Novi Sad J. Math.
Vol. XX, No. Y, 20ZZ, ??-??
ON APPROXIMATION TO FUNCTIONS IN THE
W (Lp , ξ(t)) CLASS BY A NEW MATRIX MEAN
Draft version April 11, 2016
Ug̃ur Deg̃er
1
Abstract.
In this paper we shall present results related to trigonometric approximation of functions belonging to the weighted generalized
Lipschitz class by the (C 1 · T ) matrix means of their Fourier series. The
results of Lal in [8] will be extended to a more general summability
method. Moreover, we present the results on degree of approximation
to conjugates of functions belonging to a weighted generalized Lipschitz
class by the (C 1 · T ) matrix means of their conjugate Fourier series.
AMS Mathematics Subject Classification (2010): 41A25, 42A05, 42A10,
42A24, 42A50
Key words and phrases: degree of approximation; trigonometric approximation; Fourier series; weighted generalized Lipschitz class; matrix
means.
1.
Introduction and Notations
Summability methods have been used in various fields of mathematics. For
example, summability methods are applied in function theory in connection
with the analytic continuation of holomorphic functions and the boundary behaviour of a power series, in applied analysis for the generation of iteration
methods for the solution of a linear system of equations, and for the acceleration of converge in approximation theory, in the theory of Fourier series for
both the creation and acceleration of convergence of a Fourier series, and in
other fields of mathematics like probability theory(Markov chains) and number
theory(prime number theorem) [1]. In this work we are interested in a summability method in the theory of Fourier series. For this aim, we shall give the
following notations to be use in this paper.
Let L := L(0, 2π) denote the space of functions that are 2π− periodic and
Lebesque integrable on [0, 2π] and let
(1.1)
S[f ] =
∞
∞
k=1
k=0
∑
ao ∑
+
(ak cos kx + bk sin kx) ≡
Ak (f ; x)
2
be the Fourier series of a function f ∈ L ; i.e., for any k = 0, 1, 2, · · ·
1
ak = ak (f ) =
π
∫π
−π
1
f (t) cos ktdt , bk = bk (f ) =
π
∫π
f (t) sin ktdt.
−π
1 Department of Mathematics, Faculty of Science and Literature, Mersin University,
e-mail: udeger@mersin.edu.tr, degpar@hotmail.com
2
Let
Ug̃ur Deg̃er
∑
∑
1
sn (f ; x) = a0 +
(ak cos kx + bk sin kx) ≡
Ak (f ; x)
2
n
n
k=1
k=1
denote the partial sum of the first (n + 1) terms of the Fourier series of f ∈ L
at a point x. The conjugate series of (1.1) is given by
∞
∑
(ak sin kx − bk cos kx) ≡
k=1
∞
∑
Ãk (f ; x).
k=1
Note that there is no free term in S̃[f ]. Therefore, the series conjugate to the
series S̃[f ] is the series S[f ] without free term.
The function f˜ ∈ L for which S[f˜] = S̃[f ] is called trigonometrically conjugate, or simply conjugate, to f (·). It can be shown that the functions f (·) and
f˜(·) are connected by the equality
1
f˜(x) = −
2π
1
=−
2π
(1.2)
∫π
∫π
−π
t
f (x + t)cot dt
2
t
1
η(t)cot dt = −
lim
2
2π ε→0
∫π
t
η(t)cot dt
2
ε
0
where η(t) := η(x, t) = f (x + t) − f (x − t). If f ∈ L, then equality (1.2) exists
for almost all x [25].
Let
n
∑
τn (f ; x) = τn (f, T ; x) :=
an,k sk (f ; x), ∀n ≥ 0
k=0
where T ≡ (an,k ) is a lower triangular infinite matrix satisfying the SilvermanToeplitz[24] condition of regularity such that:
{
≥ 0, k ≤ n;
an,k =
(k, n = 0, 1, 2, . . .)
0,
k>n
and
n
∑
(1.3)
an,k = 1,
(n = 0, 1, 2, . . .).
k=0
The Fourier series of a function f is said to be T -summable to s, if τn (f ; x) →
s(x) as n → ∞. The Fourier series of f is called Cesàro-T (C 1 · T ) summable
to s(x) if
tCT
n :=
n
n
m
1 ∑
1 ∑∑
τm (f ; x) =
am,k sk (f ; x) → s(x),
n + 1 m=0
n + 1 m=0
k=0
as n → ∞.
The Cesàro-T (C 1 ·T ) means give us the following means for some important
cases:
Draft version April 11, 2016
S̃[f ] =
ON APPROXIMATION TO FUNCTIONS IN THE W (Lp , ξ(t)) CLASS
3
• Cesàro- Nörlund (C 1 · Np ) means, with
{
am,k =
pm−k
Pm ,
0,
k ≤ m;
k>m
(k, m = 0, 1, 2, . . .), Pm =
pk ̸= 0;
k=0
( )
;
(
)
m m−k
1
= (1+q)
;
m
k q
• (C, 1)(E, 1) Product means, with am,k =
• (C, 1)(E, q) Product means, with am,k
• Generalized Nörlund means, with am,k =
Draft version April 11, 2016
m
∑
1 m
2m k
m
∑
pm−k qk
where rm=
pm−k qk .
rm
k=0
The degree of approximation of a function f : R → R by a trigonometric
polynomial Tn of degree n is defined by
∥Tn − f ∥∞ = sup{|Tn (x) − f (x)|, x ∈ R}
with respect to the supremum norm [25]. The degree of approximation of a
function f ∈ Lp (p ≥ 1) is given by
En (f ) = min ∥Tn − f ∥p
n
where ∥.∥p denotes the Lp -norm with respect to x and will be defined by
{
∥f ∥p :=
1
2π
∫
2π
p
} p1
|f (x)| dx
.
0
This method of approximation is called the trigonometric Fourier approximation.
We recall the following definitions:
1. A function f is said to belong to the Lipα class if |f (x + t) − f (x)| =
O(|tα |), 0 < α ≤ 1;
2. A function f is said to belong to the Lip(α, p) class if ωp (δ, f ) = O(δ α ),
where
ωp (δ, f ) = sup {
|t|≤δ
1
2π
∫2π
1
|f (x + t) − f (x)|p dx} p ,
0 < α ≤ 1;
p ≥ 1;
0
3. A function f is said to belong to the Lip(ξ(t), p) class if ωp (δ, f ) = O(ξ(t))
where ξ(t) is a positive increasing function and p ≥ 1;
4. We write f ∈ W (Lp , ξ(t)) if ∥(f (x + t) − f (x))sinβ (x/2)∥p = O(ξ(t)),
β ≥ 0 and p ≥ 1.
4
Ug̃ur Deg̃er
If β = 0, then W (Lp , ξ(t)) reduces to Lip(ξ(t), p); and, if ξ(t) = tα , the
Lip(ξ(t), p) class reduces to the Lip(α, p) class. If p → ∞ then the Lip(α, p)
class coincides with the Lipα class. Accordingly, we have the following inclusions:
Lipα ⊂ Lip(α, p) ⊂ Lip(ξ(t), p) ⊂ W (Lp , ξ(t))
for all 0 < α ≤ 1 and p ≥ 1.
Approximation by matrix means of a Fourier series
he degree of approximation, using the various summability methods in the
Lipα class, has been determined by many mathematicians such as Bernstein[25],
de la Valle-Poussin[25], Jackson[25], Mcfadden[4]. Similar problems for the
Lip(α, p) class have been studied by researchers like Quade[18], Khan[5], Qureshi
[20], Chandra[2], Leindler[11]. Other research related to the Lip(α, p) class can
also be found in [3], [12], [13], [14] and [15].
The weighted W (Lp , ξ(t)) class is a generalization of the classes Lipα,
Lip(α, p) and Lip(ξ(t), p). The degree of approximation of a function belonging
to the weighted W (Lp , ξ(t)) class has been studied by Qureshi in [21]. In [6] and
[8] Lal has considered the degree of approximation of functions belonging to
the weighted W (Lp , ξ(t)) class by the (C, 1)(E, 1) means and (C 1 · Np ) means,
respectively. Nigam has studied the same problem for the (C, 1)(E, q) means,
which are much more general than the (C, 1)(E, 1) means in [16]. Sing, Mittal
and Sonker have generalized the results of Lal[8] in [23]. Therefore, taking into
account this generalization of the function classes, we shall give two theorems
on degree of approximation to functions belonging to the classes W (Lp , ξ(t))
and Lipα by the (C 1 · T ) matrix means, being more general than (C, 1)(E, 1),
(C 1 · Np ) and (C, 1)(E, q) means given in [6], [8, 23] and [16], respectively.
Also, throughout this section, we shall use the following notations:
Ψ(x, t) := Ψ(t) = f (x + t) + f (x − t) − 2f (x)
and
KT (n, t) :=
n ∑
m
∑
sin(k + 21 )t
1
.
am,k
2π(n + 1) m=0
sin( 2t )
k=0
Before stating the theorems, we develop the following auxiliary results needed
in the proofs of both of them.
Lemma 2.1. For 0 < t ≤ π/n, we have KT (n, t) = O(n).
Proof. For 0 < t ≤ π/n, from (sin(t/2))−1 ≤ π/t and sin(n + 1)t ≤ (n + 1)t,
we have
n ∑
m
∑
sin(k + 12 )t 1
|KT (n, t)| ≤
am,k 2π(n + 1) m=0
sin( 2t ) k=0
≤
by considering (1.3).
n ∑
m
∑
2n + 1
2π(n + 1) m=0
k=0
am,k = O(n)
Draft version April 11, 2016
2.
ON APPROXIMATION TO FUNCTIONS IN THE W (Lp , ξ(t)) CLASS
5
Lemma 2.2. For π/n < t ≤ π and any n, we have
KT (n, t) = O(
t−2
) + O(t−1 ).
n+1
Proof.
KT (n, t)
n ∑
m
∑
1
1
am,k sin(k + )t
t
2
2π(n + 1)sin( 2 ) m=0 k=0
{ τ
}
n
m
∑
∑
∑
1
1
+
am,k sin(k + )t
2
2π(n + 1)sin( 2t ) m=0 m=τ +1 k=0
=
Draft version April 11, 2016
=
=
: I1 + I2 ,
where τ denotes the integer part of 1/t. Owing to (1.3) and Jordan’s inequality,
(sin(t/2))−1 ≤ π/t, for 0 < t ≤ π, we obtain
(
(2.1)
|I1 | = O
1
(n + 1)t
)∑
τ ∑
m
am,k = O(
m=0 k=0
τ t−1
t−2
) = O(
).
n+1
n+1
We now estimate I2 . By using (1.3) again and the Jordan inequality (sin( 2t ))−1
≤ π/t, for 0 < t ≤ π, we get
(
(2.2)
|I2 | = O
1
(n + 1)t
) ∑
n
m
∑
(
am,k = O
m=τ +1 k=0
(n − τ )t−1
n+1
)
= O(1/t).
Combining (2.1) and (2.2), we have
KT (n, t) = O(
t−2
) + O(t−1 ).
n+1
Theorem 2.3. Let f ∈ L and let T ≡ (an,k ) be a lower triangular regular
matrix with nonnegative entries and row sums 1. If f ∈ Lipα (0 < α ≤ 1),
then the degree of approximation by the (C 1 · T ) means of its Fourier series is
given by
{
∥tCT
n (f )
− f (x)∥∞ =
O(n−α ), 0 < α < 1;
O( logn
n ), α = 1.
Proof. We know that
(2.3)
1
sn (f, x) − f (x) =
2π
(
∫π
Ψ(t)
0
sin(n + 12 )t
sin( 2t )
)
dt.
6
Ug̃ur Deg̃er
0
0
π/n
= : J1 + J2 .
Since f ∈ Lipα, Ψ(t) belongs to the Lipα class. Therefore, from Lemma 2.1,
we obtain
π/n
π/n
∫
∫
J1 =
|Ψ(t)KT (n, t)|dt = O(n)
tα dt = O(n−α )
(2.4)
0
0
for 0 < α ≤ 1.
By using Lemma 2.2, then we have
∫π
J2
(2.5)
(
)



t−2
+ t−1 dt


n
+
1


π/n
π/n







 ∫π tα−2 

 ∫π

α−1
= O
dt + O
t
dt =: J21 + J22 .



n+1 




=
|Ψ(t)KT (n, t)|dt = O
π/n
Accordingly,


 ∫π
tα
π/n



 ∫π tα−2 
 { O(n−α ), 0 < α < 1;
1
J2 = O
dt =
O( logn

n+1 


n ), α = 1.
π/n
and
(2.6)
J22 = O


 ∫π


π/n
tα−1 dt





{
}
= O n−α .
Taking into account (2.4), (2.5) and (2.6), we obtain
{
∥tCT
n (f ) − f (x)∥∞ =
sup |tCT
n (f ) − f (x)| =
x∈[0,2π]
O(n−α ), 0 < α < 1;
O( logn
n ), α = 1.
Draft version April 11, 2016
1
Taking into account (2.3) and the definitions of tCT
n (f ) and the (C · T ) means
of sn (f ) , we write
n m
1 ∑ ∑
CT
|tn (f ) − f (x)| =
am,k (sk (f ; x) − f (x))
n + 1 m=0
k=0
π
∫
(
n ∑
m
1 ) ∑
sin(k + 2 )t
1
Ψ(t)
=
am,k
dt
t
2π(n + 1) sin(
)
2
m=0
k=0
0


π/n
∫π
∫
∫π


≤
|Ψ(t)KT (n, t)|dt = 
+
 |Ψ(t)KT (n, t)|dt
ON APPROXIMATION TO FUNCTIONS IN THE W (Lp , ξ(t)) CLASS
7
by using 1/n ≤ logn/n, for large values of n. Therefore the proof of Theorem
2.3 is completed.
Theorem 2.4. Let f ∈ L and ξ(t) be a positive increasing function. If f ∈
W (Lp , ξ(t)) with 0 ≤ β ≤ 1 − 1/p , the degree of approximation by (C 1 · T )
means of its Fourier series is given by
1
β+1/p
∥tCT
ξ( )),
n (f ) − f (x)∥p = O(n
n
Draft version April 11, 2016
provided that the function ξ(t) satisfies the following conditions:
{
ξ(t)
t
}
is a decreasing function and
(2.7)
(2.8)

π/n(

)p
∫
|Ψ(t)|sinβ (t/2)


ξ(t)
0
1/p


dt
= O(1)



1/p

 ∫π ( |Ψ(t)|t−δ )p 

dt
= O(nδ ),


ξ(t)


π/n
where δ is an arbitrary number such that q(β − δ) − 1 > 0, p−1 + q −1 = 1,
p ≥ 1, and (2.7) and (2.8) hold uniformly in x.
Proof. Proceeding as above, we have
|tCT
n (f )
(2.9)
n m
1 ∑ ∑
− f (x)| =
am,k (sk (f ; x) − f (x))
n + 1 m=0
k=0
∫π
(
n ∑
m
1 ) ∑
sin(k + 2 )t
1
Ψ(t)
dt
=
am,k
2π(n + 1) sin( 2t )
m=0
k=0
0
∫
∫π
π/n
≤ Ψ(t)KT (n, t)dt + Ψ(t)KT (n, t)dt
0
π/n
=
: J3 + J4 .
By considering Hölder’s inequality, condition (2.7), Lemma 2.1, Jordan’s in-
8
Ug̃ur Deg̃er
equality, and Ψ(t) ∈ W (Lp , ξ(t)), we get
∫
π/n Ψ(t)sinβ (t/2) ξ(t)K (n, t) T
J3 = dt
ξ(t)
sinβ (t/2)
0

1/p 
1/q
π/n
π/n
p
q
∫
∫
β
Ψ(t)sin (t/2) ξ(t)KT (n, t) 


dt
≤ 
  lim
sinβ (t/2) dt
ε→0
ξ(t)
ε

π/n(
∫

= O(1)  lim
ε→0
ξ(t)n
sinβ (t/2)
)q
1/q

dt

1/q
π/n
∫


= O(nξ(π/n))  lim
t−βq dt
ε→0
ε
ε
)
(
)
(
1
1
= O ξ( )n1+β−1/q = O nβ+1/p ξ( ) ,
n
n
(2.10)
in view of p−1 + q −1 = 1 and ξ(π/n)/(π/n) ≤ ξ(1/n)/(1/n).
Now let us estimate J4 . By using Lemma 2.2, we write




( −2 )
∫π
∫π
( ) 
t



|Ψ(t)|
(2.11) J4 = O 
dt +O 
|Ψ(t)| t−1 dt =: J41 +J42 .
n+1
π/n
π/n
We shall evaluate J41 and J42 in a manner similar to the evaluation of J3 ,
respectively. Using Hölder’s inequality, the (2.8) and Jordan’s inequality, we
have

1/p 
1/q
π (
)
)
∫π (
∫
p
q
|Ψ(t)|t−δ sinβ (t/2)
ξ(t)tδ−2

 

J41 = O(n−1 ) 
dt 
dt
ξ(t)
sinβ (t/2)
π/n


= O(nδ−1 ) 
∫π (
ξ(t)t
sinβ (t/2)
π/n

= O(n
δ−1

)
δ−2
)q
π/n
1/q

dt


=O(nδ−1 ) 
1/q
n/π
∫
(
)q

ξ(1/x)xβ−δ+2 x−2 dx
1/π
∫π
1/q
(
ξ(t)t
)
δ−β−2 q
π/n
; xξ(1/x) <
n
ξ(π/n)
π

1/q
n/π
) ∫
)
( π
( π


xβq−δq+q−2 dx
= O ξ( )nδ 
= O ξ( )nδ n1−δ+β−(1/q)
n
n
1/π
(2.12)
)
(
1
β+1/p
=O n
ξ( ) ,
n

dt
Draft version April 11, 2016
0
ON APPROXIMATION TO FUNCTIONS IN THE W (Lp , ξ(t)) CLASS
9
since p−1 + q −1 = 1 and ξ(π/n)/(π/n) ≤ ξ(1/n)/(1/n).

1/p 
1/q
)p
)q
∫π (
∫π (
−δ
β
δ−1
|Ψ(t)|t sin (t/2)
ξ(t)t

 

J42 = O(1) 
dt 
dt
ξ(t)
sinβ (t/2)
π/n

∫π
π/n
1/q
)q 
(
ξ(t)tδ−1−β dt
=

O(n ) 
=
1/q
n/π
∫
(
)
q


ξ(1/x)xβ−δ+1 x−2 dx
O(nδ ) 
δ
π/n
Draft version April 11, 2016

; xξ(1/x) <
n
ξ(π/n)
π
1/π
=
1/q

n/π
( π
( π
) ∫
)


= O ξ( )nδ+1 nβ−δ−(1/q)
O ξ( )nδ+1 
xβq−δq−2 dx
n
n
1/π
(
(2.13)
=O n
β+1/p
)
1
ξ( ) ,
n
since p−1 +q −1 = 1 and ξ(π/n)/(π/n) ≤ ξ(1/n)/(1/n). Combining (2.9)-(2.13),
we get
1
β+1/p
∥tCT
ξ( )).
n (f ) − f (x)∥p = O(n
n
3.
In case of conjugate Fourier series
As mentioned above, the problems on determining the degree of approximation by summability methods have been studied by many mathematicians.
Qureshi has determined the degree of approximation to functions which belong
to the classes Lipα and Lip(α, p) by means of conjugate series in [19] and [22],
respectively. In subsequent years, similar investigations have been made in
researches such as in [7], [9], [16] and [17].
The following two theorems are related with the degree of approximation to
conjugate of functions belonging to the classes W (Lp , ξ(t)) and Lipα by (C 1 ·T )
matrix means of conjugate of their Fourier series and are more general than
(C, 1)(E, 1) and (C, 1)(E, q). Not only for (C, 1)(E, 1) and (C, 1)(E, q) means
but also different results are obtained for other means.
The following notations will be used throughout this section and auxiliary
results:
ρ(x, t) := ρ(t) = f (x + t) + f (x − t)
and
K̃T (n, t) :=
n ∑
m
∑
cos(k + 12 )t
1
am,k
.
2π(n + 1) m=0
sin( 2t )
k=0
10
Ug̃ur Deg̃er
Lemma 3.1. For 0 < t ≤ π/n, we have K̃T (n, t) = O(1/t).
Proof. For 0 < t ≤ π/n, by (sin(t/2))−1 ≤ π/t and |cos(2k + 1)t| ≤ 1, we have
n ∑
m
∑
cos(k + 12 )t 1
.
|K̃T (n, t)| ≤
am,k 2π(n + 1) m=0
sin( 2t ) k=0
≤
n
∑
1
1 = O(1/t)
2πt(n + 1) m=0
Lemma 3.2. For π/n < t ≤ π and any n, we have
( −2 )
t
K̃T (n, t) = O
+ O(t−1 ).
n+1
Proof. This lemma can be proved by using an argument similar to that of
Lemma 2.2.
Theorem 3.3. Let f ∈ L and let T ≡ (an,k ) be a lower triangular regular
matrix with nonnegative entries and row sums 1. If f ∈ Lipα (0 < α ≤ 1),
then the degree of approximation of the conjugate function f˜ by the (C 1 · T )
means of its conjugate Fourier series is given by
(3.1)
CT (f ) − f˜(x)∥
∥tg
∞ =
n
{
O(n−α ), 0 < α < 1;
O( logn
n ), α = 1.
Proof. We have, by (1.2),
(3.2)
1
s̃n (f, x) − f˜(x) =
2π
(
∫π
ρ(t)
0
cos(n + 12 )t
sin( 2t )
)
dt.
CT (f ) that (C 1 · T ) means of s̃ (f ) , we
Taking into consideration (3.2) and tg
n
n
write
π
∫
(
n ∑
m
1 ) ∑
cos(k + 2 )t
1
CT (f ) − f (x)| =
ρ(t)
(3.3) |tg
dt .
am,k
n
2π(n + 1) sin( 2t )
m=0
0
k=0
Using Lemma 3.1 and Lemma 3.2, and proceeding as in the proof of Theorem
2.3 in (3.3) , we get (3.1).
Theorem 3.4. Let f ∈ L and ξ(t) be a positive increasing function. If f ∈
W (Lp , ξ(t)) with 0 ≤ β ≤ 1 − 1/p, then the degree of approximation of the
conjugate function f˜ by the (C 1 · T ) means of its conjugate Fourier series is
given by
Draft version April 11, 2016
by (1.3).
ON APPROXIMATION TO FUNCTIONS IN THE W (Lp , ξ(t)) CLASS
(3.4)
11
1
CT (f ) − f˜(x)∥ = O(nβ+1/p ξ( )),
∥tg
p
n
n
Draft version April 11, 2016
provided that the function ξ(t) satisfies the following conditions:
{
}
ξ(t)
t
is a decreasing function and

1/p
π/n(


)
p
∫

|ρ(t)|sinβ (t/2)
dt
(3.5)
= O(1)


ξ(t)


0
(3.6)

1/p

 ∫π ( |ρ(t)|t−δ )p 

dt
= O(nδ ),


ξ(t)


π/n
where δ is an arbitrary number such that q(β − δ) − 1 > 0, p−1 + q −1 = 1,
p ≥ 1, and (3.5) and (3.6) hold uniformly in x.
Proof. Taking into account Lemma 3.1 and Lemma 3.2, and proceeding as in
the proof of Theorem 2.4 in (3.3), we obtain (3.4).
4.
Corollaries and remarks
Using results given in Section 2 and Section 3, we observe that the following
corollaries and remarks.
Corollary 4.1. If β = 0, then the weighted class W (Lp , ξ(t)) reduces to the
class Lip(ξ(t), p). Therefore, for f ∈ Lip(ξ(t), p), we have
( )
1
CT
1/p
∥tn (f ) − f (x)∥p = O(n ξ
)
n
( )
1
)
n
with respect to Theorem 2.4 and Theorem 3.4, respectively.
and
CT (f ) − f˜(x)∥ = O(n1/p ξ
∥tg
p
n
Corollary 4.2. If β = 0 and ξ(t) = tα , (0 < α ≤ 1), then the weighted
class W (Lp , ξ(t)) reduces to the Lip(α, p) class. Therefore, for f ∈ Lip(α, p),
(1/p < α), we have
1/p−α
∥tCT
)
n (f ) − f (x)∥p = O(n
and
CT (f ) − f˜(x)∥ = O(n1/p−α )
∥tg
p
n
with respect to Theorem 2.4 and Theorem 3.4, respectively.
12
Ug̃ur Deg̃er
Corollary 4.3. If p → ∞ in Corollary 4.2, then for f ∈ Lipα, (0 < α < 1)
we have
−α
∥tCT
)
n (f ) − f (x)∥∞ = O(n
and
−α
CT (f ) − f˜(x)∥
∥tg
)
∞ = O(n
n
Remark 4.4. If T ≡ (am,k ) is a Nörlund matrix, then the (C 1 · T ) means
give us the Cesàro- Nörlund (C 1 · Np ) means. Accordingly, our main theorems
coincide with Theorem 2.3 and Theorem 2.4 in [23]. Moreover, our main results
generalize the main results in [8] and [23].
( )
1
Remark 4.5. If am,k = 21m m
k , then the (C · T ) means give us the (C, 1)(E, 1)
product means. In this case our main results are reduced to the (C, 1)(E, 1)
product means and the results given in Section 2 and Section 3 coincide with
the results in [6] and [16], respectively.
(m) m−k
1
Remark 4.6. If am,k = (1+q)
, then the (C 1 · T ) means give us the
m
k q
(C, 1)(E, q) product means. Therefore, the results mentioned in Section 2 and
Section 3 are reduced the main results in [10], [16] and [17].
Acknowledgement
This research was partially supported by the The Council of Higher Education of Turkey under a grant.
References
[1] Boos, J. and Cass, P., Classical and Modern Methods in Summability. Oxford
University Press, 2000, 586 pp.
[2] Chandra, P., Trigonometric approximation of functions in Lp -norm. Journal of
Mathematical Analysis and Applications, 275 (2002), 13–26.
[3] Deǧer, U., Daǧadur, İ. and Küc.ükaslan, M., Approximation by trigonometric
polynomials to functions in Lp -norm. Proc. Jangjeon Math. Soc., (15)2, (2012),
203-213.
[4] Fadden, L. Mc., Absolute Nörlund summability. Duke Math. J, Narosa, 9 (1942),
168–207.
[5] Khan, H. H., On degree of approximation of functions belonging to the class
Lip(α, p). Indian J. Pure Appl. Math., (5)2, (1974), 132–136.
[6] Lal, S., On degree of approximation of functions belonging to the weighted
(Lp , ξ(t)) class by (C, 1)(E, 1) means. Tamkang Journal of Mathematics, 30,
(1999), 47–52.
[7] Lal, S., On the degree of approximation of conjugate of a function belonging to
the weighted W (Lp , ξ(t)) class by matrix summability means of conjugate series
of a Fourier series. Tamkang Journal of Mathematics, (31)4, (2000), 47–52.
Draft version April 11, 2016
with respect to Theorem 2.4 and Theorem 3.4, respectively.
ON APPROXIMATION TO FUNCTIONS IN THE W (Lp , ξ(t)) CLASS
13
[8] Lal, S., Approximation of functions belonging to the generalized Lipschitz class
by (C 1 · Np ) summability method of Fourier series. Applied Mathematics and
Computation, (209)2, (2009), 346–350.
[9] Lal, S. and Kushwaha, J. K., Approximation of conjugate of functions belonging
to the generalized Lipschitz class by lower triangular matrix means. Int. Journal
of Math. Analysis, (3)21, (2009), 1031–1041.
[10] Lal, S. and Kushwaha, J. K., Degree of Approximation of Lipschitz function by
product summability method. International Mathematical Forum, (4)43, (2009),
2101–2107.
[11] Leindler, L., Trigonometric approximation in Lp -norm. Journal of Mathematical
Analysis and Applications, 302 (2005) 129–136.
Draft version April 11, 2016
[12] Mazhar, S. M. and Totik, V., Approximation of continuous functions by T means of Fourier series. J. Approx. Theory, (60)2, (1990), 174–182.
[13] Mohapatra, R. N. and Szal, B., On trigonometric approximation of functions in
the Lp -norm. arXiv:1205.5869v1 [math.CA], (2012).
[14] Mittal, M. L., Rhoades, B. E., Mishra, V. N. and Singh, U., Using infinite matrices to approximate functions of class Lip(α, p) using trigonometric polynomials.
Journal of Mathematical Analysis and Applications, (326)1, (2007), 667-676.
[15] Mittal, M. L., Rhoades, B. E., Sonker, S. and Singh, U., Approximation of signals
of class Lip(α, p) by linear operators. Applied Mathematics and Computation,
(217)9, (2011) 4483-4489.
[16] Nigam, H. K., A study on approximation of conjugate of functions belonging
to Lipschitz class and generalized Lipschitz class by product summability means
of conjugate series of Fourier series. Thai Journal of Mathematics, 10(2), (2012),
275–287.
[17] Nigam, H. K. and Sharma, A., On approximation of conjugate of functions
belonging to different classes by product means. International Journal of Pure
and Applied Mathematics, 76(2), (2012), 303–316.
[18] Quade, E. S., Trigonometric approximation in the mean. Duke Math. J. 3 (1937),
529–542.
[19] Qureshi, K., On the degree of approximation of a function belonging to the
Lipschitz class by means of conjugate series. Indian J. Pure Appl. Math., (12)9,
(1981), 1120-1123.
[20] Qureshi, K., On the degree of approximation of a function belonging to the class
Lipα. Indian J. Pure Appl. Math., (13)8, (1982), 560–563.
[21] Qureshi, K., On the degree of approximation of a function belonging to weighted
W (Lr , ξ(t)) class. Indian J. Pure Appl. Math., 13, (1982), 471-475.
[22] Qureshi, K., On the degree of approximation of a function belonging to the class
Lip(α, p) by means of conjugate series. Indian J. Pure Appl. Math., (13)5, (1982),
560-563.
[23] Sing, U., Mittal, M. L. and Sonker, S., Trigonometric Approximation of Signals(Functions) Belonging to W (Lr , ξ(t)) Class by Matrix (C 1 · Np ) Operator.
International Journal of Mathematics and Mathematical Sciences, Vol. (2012)
Article ID 964101 (2012), 11 p.
14
Ug̃ur Deg̃er
[24] Toeplitz, O., Uberallagemeine lineara Mittel bil dunger. P.M.F., 22, (1913), 113–
119.
Draft version April 11, 2016
[25] Zygmund, A., Trigonometric Series. Vol. I, Cambridge: Cambridge University
Press, 1959.