A matrix application of power increasing sequences to infinite series and Fourier series
UDC 517.54 The aim of the paper is a generalization, under weaker conditions, of the main theorem on quasi-$\sigma$-power increasing sequences applied to $|A,\theta_{n}|_{k}$ summability factors of infinite series and Fourier series. We obtain some new and known results related to basic summability...
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| author | Yıldız, Şebnem Yıldız, Şebnem Yıldız, Şebnem |
| author_facet | Yıldız, Şebnem Yıldız, Şebnem Yıldız, Şebnem |
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| description | UDC 517.54
The aim of the paper is a generalization, under weaker conditions, of the main theorem on quasi-$\sigma$-power increasing sequences applied to $|A,\theta_{n}|_{k}$ summability factors of infinite series and Fourier series. We obtain some new and known results related to basic summability methods. |
| doi_str_mv | 10.37863/umzh.v72i5.6016 |
| first_indexed | 2026-03-24T03:25:15Z |
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DOI: 10.37863/umzh.v72i5.6016
UDC 517.54
Ş. Yıldız (Dep. Math., Ahi Evran Univ., Kırşehir, Turkey)
A MATRIX APPLICATION OF POWER INCREASING SEQUENCES
TO INFINITE SERIES AND FOURIER SERIES
МАТРИЧНЕ ЗАСТОСУВАННЯ ЗРОСТАЮЧИХ СТЕПЕНЕВИХ
ПОСЛIДОВНОСТЕЙ ДО НЕСКIНЧЕННИХ РЯДIВ I РЯДIВ ФУР’Є
The aim of the paper is a generalization, under weaker conditions, of the main theorem on quasi-\sigma -power increasing
sequences applied to | A, \theta n| k summability factors of infinite series and Fourier series. We obtain some new and known
results related to basic summability methods.
Метою даної роботи є узагальнення основної теореми про застосування зростаючих квазi-\sigma -степеневих послiдов-
ностей до коефiцiєнтiв пiдсумовування | A, \theta n| k нескiнченних рядiв i рядiв Фур’є при слабших умовах. Отримано
деякi новi та вiдомi результати, що вiдносяться до базових методiв пiдсумовування.
1. Introduction.
Definition 1.1. A positive sequence (bn) is said to be an almost increasing sequence if there
exists a positive increasing sequence (cn) and two positive constants M and N such that Mcn \leq
\leq bn \leq Ncn (see [1]).
Definition 1.2. A positive sequence (Xn) is said to be quasi-\sigma -power increasing sequence if
there exists a constant K = K(\sigma ,X) \geq 1 such that Kn\sigma Xn \geq m\sigma Xm for all n \geq m \geq 1.
Every almost increasing sequence is a quasi-\sigma -power increasing sequence for any nonnegative
\sigma , but the converse is not true for \sigma > 0 (see [13]). For any sequence (\lambda n) we write that \Delta 2\lambda n =
= \Delta \lambda n - \Delta \lambda n+1 and \Delta \lambda n = \lambda n - \lambda n+1.
Definition 1.3. The sequence (\lambda n) is said to be of bounded variation, denoted by (\lambda n) \in \scrB \scrV ,
if
\sum \infty
n=1
| \Delta \lambda n| < \infty .
Let
\sum
an be a given infinite series with the partial sums (sn). By u\alpha n and t\alpha n we denote the nth
Cesàro means of order \alpha , with \alpha > - 1, of the sequence (sn) and (nan), respectively, that is (see
[8])
u\alpha n =
1
A\alpha
n
n\sum
v=0
A\alpha - 1
n - vsv and t\alpha n =
1
A\alpha
n
n\sum
v=0
A\alpha - 1
n - vvav,
where
A\alpha
n =
(\alpha + 1)(\alpha + 2) . . . (\alpha + n)
n!
= O(n\alpha ), A\alpha
- n = 0 for n > 0.
Definition 1.4. The series
\sum
an is said to be summable | C,\alpha | k, k \geq 1, if (see [10, 12])
\infty \sum
n=1
nk - 1| u\alpha n - u\alpha n - 1| k =
\infty \sum
n=1
1
n
| t\alpha n| k < \infty .
c\bigcirc Ş. YILDIZ, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5 635
636 Ş. YILDIZ
If we take \alpha = 1, then | C,\alpha | k summability reduces to | C, 1| k summability. Let (pn) be a
sequence of positive real numbers such that
Pn =
n\sum
v=0
pv \rightarrow \infty as n \rightarrow \infty (P - i = p - i = 0, i \geq 1).
The sequence-to-sequence transformation
wn =
1
Pn
n\sum
v=0
pvsv
defines the sequence (wn) of the Riesz mean or simply the ( \=N, pn) mean of the sequence (sn)
generated by the sequence of coefficients (pn) (see [11]).
Definition 1.5. The series
\sum
an is said to be summable | \=N, pn| k, k \geq 1, if (see [2])
\infty \sum
n=1
\biggl(
Pn
pn
\biggr) k - 1
| wn - wn - 1| k < \infty .
In the special case when pn = 1 for all values of n (resp., k = 1), | \=N, pn| k summability is the
same as | C, 1| k (resp., | \=N, pn| ) summability.
2. Known results. The following theorem is dealing with | \=N, pn| k summability factors of
infinite series under weaker conditions.
Theorem 2.1 [7]. Let (Xn) be a quasi-\sigma -power increasing sequence. If the sequences (Xn),
(\lambda n) and (pn) satisfy the conditions
\lambda mXm = O(1) as m \rightarrow \infty , (2.1)
m\sum
n=1
nXn| \Delta 2\lambda n| = O(1) as m \rightarrow \infty , (2.2)
m\sum
n=1
Pn
n
= O(Pm), (2.3)
m\sum
n=1
pn
Pn
| tn| k
Xk - 1
n
= O(Xm) as m \rightarrow \infty , (2.4)
m\sum
n=1
| tn| k
nXk - 1
n
= O(Xm) as m \rightarrow \infty , (2.5)
then the series
\sum
an\lambda n is summable | \=N, pn| k, k \geq 1.
3. An application of absolute matrix summability to infinite series. Let A = (anv) be
a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. Then A defines the
sequence-to-sequence transformation, mapping the sequence s = (sn) to As = (An(s)), where
An(s) =
n\sum
v=0
anvsv, n = 0, 1, . . . .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
A MATRIX APPLICATION OF POWER INCREASING SEQUENCES TO INFINITE SERIES . . . 637
Definition 3.1. Let (\theta n) be any sequence of positive real numbers. The series
\sum
an is said to
be summable | A, \theta n| k, k \geq 1, if (see [14, 15])
\infty \sum
n=1
\theta k - 1
n
\bigm| \bigm| \=\Delta An(s)
\bigm| \bigm| k < \infty ,
where
\=\Delta An(s) = An(s) - An - 1(s).
If we take \theta n =
Pn
pn
, then we obtain | A, pn| k summability (see [16]), and if we take \theta n = n,
then we have | A| k summability (see [18]). Also, if we take \theta n =
Pn
pn
and anv =
pv
Pn
, then we have
| \=N, pn| k summability. Furthermore, if we take \theta n = n, anv =
pv
Pn
and pn = 1 for all values of n,
then | A, \theta n| k summability reduces to | C, 1| k summability (see [10]). Finally, if we take \theta n = n and
anv =
pv
Pn
, then we obtain | R, pn| k summability (see [3]).
4. Main results. The Fourier series play an important role in many areas of applied mathematics
and mechanics. Recently some papers have been done concerning absolute matrix summability of
infinite series and Fourier series (see [5, 6, 19 – 21]). The aim of this paper is to generalize Theorem
2.1 for | A, \theta n| k summability method for these series.
Given a normal matrix A = (anv), we associate two lower semimatrices \=A = (\=anv) and \^A =
= (\^anv) as follows:
\=anv =
n\sum
i=v
ani, n, v = 0, 1, . . . ,
and
\^a00 = \=a00 = a00, \^anv = \=anv - \=an - 1,v, n = 1, 2, . . . .
It may be noted that \=A and \^A are the well-known matrices of series-to-sequence and series-to-series
transformations, respectively. Then we have
An(s) =
n\sum
v=0
anvsv =
n\sum
v=0
\=anvav (4.1)
and
\=\Delta An(s) =
n\sum
v=0
\^anvav. (4.2)
Using this notation we have the following theorem.
Theorem 4.1. Let k \geq 1 and A = (anv) be a positive normal matrix such that
an0 = 1, n = 0, 1, . . . , (4.3)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
638 Ş. YILDIZ
an - 1,v \geq anv for n \geq v + 1, (4.4)
n - 1\sum
v=1
1
v
\^an,v+1 = O(ann). (4.5)
Let (Xn) be a quasi-\sigma -power increasing sequence and let (\theta nann) be a non increasing sequence. If
the sequences (Xn), (\lambda n) and (pn) satisfy the conditions (2.1) – (2.3) of Theorem 2.1, and
m\sum
n=1
\theta k - 1
n aknn
| tn| k
Xk - 1
n
= O(Xm) as m \rightarrow \infty , (4.6)
m\sum
n=1
(\theta nann)
k - 1 | tn| k
nXk - 1
n
= O(Xm) as m \rightarrow \infty , (4.7)
then the series
\sum
an\lambda n is summable | A, \theta n| k, k \geq 1.
It may be remarked that if we take A = ( \=N, pn) and \theta n =
Pn
pn
, then the conditions (4.6), (4.7)
are reduced to (2.4), (2.5). Also, the condition (4.5) satisfied by condition (2.3). Therefore, we have
Theorem 2.1.
We need the following lemmas for the proof of our theorem.
Lemma 4.1 [17]. From the conditions (4.3) and (4.4) of Theorem 4.1, we have
n - 1\sum
v=0
| \=\Delta anv| \leq ann,
\^an,v+1 \geq 0,
m+1\sum
n=v+1
\^an,v+1 = O(1).
Lemma 4.2 [4]. Under the conditions of Theorem 2.1 we have that
nXn| \Delta \lambda n| = O(1) as n \rightarrow \infty ,
\infty \sum
n=1
Xn| \Delta \lambda n| < \infty .
Proof of Theorem 4.1. Let (In) denotes the A-transform of the series
\sum \infty
n=1
an\lambda n. Then, by
(4.1) and (4.2), we have
\=\Delta In =
n\sum
v=1
\^anvav\lambda v.
Applying Abel’s transformation to this sum, we obtain
\=\Delta In =
n\sum
v=1
\^anvav\lambda v
v
v
=
n - 1\sum
v=1
\Delta
\biggl(
\^anv\lambda v
v
\biggr) v\sum
r=1
rar +
\^ann\lambda n
n
n\sum
r=1
rar =
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
A MATRIX APPLICATION OF POWER INCREASING SEQUENCES TO INFINITE SERIES . . . 639
=
n - 1\sum
v=1
\Delta
\biggl(
\^anv\lambda v
v
\biggr)
(v + 1)tv + \^ann\lambda n
n+ 1
n
tn =
=
n - 1\sum
v=1
\=\Delta anv\lambda vtv
v + 1
v
+
n - 1\sum
v=1
\^an,v+1\Delta \lambda vtv
v + 1
v
+
n - 1\sum
v=1
\^an,v+1\lambda v+1
tv
v
+ ann\lambda ntn
n+ 1
n
=
= In,1 + In,2 + In,3 + In,4.
To complete the proof of Theorem 4.1, by Minkowski’s inequality, it is sufficient to show that
\infty \sum
n=1
\theta k - 1
n | In,r| k < \infty for r = 1, 2, 3, 4.
First, by applying Hölder’s inequality with indices k and k\prime , where k > 1 and
1
k
+
1
k\prime
= 1, we get
m+1\sum
n=2
\theta k - 1
n | In,1| k \leq
m+1\sum
n=2
\theta k - 1
n
\Biggl\{
n - 1\sum
v=1
\bigm| \bigm| \bigm| \bigm| v + 1
v
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \=\Delta anv
\bigm| \bigm| | \lambda v| | tv|
\Biggr\} k
=
= O(1)
m+1\sum
n=2
\theta k - 1
n
n - 1\sum
v=1
\bigm| \bigm| \=\Delta anv
\bigm| \bigm| | \lambda v| k| tv| k
\Biggl\{
n - 1\sum
v=1
\bigm| \bigm| \=\Delta anv
\bigm| \bigm| \Biggr\} k - 1
.
By using
\Delta \^anv = \^anv - \^an,v+1 = \=anv - \=an - 1,v - \=an,v+1 + \=an - 1,v+1 = anv - an - 1,v,
and (4.3) and (4.4), we have
n - 1\sum
v=1
| \=\Delta anv| =
n - 1\sum
v=1
| anv - an - 1,v| =
n - 1\sum
v=1
(an - 1,v - anv) =
=
n - 1\sum
v=0
an - 1,v - an - 1,0 -
n\sum
v=0
anv + an0 + ann =
= 1 - an - 1,0 - 1 + an0 + ann \leq ann.
By using
\sum m+1
n=v+1
| \=\Delta anv| \leq avv, we obtain
m+1\sum
n=2
\theta k - 1
n | In,1| k = O(1)
m+1\sum
n=2
\theta k - 1
n ak - 1
nn
\Biggl\{
n - 1\sum
v=1
| \=\Delta anv| | \lambda v| k| tv| k
\Biggr\}
=
= O(1)
m\sum
v=1
| \lambda v| k - 1| \lambda v| | tv| k
m+1\sum
n=v+1
(\theta nann)
k - 1| \=\Delta anv| =
= O(1)
m\sum
v=1
(\theta vavv)
k - 1| \lambda v| k - 1| \lambda v| | tv| k
m+1\sum
n=v+1
| \=\Delta anv| =
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
640 Ş. YILDIZ
= O(1)
m\sum
v=1
(\theta vavv)
k - 1 1
Xk - 1
v
| \lambda v| | tv| kavv =
= O(1)
m - 1\sum
v=1
\Delta | \lambda v|
v\sum
r=1
\theta k - 1
r akrr
| tr| k
Xk - 1
r
+O(1)| \lambda m|
m\sum
v=1
\theta k - 1
v akvv
| tv| k
Xk - 1
v
=
= O(1)
m - 1\sum
v=1
| \Delta \lambda v| Xv +O(1)| \lambda m| Xm = O(1) as m \rightarrow \infty ,
by virtue of the hypotheses of Theorem 4.1, Lemmas 4.1 and 4.2. Also, we have
m+1\sum
n=2
\theta k - 1
n | In,2| k \leq
m+1\sum
n=2
\theta k - 1
n
\Biggl\{
n - 1\sum
v=1
| v + 1
v
| | \^an,v+1| | \Delta \lambda v| | tv|
\Biggr\} k
=
= O(1)
m+1\sum
n=2
\theta k - 1
n
\Biggl\{
n - 1\sum
v=1
\^an,v+1| \Delta \lambda v| | tv|
Xv
Xv
\Biggr\} k
=
= O(1)
m+1\sum
n=2
\theta k - 1
n
\Biggl\{
n - 1\sum
v=1
\^an,v+1| \Delta \lambda v| Xv
1
Xk
v
| tv| k
\Biggr\} \Biggl\{
n - 1\sum
v=1
\^an,v+1| \Delta \lambda v| Xv
\Biggr\} k - 1
=
= O(1)
m+1\sum
n=2
\theta k - 1
n ak - 1
nn
\Biggl\{
n - 1\sum
v=1
\^an,v+1| \Delta \lambda v| Xv
1
Xk
v
| tv| k
\Biggr\} \Biggl\{
m - 1\sum
v=1
| \Delta \lambda v| Xv
\Biggr\} k - 1
=
= O(1)
m\sum
v=1
v| \Delta \lambda v|
1
Xk - 1
v
1
v
| tv| k
m+1\sum
n=v+1
(\theta nann)
k - 1\^an,v+1 =
= O(1)
m\sum
v=1
(\theta vavv)
k - 1v| \Delta \lambda v|
1
Xk - 1
v
1
v
| tv| k
m+1\sum
n=v+1
\^an,v+1 =
= O(1)
m\sum
v=1
v(\theta vavv)
k - 1| \Delta \lambda v|
1
vXk - 1
v
| tv| k =
= O(1)
m - 1\sum
v=1
\Delta (v| \Delta \lambda v| )
v\sum
r=1
(\theta rarr)
k - 1 | tr| k
rXk - 1
r
+O(1)m| \Delta \lambda m|
m\sum
r=1
(\theta rarr)
k - 1 | tr| k
rXk - 1
r
=
= O(1)
m - 1\sum
v=1
| \Delta (v| \Delta \lambda v| )| Xv +O(1)m| \Delta \lambda m| Xm =
= O(1)
m - 1\sum
v=1
vXv| \Delta 2\lambda v| +O(1)
m - 1\sum
v=1
Xv| \Delta \lambda v| +O(1)m| \Delta \lambda m| Xm =
= O(1) as m \rightarrow \infty ,
by virtue of the hypotheses of Theorem 4.1, Lemmas 4.1 and 4.2.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
A MATRIX APPLICATION OF POWER INCREASING SEQUENCES TO INFINITE SERIES . . . 641
Furthermore, as in In,1, we get
m+1\sum
n=2
\theta k - 1
n | In,3| k \leq
m+1\sum
n=2
\theta k - 1
n
\Biggl\{
n - 1\sum
v=1
| \^an,v+1| | \lambda v+1|
| tv|
v
\Biggr\} k
=
= O(1)
m+1\sum
n=2
\theta k - 1
n
\Biggl\{
n - 1\sum
v=1
| \^an,v+1| | \lambda v+1| k
| tv| k
v
\Biggr\} \Biggl\{
n - 1\sum
v=1
1
v
\^an,v+1
\Biggr\} k - 1
=
= O(1)
m+1\sum
n=2
\theta k - 1
n ak - 1
nn
n - 1\sum
v=1
| \lambda v+1| | \lambda v+1| k - 1 | tv| k
v
\^an,v+1 =
= O(1)
m\sum
v=1
| tv| k
v
1
Xk - 1
v
| \lambda v+1|
m+1\sum
n=v+1
(\theta nann)
k - 1\^an,v+1 =
= O(1)
m\sum
v=1
(\theta vavv)
k - 1 | tv| k
v
1
Xk - 1
v
| \lambda v+1|
m+1\sum
n=v+1
\^an,v+1 =
= O(1)
m\sum
v=1
(\theta vavv)
k - 1 1
Xk - 1
v
| \lambda v+1|
| tv| k
v
=
= O(1) as m \rightarrow \infty ,
by virtue of the hypotheses of Theorem 4.1, Lemmas 4.1 and 4.2.
Again, as in In,1, we obtain
m\sum
n=1
\theta k - 1
n | In,4| k = O(1)
m\sum
n=1
\theta k - 1
n aknn| \lambda n| k| tn| k = O(1)
m\sum
n=1
\theta k - 1
n aknn| \lambda n| k - 1| \lambda n| | tn| k =
= O(1)
m\sum
n=1
\theta k - 1
n aknn
1
Xk - 1
n
| \lambda n| | tn| k = O(1) as m \rightarrow \infty ,
by virtue of hypotheses of the Theorem 4.1, Lemmas 4.1 and 4.2.
Theorem 4.1 is proved.
5. An application of absolute matrix summability to Fourier series. Let f be a periodic
function with period 2\pi and integrable (L) over ( - \pi , \pi ). Without any loss of generality the constant
term in the Fourier series of f can be taken to be zero, so that
f(x) \sim
\infty \sum
n=1
(an \mathrm{c}\mathrm{o}\mathrm{s}nx+ bn \mathrm{s}\mathrm{i}\mathrm{n}nx) =
\infty \sum
n=1
Cn(x),
where
a0 =
1
\pi
\pi \int
- \pi
f(x)dx, an =
1
\pi
\pi \int
- \pi
f(x) \mathrm{c}\mathrm{o}\mathrm{s}(nx)dx, bn =
1
\pi
\pi \int
- \pi
f(x) \mathrm{s}\mathrm{i}\mathrm{n}(nx)dx.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
642 Ş. YILDIZ
We write
\phi (t) =
1
2
\{ f(x+ t) + f(x - t)\} ,
\phi \alpha (t) =
\alpha
t\alpha
t\int
0
(t - u)\alpha - 1\phi (u) du, \alpha > 0.
It is well-known that if \phi 1(t) \in \scrB \scrV (0, \pi ), then tn(x) = O(1), where tn(x) is the (C, 1) mean of
the sequence (nCn(x)) (see [9]).
Using this fact, Bor has obtained the following main result dealing with the trigonometric Fourier
series.
Theorem 5.1 [7]. Let (Xn) be a quasi-\sigma -power increasing sequence. If \phi 1(t) \in \scrB \scrV (0, \pi )
and the sequences (pn), (\lambda n), and (Xn) satisfy the conditions of Theorem 2.1, then the series\sum
Cn(x)\lambda n is summable | \=N, pn| k, k \geq 1.
By using the above theorem, we have obtained the following result for | A, \theta n| k summability.
Theorem 5.2. Let A be a positive normal matrix satisfying the conditions of Theorem 4.1. Let
(Xn) be a quasi-\sigma -power increasing sequence. If \phi 1(t) \in \scrB \scrV (0, \pi ) and the sequences (pn), (\lambda n),
and (Xn) satisfy the conditions of Theorem 4.1, then the series
\sum
Cn(x)\lambda n is summable | A, \theta n| k,
k \geq 1.
6. Applications. We can apply Theorems 4.1 and 5.2 to the weighted mean in which A = (anv)
is defined as anv =
pv
Pn
when 0 \leq v \leq n, where Pn = p0 + p1 + . . .+ pn. We have
\=anv =
Pn - Pv - 1
Pn
and \^an,v+1 =
pnPv
PnPn - 1
.
So, the following results can be easily verified.
7. Conclusions.
1. If we take \theta n =
Pn
pn
in Theorems 4.1 and 5.2, then we have a result dealing with | A, pn| k
summability.
2. If we take \theta n = n in Theorems 4.1 and 5.2, then we have a result dealing with | A| k
summability.
3. If we take \theta n =
Pn
pn
and anv =
pv
Pn
in Theorems 4.1 and 5.2, then we have Theorems 2.1 and
5.1, respectively.
4. If we take \theta n = n, anv =
pv
Pn
and pn = 1 for all values of n in Theorems 4.1 and 5.2, then
we have a new result concerning | C, 1| k summability.
5. If we take \theta n = n and anv =
pv
Pn
in Theorems 4.1 and 5.2, then we have | R, pn| k summability.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
A MATRIX APPLICATION OF POWER INCREASING SEQUENCES TO INFINITE SERIES . . . 643
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Received 18.04.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
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| id | umjimathkievua-article-6016 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:15Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/cf/155796659b55ec3a4865ad765c651fcf.pdf |
| spelling | umjimathkievua-article-60162022-03-26T11:01:44Z A matrix application of power increasing sequences to infinite series and Fourier series A matrix application of power increasing sequences to infinite series and Fourier series A matrix application of power increasing sequences to infinite series and Fourier series Yıldız, Şebnem Yıldız, Şebnem Yıldız, Şebnem UDC 517.54 The aim of the paper is a generalization, under weaker conditions, of the main theorem on quasi-$\sigma$-power increasing sequences applied to $|A,\theta_{n}|_{k}$ summability factors of infinite series and Fourier series. We obtain some new and known results related to basic summability methods. Метою даної роботи є узагальнення основної теореми про застосування зростаючих квазі-$\sigma $-степеневих послідовностей до коефіцієнтів підсумовування $|A,\theta_{n}|_{k}$ нескінченних рядів і рядів Фур'є при слабших умовах.&nbsp;Отримано деякі нові та відомі результати, що відносяться до базових методів підсумовування. Institute of Mathematics, NAS of Ukraine 2020-04-29 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6016 10.37863/umzh.v72i5.6016 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 5 (2020); 635–643 Український математичний журнал; Том 72 № 5 (2020); 635–643 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6016/8692 |
| spellingShingle | Yıldız, Şebnem Yıldız, Şebnem Yıldız, Şebnem A matrix application of power increasing sequences to infinite series and Fourier series |
| title | A matrix application of power increasing sequences to infinite series and Fourier series |
| title_alt | A matrix application of power increasing sequences to infinite series and Fourier series A matrix application of power increasing sequences to infinite series and Fourier series |
| title_full | A matrix application of power increasing sequences to infinite series and Fourier series |
| title_fullStr | A matrix application of power increasing sequences to infinite series and Fourier series |
| title_full_unstemmed | A matrix application of power increasing sequences to infinite series and Fourier series |
| title_short | A matrix application of power increasing sequences to infinite series and Fourier series |
| title_sort | matrix application of power increasing sequences to infinite series and fourier series |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6016 |
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