A note on degree of approximation by matrix means in generalized Hölder metric
The aim of the paper is to determine the degree of approximation of functions by matrix means of their Fourier series in a new space of functions introduced by Das, Nath, and Ray. In particular, we extend some results of Leindler and some other results by weakening the monotonicity conditions in res...
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| author | Değer, U. Дегер, У. |
| author_facet | Değer, U. Дегер, У. |
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| description | The aim of the paper is to determine the degree of approximation of functions by matrix means of their Fourier series in a new space of functions introduced by Das, Nath, and Ray. In particular, we extend some results of Leindler and some other results by weakening the monotonicity conditions in results obtained by Singh and Sonker for some classes of numerical sequences introduced by Mohapatra and Szal and present new results by using matrix means. |
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UDC 517.5
U. Deǧer (Mersin Univ., Turkey)
A NOTE ON DEGREE OF APPROXIMATION BY MATRIX MEANS
IN GENERALIZED HÖLDER METRIC
ПРО СТУПIНЬ АПРОКСИМАЦIЇ МАТРИЧНИМИ СЕРЕДНIМИ
В УЗАГАЛЬНЕНIЙ МЕТРИЦI ГЕЛЬДЕРА
The aim of the paper is to determine the degree of approximation of functions by matrix means of their Fourier series in a
new space of functions introduced by Das, Nath, and Ray. In particular, we extend some results of Leindler and some other
results by weakening the monotonicity conditions in results obtained by Singh and Sonker for some classes of numerical
sequences introduced by Mohapatra and Szal and present new results by using matrix means.
Визначено ступiнь апроксимацiї функцiй матричними середнiми їх рядiв Фур’є в новому просторi функцiй, уведе-
них Дасом, Насом та Реєм. Зокрема, розширено деякi результати Лейндлера, а також деякi iншi результати шляхом
послаблення умов монотонностi в результатах, отриманих Сiнгхом та Сонкером для деяких класiв числових послi-
довностей, що були введенi Мoгапатра та Сaлoм, тa наведено новi результати, отриманi за допомогою матричних
сeреднiх.
1. Notations and background. Let f be a 2\pi -periodic function and f \in Lp := Lp (0, 2\pi ) for p \geq 1.
Then we write
sn(f ;x) =
1
2
a0 +
n\sum
k=1
(ak \mathrm{c}\mathrm{o}\mathrm{s} kx+ bk \mathrm{s}\mathrm{i}\mathrm{n} kx) \equiv
n\sum
k=0
Uk(f ;x)
partial sum of the first (n+ 1) terms of the Fourier series of f \in Lp, p \geq 1, at a point x. There
are numerous papers devoted to the approximations by partial sums of Fourier series by many
mathematicians, as Quade [8], Chandra [1], Das, Nath and Ray [2], Leindler [4, 5].
Throughout this work \| .\| p will denote Lp-norm with respect to x and will be defined by
\| f\| p :=
\left\{ 1
2\pi
2\pi \int
0
| f(x)| p dx
\right\}
1/p
.
Moreover, modulus of continuity of f \in C2\pi is defined by
\omega (f, \delta ) := \mathrm{s}\mathrm{u}\mathrm{p}
| h| \leq \delta
| f(x+ h) - f(x)| ,
where C2\pi is space of all 2\pi -periodic and continuous functions defined on \BbbR with the supremum
norm. The class of functions H\omega is defined as the following:
H\omega := \{ f \in C2\pi : \omega (f, \delta ) = O(\omega (\delta ))\} ,
where \omega (\delta ) is a modulus of continuity. The degree of approximation of functions from classes H\omega
in various spaces has been studied by Leindler [5], Mazhar and Totik [6], Das, Nath and Ray [2]. A
further generalization of H\omega space has been given by Das, Nath and Ray in [2]. They defined the
following notations: If f \in Lp(0, 2\pi ), p \geq 1, then denote
c\bigcirc U. DEǦER, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4 485
486 U. DEǦER
H(\omega )
p := \{ f \in Lp(0, 2\pi ), p \geq 1 : A(f, \omega ) < \infty \} ,
where \omega is a modulus of continuity and
A(f, \omega ) := \mathrm{s}\mathrm{u}\mathrm{p}
t\not =0
| | f(\cdot + t) - f(\cdot )| | p
\omega (| t| )
.
The norm in the space H
(\omega )
p is defined by
| | f | | (\omega )p := | | f | | p +A(f, \omega ).
In [2] they proved the following theorem.
Theorem 1.1. Let v and \omega be moduli of continuity such that
\omega (t)
v(t)
is nondecreasing. Moreover,
if f \in H
(\omega )
p , p \geq 1, then
\| sn - f\| (v)p = O
\left( \omega (\pi /n)
v(\pi /n)
\mathrm{l}\mathrm{o}\mathrm{g} n+
1
n
\pi \int
\pi /n
\omega (t)
t2v(t)
dt
\right) .
In [5], Leindler has established the following result improving Theorem 1.1.
Theorem 1.2. Let v and \omega be moduli of continuity such that
\omega (t)
v(t)
is nondecreasing. Moreover,
let the function
\eta (t) := \eta (v, \omega , \varepsilon ; t) := t - \varepsilon \omega (t)
v(t)
be nonincreasing for some 0 < \varepsilon \leq 1. If f \in H
(\omega )
p , p \geq 1, then
\| sn - f\| (v)p \leq \omega (\pi /n)
v(\pi /n)
\mathrm{l}\mathrm{o}\mathrm{g} n (1.1)
for all n \geq 2.
Due to this theorem, Leindler showed that if there exists \varepsilon > 0 such that t - \varepsilon \omega (t)
v(t)
is also
nonincreasing, then in Theorem 1.1 the second term can be omitted. In this paper of Leindler, he
also considered the degree of approximation of f \in H
(\omega )
p by Nörlund1, Riesz and generalized de la
Vallée Poussin means defined as follows:
Nn(f ;x) :=
1
Pn
n\sum
m=0
pn - msm(f ;x),
Rn(f ;x) :=
1
Pn
n\sum
m=0
pmsm(f ;x),
where Pn = p0 + p1 + p2 + ...+ pn \not = 0, n \geq 0, and by convention p - 1 = P - 1 = 0; and
1Called the "Woronoi’s transformations" instead of "Nörlund’s transformations (or Nörlund means)" by Russian math-
ematicians. See [10] for more detailed information.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
A NOTE ON DEGREE OF APPROXIMATION BY MATRIX MEANS IN GENERALIZED HÖLDER METRIC 487
Vn(x) := Vn(\lambda , f ;x) :=
1
\lambda n+1
n\sum
m=n - \lambda n
sm(f ;x),
where (\lambda n) is a nondecreasing sequence of positive integers with \lambda 0 = 1 and \lambda n+1 \leq \lambda n + 1. In
this studying, we shall consider the degree of approximation of f \in H
(\omega )
p with the norm in the space
H
(\omega )
p by trigonometrical polynomials \tau n(f ;x), where
\tau n(f ;x) = \tau n(f, T ;x) :=
n\sum
m=0
an,msm(f ;x) \forall n \geq 0.
Throughout T \equiv (an,m) is a lower triangular infinite matrix of nonnegative real numbers such that:
an,m =
\left\{ \geq 0, m \leq n,
0, m > n,
n,m = 0, 1, 2, . . . , (1.2)
n\sum
m=0
an,m = 1, n = 0, 1, 2, . . . . (1.3)
The Fourier series of signal f is called to be T -summable to s, if \tau n(f ;x) \rightarrow s as n \rightarrow \infty . On the
other hand, we know that a summability method is regular, if for every convergent sequence (sn),
\mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty sn = s \Rightarrow \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \tau n = s. If we take
an,m =
\left\{
1
n+ 1
, m \leq n,
0, m > n,
n,m = 0, 1, 2, . . . ,
\tau n(f ;x) gives us
\sigma n(f ;x) =
1
n+ 1
n\sum
m=0
sm(f ;x) \forall n \geq 0.
Throughout this studying, we shall use notations D \ll R (R \ll D) in inequalities if there exists
a positive constant K such that D \leq KR (R \leq KD) where D and R are depend on n. However,
K may be different in different occurrences of "" \ll "".
Now, let’s recall the definitions of some classes of numerical sequences discussed in detail in [5]
and [7]. Let u := (un) be a nonnegative infinite sequence and C := (Cn) =
1
n+ 1
\sum n
m=0
um.
A sequence u is called almost monotone decreasing (briefly u \in AMDS) (increasing (briefly
u \in AMIS)), if there exists a constant K := K(u) which only depends on u such that
un \leq Kum (un \geq Kum)
for all n \geq m.
If C \in AMDS (C \in AMIS), then we say that the sequence u is almost monotone decreasing
(increasing) mean sequence and denoted by C \in AMDMS (C \in AMIMS).
A sequence u tending to zero is called a rest bounded variation sequence (RBV S) (rest bounded
variation mean sequence (RBVMS)), if it has the property
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
488 U. DEǦER
\infty \sum
m=k
| \Delta um| \leq K(u)uk
\Biggl( \infty \sum
m=k
| \Delta Cm| \leq K(u)Ck
\Biggr)
for all natural numbers k, where \Delta un = un - un+1.
A sequence u is called a head bounded variation sequence (HBV S) (head bounded variation
mean sequence (HBVMS)), if it has the property
k - 1\sum
m=0
| \Delta um| \leq K(u)uk
\Biggl(
k - 1\sum
m=0
| \Delta Cm| \leq K(u)Ck
\Biggr)
for all natural numbers k, or only for all k \leq N if the sequence u has only finite nonzero terms and
the last nonzero term uN .
When a matrix T = (an,m) belongs to one of the above classes, it means that it satisfies the
required conditions from the above definitions with respect to m = 0, 1, 2, . . . , n for all n. Accordingly
if (an,m)\infty m=0 belongs to RBV S (RBVMS) or HBV S (HBVMS), respectively then
\infty \sum
m=k
| \Delta man,m| \leq Kan,k
\Biggl( \infty \sum
m=k
| \Delta mAn,m| \leq KAn,k
\Biggr)
,
k - 1\sum
m=0
| \Delta man,m| \leq Kan,k
\Biggl(
k - 1\sum
m=0
| \Delta mAn,m| \leq KAn,k
\Biggr)
,
where
An,m =
1
m+ 1
m\sum
k=0
an,k
for all n (0 \leq k \leq n) and \Delta man,m = an,m - an,m+1.
It is clear that the following inclusions are true for the above classes of numerical sequences:
RBV S \subset AMDS , RBVMS \subset AMDMS
and
HBV S \subset AMIS , HBVMS \subset AMIMS.
Moreover, Mohapatra and Szal in [7] showed that the following embedding relations are true:
AMDS \subset AMDMS
and
AMIS \subset AMIMS.
Taking into these inclusions, we will extend the some results given in [9] by weakening the
monotonicity conditions. Furthermore we shall give the degree of approximation of functions by
matrix means of their Fourier series under the new conditions. We see that the results obtained in
this paper generalize the some results in [3, 5, 9].
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
A NOTE ON DEGREE OF APPROXIMATION BY MATRIX MEANS IN GENERALIZED HÖLDER METRIC 489
2. Main result. The following theorem shall extend the some results of Leindler in [5] and the
some of the their results by weakening the monotonicity conditions in results of Singh and Sonker [9]
for some classes of numerical sequences that given by Mohapatra and Szal in [7]. Also it includes
some result under a new condition. Accordingly, the main result is as follows:
Theorem 2.1. Let v and \omega be moduli of continuity such that
\omega (t)
v(t)
is nondecreasing. Moreover,
let the function
\eta (t) := \eta (v, \omega , \varepsilon ; t) := t - \varepsilon \omega (t)
v(t)
be nonincreasing for some 0 < \varepsilon \leq 1 and T := (an,m) be a lower triangular infinite regular matrix.
If one of the following additional conditions is satisfied:
(i) (an,m) \in AMIMS,
(ii) (an,m) \in AMDMS and (n+ 1)an,0 \ll 1,
(iii)
\sum n - 1
m=0
| \Delta m(An,m)| \ll n - 1,
(iv)
\sum n - 1
m=1
m1 - \varepsilon | \Delta m(an,m)| \ll n - \varepsilon , n| \Delta m(an,m)| \ll 1(m = 0, 1) and (n+ 1)an,n \ll 1,
then for f \in H
(\omega )
p , p \geq 1
\| f(x) - \tau n(f ;x)\| (v)p \leq w(1/n)
v(1/n)
\mathrm{l}\mathrm{o}\mathrm{g} n (2.1)
for all n \geq 2.
3. Known results. In this section we will give some known results to use in proof of the
Theorem 2.1.
Lemma 3.1 [7]. Let (1.2) and (1.3) hold. If (an,m) \in AMIMS or (an,m) \in AMDMS and
(n+ 1)an,0 \ll 1, then
n\sum
m=0
(m+ 1) - \alpha an,m \ll (n+ 1) - \alpha
holds for 0 < \alpha < 1.
Lemma 3.2 [5]. If the conditions of Theorem 1.2 are satisfied with 0 < \varepsilon < 1,
\| \sigma n - f\| (v)p \leq w(1/n)
v(1/n)
\mathrm{l}\mathrm{o}\mathrm{g} n, n \geq 2, (3.1)
and
\| \sigma n - sn\| (v)p \leq w(1/n)
v(1/n)
\mathrm{l}\mathrm{o}\mathrm{g} n, n \geq 2. (3.2)
4. Proof of Theorem 2.1. First of all, let us give the following inequalities will be used in proof
of Theorem 2.1. Since \eta (t) is nonincreasing, the sequence
\eta
\biggl(
1
n
\biggr)
= n\varepsilon w(1/n)
v(1/n)
(4.1)
is nondecreasing. Accordingly,
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
490 U. DEǦER
w(1/n)
v(1/n)
\gg 1
n\varepsilon
\geq 1
n
, 0 < \varepsilon \leq 1, (4.2)
and based on the knowledge of nonincreasing of sequence
\biggl(
w(1/n)
v(1/n)
\biggr)
we have
w(1/\~n)
v(1/\~n)
\ll w(1/n)
v(1/n)
, (4.3)
where \~n denotes integer part of
n
2
. The method of proof will be similar with some parts in [9].
Proof Theorem 2.1. Let us prove the theorem for all the cases, respectively.
Case 1. Let (an,k) \in AMIMS. By the definition of \tau n(f, x), we have
\tau n(f ;x) - f(x) =
n\sum
m=0
an,m \{ sm(f ;x) - f(x)\} .
So, from hypothesis and (1.1) we obtain
\| \tau n(f ;x) - f(x)\| (v)p \leq
\Biggl(
\~n\sum
m=0
+
n\sum
m=\~n+1
\Biggr)
an,m \| sm(f ;x) - f\| (v)p \ll
\ll \{ an,0 \| s0(f ;x) - f(x)\| (v)p + an,1 \| s1(f ;x) - f(x)\| (v)p \} +
+
\~n\sum
m=2
an,m
w(1/m)
v(1/m)
\mathrm{l}\mathrm{o}\mathrm{g}m+
n\sum
m=\~n+1
an,m
w(1/m)
v(1/m)
\mathrm{l}\mathrm{o}\mathrm{g}m =: I1 + I2 + I3.
Now let’s estimate I1, I2 and I3, respectively: since (an,k) \in AMIMS, we get
an,0 = C0 \ll Cn =
1
n+ 1
n\sum
m=0
an,m =
1
n+ 1
\ll n - 1
and
C1 =
1
2
\{ an,0 + an,1\} =
1
2
\{ C0 + an,1\} \Rightarrow an,1 \ll n - 1,
where Ck := An,k =
1
k + 1
\sum k
m=0
an,m. Therefore, we have
I1 \ll
w(1/n)
v(1/n)
in view of (4.2) for all n \geq 2
I2 =
\~n\sum
m=1
an,m
(m+ 1)\varepsilon
(m+ 1)\varepsilon
w(1/m)
v(1/m)
\mathrm{l}\mathrm{o}\mathrm{g}m \ll (\~n)\varepsilon
w(1/\~n)
v(1/\~n)
\mathrm{l}\mathrm{o}\mathrm{g} \~n
\~n\sum
m=1
an,m(m+ 1) - \varepsilon .
From Lemma 3.1 and (4.3), we obtain
I2 \ll
w(1/n)
v(1/n)
\mathrm{l}\mathrm{o}\mathrm{g} n
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
A NOTE ON DEGREE OF APPROXIMATION BY MATRIX MEANS IN GENERALIZED HÖLDER METRIC 491
for all n \geq 2. With a simple analysis, we get
I3 =
n\sum
m=\~n+1
an,m
w(1/m)
v(1/m)
\mathrm{l}\mathrm{o}\mathrm{g}m \ll w(1/\~n)
v(1/\~n)
\mathrm{l}\mathrm{o}\mathrm{g} n
n\sum
m=\~n+1
an,m \ll w(1/\~n)
v(1/\~n)
\mathrm{l}\mathrm{o}\mathrm{g} n
n\sum
m=0
an,m
by considering (4.3) for all n \geq 2. Therefore, we obtain (2.1) for the case (i).
Case 2. Let (an,k) \in AMDMS and (n+ 1)an,0 \ll 1. As the first case, we have
\| \tau n(f ;x) - f(x)\| (v)p \leq
\Biggl(
\~n\sum
m=0
+
n\sum
m=\~n+1
\Biggr)
an,m \| sm(f ;x) - f\| (v)p \ll
\ll \{ an,0 \| s0(f ;x) - f(x)\| (v)p + an,1 \| s1(f ;x) - f(x)\| (v)p \} +
+
\Biggl(
\~n\sum
m=2
+
n\sum
m=\~n+1
\Biggr)
an,m
w(1/m)
v(1/m)
\mathrm{l}\mathrm{o}\mathrm{g}m =: J1 + J2.
Taking into account (4.3) and an,0 = C0 \gg C1 =
1
2
\{ an,0 + an,1\} \Rightarrow an,0 \gg an,1, we get J1 \ll
\ll w(1/n)
v(1/n)
\mathrm{l}\mathrm{o}\mathrm{g} n for all n \geq 2. By using Lemma 3.1 and (4.3), the evaluation of J2 is obtained
similarly to the case (i). Therefore, (2.1) is proved.
Case 3. By applying two times Abel’s transformation and using
\sum n
m=0
an,m = 1, we have
\tau n(f ;x) - f(x) =
n\sum
m=0
an,m \{ sm(f ;x) - f(x)\} =
=
n - 1\sum
m=0
(sm(f ;x) - sm+1(f ;x))
m\sum
k=0
an,k + \{ sn(f ;x) - f(x)\} =
= \{ sn(f ;x) - f(x)\} -
n - 1\sum
m=0
(m+ 1)Um+1(f ;x)An,m =
= \{ sn(f ;x) - f(x)\} -
n - 2\sum
m=0
(An,m - An,m+1)
m\sum
k=0
(k + 1)Uk+1(f ;x) -
- 1
n
n - 1\sum
k=0
an,k
n - 1\sum
m=0
(m+ 1)Um+1(f ;x).
Here
\| \tau n(f ;x) - f(x)\| (v)p \leq \| sm(f ;x) - f\| (v)p +
n - 2\sum
m=0
| An,m - An,m+1|
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
m+1\sum
k=1
kUk(f ;x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
(v)
p
+
+
1
n
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
n\sum
m=1
mUm(f ;x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
(v)
p
\ll
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
492 U. DEǦER
\ll \| sm(f ;x) - f\| (v)p + | An,0 - An,1| + | An,1 - An,2| +
+
n - 2\sum
m=2
| An,m - An,m+1|
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
m+1\sum
k=1
kUk(f ;x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
(v)
p
+
1
n
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
n\sum
m=1
mUm(f ;x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
(v)
p
. (4.4)
Since
sn(f ;x) - \sigma n(f ;x) =
1
n+ 1
n\sum
k=1
kUk(f ;x),
by (3.2) we get\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
n\sum
k=1
kUk(f ;x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
(v)
p
= (n+ 1) \| sn(f ;x) - \sigma n(f ;x)\| (v)p \ll n
w(1/n)
v(1/n)
\mathrm{l}\mathrm{o}\mathrm{g} n. (4.5)
So, combining (1.1), (4.4) and (4.5), we get that
\| \tau n(f ;x) - f(x)\| (v)p \ll w(1/n)
v(1/n)
\mathrm{l}\mathrm{o}\mathrm{g} n+
n - 1\sum
m=0
| An,m - An,m+1| +
+n
w(1/n)
v(1/n)
\mathrm{l}\mathrm{o}\mathrm{g} n
n - 1\sum
m=0
| An,m - An,m+1| . (4.6)
Taking into account (4.2), (4.6) and the condition (iii) of Theorem 2.1, then
\| \tau n(f ;x) - f(x)\| (v)p \ll w(1/n)
v(1/n)
\mathrm{l}\mathrm{o}\mathrm{g} n
holds for all n \geq 2.
Case 4. By applying Abel’s transformation, we obtain
\tau n(f ;x) - f(x) =
n\sum
m=0
an,m \{ sm(f ;x) - f(x)\} =
=
n - 1\sum
m=0
\Delta man,m
m\sum
k=0
(sk(f ;x) - f(x)) + an,n
n\sum
k=0
\{ sk(f ;x) - f(x)\} =
=
n - 1\sum
m=0
(m+ 1)(\Delta man,m)(\sigma m(f ;x) - f(x)) + (n+ 1)an,n(\sigma n(f ;x) - f(x)).
Thus, owing to (3.1), (4.1), (4.2) and the condition (iv) of Theorem 2.1, we have
\| \tau n(f ;x) - f(x)\| (v)p \leq
n - 1\sum
m=0
| (\Delta man,m)| (m+ 1)\| \sigma m(f ;x) - f(x)\| (v)p +
+(n+ 1)an,n\| \sigma n(f ;x) - f(x)\| (v)p \ll
\ll | (\Delta man,0)| \| \sigma 0(f ;x) - f(x)\| (v)p +
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
A NOTE ON DEGREE OF APPROXIMATION BY MATRIX MEANS IN GENERALIZED HÖLDER METRIC 493
+2| (\Delta man,1)| \| \sigma 1(f ;x) - f(x)\| (v)p +
+
n - 1\sum
m=2
m| (\Delta man,m)| \| \sigma m(f ;x) - f(x)\| (v)p +
w(1/n)
v(1/n)
\mathrm{l}\mathrm{o}\mathrm{g} n \ll
\ll 1
n
+
n - 1\sum
m=2
m1 - \varepsilon | (\Delta man,m)| m\varepsilon w(1/m)
v(1/m)
\mathrm{l}\mathrm{o}\mathrm{g}m+
w(1/n)
v(1/n)
\mathrm{l}\mathrm{o}\mathrm{g} n \ll
\ll n\varepsilon w(1/n)
v(1/n)
\mathrm{l}\mathrm{o}\mathrm{g} n
n - 1\sum
m=1
m1 - \varepsilon | (\Delta man,m)| + w(1/n)
v(1/n)
\mathrm{l}\mathrm{o}\mathrm{g} n \ll
\ll w(1/n)
v(1/n)
\mathrm{l}\mathrm{o}\mathrm{g} n.
Therefore, Theorem 2.1 is proved.
5. Corollaries and remarks. Theorem 2.1 gives us some corollaries and remarks which related
with the results in [3, 5, 9].
Corollary 5.1. If the conditions of Theorem 2.1 and the following additional conditions are
satisfied:
(i) (an,m) \in HBVMS,
(ii) (an,m) \in RBVMS and (n+ 1)an,0 \ll 1,
then for f \in H
(\omega )
p , p \geq 1,
\| f(x) - \tau n(f ;x)\| (v)p \leq w(1/n)
v(1/n)
\mathrm{l}\mathrm{o}\mathrm{g} n
for all n \geq 2.
Proof. Since HBVMS \subset AMIMS and RBVMS \subset AMDMS, the proof is obvious.
Since AMDS \subset AMDMS and AMIS \subset AMIMS, we can write the following result which
coincide with the conditions (i) and (ii) of Theorem 6 in [9].
Corollary 5.2. If the conditions of Theorem 2.1 and the following additional conditions are
satisfied:
(i) (an,m) \in AMIS,
(ii) (an,m) \in AMDS and (n+ 1)an,0 \ll 1,
then for f \in H
(\omega )
p , p \geq 1,
\| f(x) - \tau n(f ;x)\| (v)p \leq w(1/n)
v(1/n)
\mathrm{l}\mathrm{o}\mathrm{g} n
for all n \geq 2.
Remark 5.1. Theorem 2.1 generalizes Theorem 6 in [9] under the conditions (i) and (ii).
Remark 5.2. In addition, the above corollary is also true for the classes of sequence HBV S and
RBV S owing to the inclusions HBV S \subset AMIS and RBV S \subset AMDS.
Remark 5.3. If T \equiv (an,m) is a Nörlund matrix, then the conditions (i) and (ii) of Corollary 5.2
is reduced to the conditions of Theorem 2 in [5] and to more general case of Theorem 3.1 in [3],
respectively. Moreover, Corollary 5.1 generalizes Theorem 2 in [5].
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
494 U. DEǦER
Remark 5.4. If we take an,m =
pn - mqm
rn
, where rn =
\sum n
m=0
pn - mqm, then the condition (iv)
of Theorem 2.1 in this studying is reduced the conditions of Theorem 3.2 in [3]. Therefore, Theorem
2.1 generalizes Theorem 3.2 in [3] under the condition (iv). Here, if pn - m =
1
\lambda n+1
, 0 \leq m \leq n, and
qm =
\Biggl\{
0, 0 \leq m < n - \lambda n,
1, n - \lambda n \leq m \leq n,
then it gives us generalized de la Vallée Poussin means where (\lambda n) is a nondecreasing sequence of
positive integers with \lambda 0 = 1 and \lambda n+1 \leq \lambda n + 1.
Remark 5.5. If we take an,m =
p\alpha n - m
P\alpha
n
, where P\alpha
n =
\sum n
m=0
p\alpha m; p\alpha m =
\sum m
i=0
A\alpha - 1
m - vpv and
A\alpha
n =
\biggl(
\alpha + n
n
\biggr)
, \alpha > - 1, n = 1, 2, 3, . . . , then the conditions of Corollary 5.2 is reduced the
conditions (i) and (ii) of Theorem 3.3 in [3]. Furthermore, Corollary 5.1 and Corollary 5.2 generalize
Theorem 3.3 and Theorem 3.4 in [3] by taking into account Remark 5.2 under the conditions (i) and
(ii).
Remark 5.6. The condition (iv) of Theorem 2.1 is different from the condition (iv) of Theorem
6 in [9] with the additional condition n| \Delta m(an,m)| \ll 1, m = 0, 1.
Acknowledgment. The author is thankful to The Council of Higher Education of Turkey for the
Scientific Research Grant.
References
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Received 27.02.13,
after revision — 15.01.16
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
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| id | umjimathkievua-article-1854 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:13:57Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-18542019-12-05T09:29:54Z A note on degree of approximation by matrix means in generalized Hölder metric Про ступiнь апроксимацiї матричними середнiми в узагальненiй метрицi Гельдера Değer, U. Дегер, У. The aim of the paper is to determine the degree of approximation of functions by matrix means of their Fourier series in a new space of functions introduced by Das, Nath, and Ray. In particular, we extend some results of Leindler and some other results by weakening the monotonicity conditions in results obtained by Singh and Sonker for some classes of numerical sequences introduced by Mohapatra and Szal and present new results by using matrix means. Визначено ступiнь апроксимацiї функцiй матричними середнiми їх рядiв Фур’є в новому просторi функцiй, уведених Дасом, Насом та Реєм. Зокрема, розширено деякi результати Лейндлера, а також деякi iншi результати шляхом послаблення умов монотонностi в результатах, отриманих Сiнгхом та Сонкером для деяких класiв числових послiдовностей, що були введенi Мoгапатра та Сaлoм, тa наведено новi результати, отриманi за допомогою матричних сeреднiх. Institute of Mathematics, NAS of Ukraine 2016-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1854 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 4 (2016); 485-494 Український математичний журнал; Том 68 № 4 (2016); 485-494 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1854/836 Copyright (c) 2016 Değer U. |
| spellingShingle | Değer, U. Дегер, У. A note on degree of approximation by matrix means in generalized Hölder metric |
| title | A note on degree of approximation by matrix means in generalized Hölder metric |
| title_alt | Про ступiнь апроксимацiї матричними середнiми в узагальненiй метрицi Гельдера |
| title_full | A note on degree of approximation by matrix means in generalized Hölder metric |
| title_fullStr | A note on degree of approximation by matrix means in generalized Hölder metric |
| title_full_unstemmed | A note on degree of approximation by matrix means in generalized Hölder metric |
| title_short | A note on degree of approximation by matrix means in generalized Hölder metric |
| title_sort | note on degree of approximation by matrix means in generalized hölder metric |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1854 |
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