyam J. Math. 1 (2015), no. 2, 243–252
ON THE DEGREE OF APPROXIMATION OF FUNCTIONS
BELONGING TO THE LIPSCHITZ CLASS BY (E, q)(C, α, β)
MEANS
XHEVAT Z. KRASNIQI
Communicated by T. Riedel
Abstract. In this paper two generalized theorems on the degree of approximation of conjugate functions belonging to the Lipschitz classes of the type
Lipα, 0 < α ≤ 1, and W (Lp , ξ(t)) are proved. The first one gives the degree of
approximation with respect to the L∞ -norm, and the second one with respect
to Lp -norm, p ≥ 1. In addition, a correct condition in proving of the second
mentioned theorem is employed.
1. Introduction and preliminaries
P∞
un be a given infinite series with its partial sums sn . We denote by
the n-th Cesáro means of order (θ, β), with θ + β > −1 of the sequence
(sn ), i.e. (see [2])
Let
(θ,β)
Cn
n=0
Cn(θ,β)
=
1
n
X
Anθ+β
v=0
θ−1 β
An−v
Av sv ,
where Aθ+β
=PO(nθ+β ), θ + β > −1 and Aθ+β
= 1.
0
n
The series ∞
u
is
said
to
be
(C,
θ,
β)
summable
to the definite number s if
n=0 n
Cn(θ,β)
=
1
n
X
Aθ+β
n
v=0
β
Aθ−1
n−v Av sv → s,
as n → ∞.
Date: Received: 4 June 2015; Accepted: 22 December 2015.
2010 Mathematics Subject Classification. Primary 42B05; Secondary 42B08.
Key words and phrases. Lipschitz classes, Fourier series, summability,
approximation.
243
degree of
244
X.Z. KRASNIQI
Then, for q > 0 a real number the Euler means (E, q) of the sequence (sn ) are
defined to be (see for example [2])
n X
1
n n−v
q
q sv .
En =
n
(1 + q) v=0 v
P
The series ∞
n=0 un is said to be (E, q) summable to the definite number s if
n X
1
n n−v
q
En =
q sv → s, as n → ∞.
n
(1 + q) v=0 v
The (E, q) transform of the (C, θ, β) transform, defines (E, q)(C, θ, β) transform
and we shall denote it by (EC)q,θ,β
.
n
Moreover, if
n X
1
n n−k (θ,β)
q,θ,β
(EC)n =
q Ck
→ s, as n → ∞,
n
(1 + q) k=0 k
P
then we shall say that the infinite series ∞
n=0 un is (E, q)(C, θ, β) summable to
the definite number s.
We note that for q = 1, θ = 1 and β = 0 the concept of (E, q)(C, θ, β)
summability reduces to the (E, 1)(C, 1) summability introduced in [10].
Let f (x) be a 2π periodic function and integrable in the sense of Lebesgue.
Then, let
∞
a0 X
f (x) ∼
(an cos nx + bn sin nx)
+
2
n=1
be its Fourier series with n-th partial sum sn (f ; x).
The conjugate series of the above Fourier series is given by
∞
X
(an cos nx − bn sin nx).
(1.1)
n=1
For a function f : R → R the equalities
kf k∞ = sup{|f (x)| : x ∈ R}
and
Z
kf kp =
2π
p
|f (x)| dx
1/p
,
p≥1
0
denote the L∞ -norm and Lp -norm, respectively.
The degree of approximation of a function f by a trigonometric polynomial tn
of order n under the norm k · k∞ is defined by Zygmund [16] with
kf − tn k∞ = sup{|f (x) − tn (x)| : x ∈ R}
and the best approximation En (f ) of a function f ∈ Lp is defined by the equality
En (f ) = min kf − tn kp .
tn
A function f ∈ Lipα or f ∈ Lip(α, p) if
|f (x + t) − f (x)| = O(|t|α ) for 0 < α ≤ 1
ON THE DEGREE OF APPROXIMATION OF FUNCTIONS BELONGING
245
or
2π
Z
1/p
|f (x + t) − f (x)| dx
= O(|t|α ) for 0 < α ≤ 1 and p ≥ 1,
p
0
respectively.
For a given positive increasing function ξ(t) and an integer p ≥ 1, f ∈
Lip(ξ(t), p) (see [14]) if
1/p
Z 2π
p
|f (x + t) − f (x)| dx
= O(ξ(t))
0
and f ∈ W (Lp , ξ(t)) if
1/p
Z 2π
γ
p
|[f (x + t) − f (x)] sin x| dx
= O(ξ(t)),
γ ≥ 0,
p ≥ 1.
0
We note here in these definitions that for β = 0 the class W (Lp , ξ(t)) reduces
to the class Lip(ξ(t), p) and if ξ(t) = tα then the class W (Lp , ξ(t)) reduces to the
class Lip(α, p), and if p → ∞ then the class Lip(α, p) reduces to the class Lipα.
A lot of authors have determined the degree of approximation of functions from
above mentioned classes, using Cesáro and generalized Nörlund means (we refer
the reader for details to the papers [1], and [3]–[15]. Very recently H. K. Nigam
and K. Sharma [10] have established two theorems on determining the degree of
approximation of conjugate functions using (E, 1)(C, 1) means. The condition
)1/p
(Z 1 p
n+1
t|φx (t)|
1
γp
sin tdt
=O
ξ(t)
n+1
0
assumed in Theorem 2, of their result, is not sufficient for the validity of it. This
condition leads to the divergent integral of type (see for details [6], page 14)
!1/p
Z π
n
t−(2+γ)p dt
.
0
Here in this paper we shall generalize their theorems using (E, q)(C, θ, β) means
instead of (E, 1)(C, 1) means that are obviously particular cases of them. Moreover, we employ a correct condition in our result. To verify the main results we
need first to prove some helpful statements given in the next section. Everywhere
in this paper, we write u = O(v) if there exists a positive constant C such that
u ≤ Cv.
2. Auxiliary Lemma
Throughout this paper we shall use notations
φx (t) := f (x + t) + f (x − t),
n k
θ−1 β
1
X
X
A
A
cos
v
+
t
1
n
1
k−v v
2
e nq;θ,β (t) :=
D
q n−k θ+β
,
t
n
π(1 + q) k=0 k
2 sin 2
Ak v=0
246
X.Z. KRASNIQI
and we prove a lemma which plays a key role in the proof of the main results.
e nq;θ,β (t)| = O 1 holds true for 0 ≤ t ≤ π.
Lemma 2.1. The estimate |D
t
Proof. First proof: Since for 0 ≤ t ≤ π, sin(t/2) ≥ t/π, then
n n−k X
k
θ−1 β
X
1
A
A
cos
v
+
t
1
n
q
k−v v
2
e nq;θ,β (t)| =
|D
t
θ+β
n
π(1 + q) k=0 k Ak v=0
2 sin 2
n n−k X
k
cos v + 1 t X
1
n q
2
≤
Aθ−1 Aβ π(1 + q)n k=0 k Akθ+β v=0 k−v v 2 sin 2t
k
n n−k X
X
1
n q
≤
Aθ−1 Aβ
2t(1 + q)n k=0 k Akθ+β v=0 k−v v
n X
1
n k n−k
=
1 q
n
2t(1 + q) k=0 k
1
,
= O
t
because of
k
X
β
Aθ−1
k−v Av
=
Aθ+β
k
and
v=0
n X
n
k=0
k
q n−k = (1 + q)n .
The first proof of this lemma is completed.
Second proof: Applying the well-known inequality sin(t/2) ≥ t/π for
0 ≤ t ≤ π, we obtain
n n−k X
k
θ−1 β
X
1
A
cos
v
+
t
A
1
n
q
k−v v
2
e q;θ,β (t)| =
|D
n
t
θ+β
n
π(1 + q) k=0 k Ak v=0
2 sin 2
k
n n−k
X
X
1
n q
θ−1 β i(v+ 12 )t ≤
<
A
A
e
2t(1 + q)n k=0 k Akθ+β v=0 k−v v
n n−k
k
X
X
1
n q
θ−1 β ivt it
≤
<
A A e e2
2t(1 + q)n k=0 k Akθ+β v=0 k−v v n k
X n q n−k X
1
θ−1 β ivt ≤
<
A
A
e
2t(1 + q)n k=0 k Akθ+β v=0 k−v v τ −1 k
n−k
X
X
1
n q
θ−1 β ivt ≤
<
A A e 2t(1 + q)n k=0 k Akθ+β v=0 k−v v n n−k
k
X
X
1
n q
θ−1 β ivt +
<
Ak−v Av e 2t(1 + q)n k Aθ+β
k
k=τ
=: J1 + J2 .
v=0
ON THE DEGREE OF APPROXIMATION OF FUNCTIONS BELONGING
247
For the quantity J1 , we have
J1
k
τ −1 n−k
X
X
n
q
θ−1 β ivt <
A
A
e
k−v v
k Aθ+β
k
v=0
k=0
k
τ −1 n−k X
X
1
n q
θ−1 β ivt ≤
A
A
k−v v e
2t(1 + q)n k=0 k Aθ+β
k
v=0
τ −1 X
n n−k
1
≤
q .
n
2t(1 + q) k=0 k
1
≤
2t(1 + q)n
(2.1)
The use of Abel’s lemma leads to
k
n n−k
X
X
1
n
q
θ−1 β ivt <
A
A
e
J2 ≤
k−v v
2t(1 + q)n k=τ k Aθ+β
k
v=0
j
n n−k
X
X
1
n q
θ−1 β ivt ≤
Ak−v Av e max 0≤j≤k 2t(1 + q)n k=τ k Aθ+β
k
v=0
n X
n k n−k
1
1 q .
≤
n
2t(1 + q) k=τ k
(2.2)
Thus, the estimations (2.1) and (2.2) give
τ −1 n X
X
1
n n−k
1
n k n−k
1
q;θ,β
e
|Dn (t)| ≤
q
+
1 q
=O
,
n
n
2t(1 + q) k=0 k
2t(1 + q) k=τ k
t
which as well verifies the statement of the lemma.
3. Main Results
At first, we prove the following theorem.
Theorem 3.1. If a function f , conjugate to a 2π periodic function f , belongs to
Lipα class, then its degree of approximation by (E, q)(C, θ, β) means of conjugate
Fourier series is given by
q;θ,β
sup (EC)q;θ,β
(f
(x))
−
f
(x)
=
(EC)
(f
)
−
f
n
n
0<x<2π
∞
=O
1
(n + 1)α
, 0 < α < 1,
where (EC)q;θ,β
(f (x)) denotes the (E, q)(C, θ, β) transform of partial sums of the series
n
(1.1).
Proof. Let sv (x) be the partial sums of the series (1.1). Then, in [7] it is verified
that
Z π
cos v + 12 t
1
sv (x) − f (x) =
φx (t)
dt.
2π 0
sin 2t
Thus, the (C, θ, β) transform Ckθ,β (x) of sv (x) is
Z π
k
θ−1 β
X
Ak−v
Av cos v + 12 t
1
θ,β
Ck (x) − f (x) =
φx (t)
dt.
t
sin
2πAθ+β
0
2
k
v=0
(3.1)
248
X.Z. KRASNIQI
Further, denoting the (E, q)(C, θ, β) transform of sv (x) by (EC)q;θ,β
, we have
n
(EC)q;θ,β
(f (x)) − f (x) =
n
Z
n k
1
β
X
1
n n−k π φx (t) X Aθ−1
k−v Av cos v + 2 t
=
q
dt
t
θ+β
(1 + q)n k=0 k
2
sin
0 πAk
2
v=0
Z π
e q;θ,β (t)dt
φx (t)D
=
n
0
Z π !
Z 1
n+1
e q;θ,β (t)dt
=
+
φx (t)D
n
1
n+1
0
=: I1 + I2 .
(3.2)
We apply Lemma 2.1 in order to estimate I1 :
Z 1
n+1
e nq;θ,β (t)|dt
|φx (t)||D
|I1 | ≤
0
!
Z 1
n+1
= O
α−1
t
dt
=O
0
1
(n + 1)α
.
(3.3)
.
(3.4)
Also, applying again Lemma 2.1, we have
Z π
e q;θ,β (t)|dt
|I2 | ≤
|φx (t)||D
n
1
n+1
!
π
Z
α−1
= O
t
dt
=O
1
n+1
1
(n + 1)α
Based on (3.3), (3.4), and (3.2), the required estimation is an immediate result.
The proof of the theorem is completed.
The following result gives the degree of approximation of conjugate functions
with respect to Lp -norm, 1 ≤ p < ∞.
Theorem 3.2. If f , conjugate to a 2π periodic function f , belongs to W (Lp , ξ(t))
class, then its degree of approximation by (E, q)(C, θ, β) means of conjugate Fourier
series is given by
1
q;θ,β
γ+ p1
k(EC)n (f ) − f kp = O (n + 1)
ξ
n+1
provided that ξ(t) satisfies the following conditions:
(Z
0
1
n+1
ξ(t)
t
|φx (t)|
ξ(t)
is a decreasing sequence,
p
sinγp
)1/p
t
1
dt
=O
,
2
(n + 1)1/p
(3.5)
(3.6)
ON THE DEGREE OF APPROXIMATION OF FUNCTIONS BELONGING
249
and
(Z
π
1
n+1
t−δ |φx (t)|
ξ(t)
)1/p
p
= O (n + 1)δ
dt
(3.7)
where δ is an arbitrary number such that s(1 − δ) − 1 > 0, 1/p + 1/s = 1,
1 ≤ p < ∞, conditions (3.6) and (3.7) hold uniformly in x and (EC)q;θ,β
are
n
(E, q)(Cnθ,β ) means of the series (1.1), and
1
f (x) = −
2π
2π
Z
0
t
φx (t) cot
dt.
2
Proof. We shall use the equality
(EC)q;θ,β
(f (x)) − f (x) =
n
Z
1
n+1
Z
π
+
0
1
n+1
!
e q;θ,β (t)dt =: J1 + J2 ,
φx (t)D
n
(3.8)
obtained earlier in the proof of theorem 3.1.
Moreover, using Hölder’s inequality and the fact that φ ∈ W (Lp , ξ(t)), condition (3.6), sin t ≥ 2t
, Lemma 2.1, and second mean value theorem for integrals,
π
we have
)1/p (Z 1
!s )1/s
e nq;θ,β (t)|
n+1
t
ξ(t)|
D
sinγp dt
dt
2
sinγ 2t
0
0
(Z 1 s )1/s
n+1
1
ξ(t)
O
dt
(n + 1)1/p
t1+γ
0
)1/s
(Z 1
n+1
1
1
1
dt
O
ξ
dt
,
0<ε<
(n + 1)1/p
n+1
t(1+γ)s
n+1
ε
1
1
O (n + 1)− p ξ
(n + 1)1+γ
n+1
1
γ
O (n + 1) ξ
, because of 1/p + 1/s = 1.
(3.9)
n+1
(Z
|J1 | ≤
=
=
=
=
1
n+1
|φ(t)|
ξ(t)
p
250
X.Z. KRASNIQI
Again, using Hölder’s inequality, | sin t| ≤ 1, sin t ≥ 2t
, conditions (3.5) and
π
(3.7), Lemma 2.1, and second mean value theorem for integrals, we obtain
(Z
!s )1/s
)1/p (Z
p
π −δ
π
e q;θ,β (t)|
t |φ(t)|
t
ξ(t)|
D
n
|J2 | ≤
dt
sinγp dt
−δ sinγ t
1
1
ξ(t)
2
t
2
n+1
n+1
(Z
s )1/s
π ξ(t)
δ
= O (n + 1)
dt
1
tγ+1−δ
n+1
(Z
s )1/s
n+1 du
ξ(1/u)
= O (n + 1)δ
1
uδ−1−γ
u2
π
)1/s
(Z n+1
1
du
= O (n + 1)δ ξ
1
n+1
us(δ−1−γ)+2
π
1/s
1
(n + 1)s(γ+1−δ)−1 − π s(δ−1−γ)+1
δ
= O (n + 1) ξ
n+1
s(γ + 1 − δ) − 1
1
= O (n + 1)δ ξ
(n + 1)γ+1−δ−1/s
n+1
1
γ+ p1
ξ
, where 1/p + 1/s = 1.
(3.10)
= O (n + 1)
n+1
Inserting (3.9) and (3.10) into (3.8), we obtain
1
q;θ,β
γ+ p1
ξ
,
(EC)n (f (x)) − f (x) = O (n + 1)
n+1
and whence,
(f )
k(EC)q;θ,β
n
− f kp = O (n + 1)
γ+ p1
ξ
1
n+1
.
The proof is completed.
4. Corollaries
In this section we give some direct consequences of the main results. First it
is clear that (E, q)(C, θ, β) means can be reduced to the following means:
1. If β = 0 then we obtain (E, q)(C, θ, β) ≡ (E, q)(C, θ, 0) ≡ (E, q)(C, θ)
means.
2. If θ = 1 then we obtain (E, q)(C, θ, β) ≡ (E, q)(C, 1, β) means.
3. If β = 0, q = 1 then we obtain (E, q)(C, θ, β) ≡ (E, 1)(C, θ, 0) ≡ (E, 1)(C, θ)
means.
4. If θ = 1, q = 1 then we obtain (E, q)(C, θ, β) ≡ (E, 1)(C, 1, β) means.
5. If θ = q = 1, β = 0 then we obtain (E, q)(C, θ, β) ≡ (E, 1)(C, 1) means.
ON THE DEGREE OF APPROXIMATION OF FUNCTIONS BELONGING
251
Denoting (E, q)(C, θ), (E, q)(C, 1, β), (E, 1)(C, θ), (E, 1)(C, 1, β), (E, 1)(C, 1)
means of
sn (f ; x),
(1;θ,0)
(EC)n
(f ; x),
respectively,
(1;1,β)
(EC)n
(f ; x)
(q;θ,0)
by (EC)n
(1;1,0)
(EC)n
(f ; x),
and
and 3.2 lots of corollaries can be derived.
We shall formulate below only some of them.
(f ; x),
(q;1,β)
(EC)n
(f ; x),
then from theorems 3.1
Corollary 4.1 ([10]). If θ = q = 1, β = 0 and all conditions of Theorem 3.1 are
satisfied, then
1
1;1,0
, 0 < α < 1.
k(EC)n (f ) − f k∞ = O
(n + 1)α
Corollary 4.2 ([10]). If θ = q = 1, β = 0 and all conditions of Theorem 3.2 are
satisfied (with corrected condition (3.6)), then
1
1;1,0
γ+ p1
k(EC)n (f ) − f kp = O (n + 1)
ξ
.
n+1
Corollary 4.3 ([10]). If γ = β = 0, θ = q = 1, and ξ(t) = tα , then the degree of
approximation of a function f , conjugate to a 2π-periodic function f ∈ Lip(α, p),
1/p ≤ α ≤ 1, is given by
!
1
1;1,0
k(EC)n (f ) − f kp = O
.
1
(n + 1)α− p
Corollary 4.4 ([10]). Let γ = β = 0, θ = q = 1, and ξ(t) = tα . If p → ∞ in
Corollary 4.3, then f ∈ Lip(α, p) reduces to Lipα for 0 < α < 1, and we have
1
1;1,0
k(EC)n (f ) − f k∞ = O
.
(n + 1)α
Acknowledgement. The author wishes to thank the referee for her/his suggestions which definitely improved the final form of this paper.
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1
University of Prishtina “Hasan Prishtina”, Faculty of Education, Department of Mathematics and Informatics, Avenue “Mother Theresa” 5, Prishtinë,
Kosovo.
E-mail address: xhevat.krasniqi@uni-pr.edu