On spliced sequences and the density of points with respect to a matrix constructed by using a weight function
UDC 517.5 Following the line of investigation in [Linear Algebra and Appl. -- 2015. -- {\bf 487}. -- P. 22--42], for $y\in\mathbb{R}$ and a sequence $x=(x_n)\in\ell^\infty$ we define а new notion of density $\delta_{g}$ with respect to a weight function $g$ of indices of the elements $x_n$ close t...
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| author | Bose, K. Das, P. Sengupta, S. Бозе, К. Дас, П. Сенгупта, С. |
| author_facet | Bose, K. Das, P. Sengupta, S. Бозе, К. Дас, П. Сенгупта, С. |
| author_sort | Bose, K. |
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| datestamp_date | 2019-12-05T08:57:49Z |
| description | UDC 517.5
Following the line of investigation in
[Linear Algebra and Appl. -- 2015. -- {\bf 487}. -- P. 22--42],
for $y\in\mathbb{R}$ and a sequence $x=(x_n)\in\ell^\infty$ we define
а new notion of density $\delta_{g}$ with respect to a weight function $g$ of indices of the elements $x_n$ close to $y,$ where $ g\colon \mathbb{N}\to[
{0,\infty })$ is such that $ g(n) \to \infty $ and
$ n / g(n) \nrightarrow 0.$
We present the relationships between the densities
$\delta_{g}$ of indices of $(x_n)$ and the variation of the Ces\`aro-limit of $(x_n).$
Our main result states that if the set of limit points of $(x_n)$
is countable and $\delta_g(y)$ exists for any $y\in\mathbb{R},$ then $ \lim\nolimits_{n\to\infty}
\dfrac{1}{g(n)}\displaystyle\sum\nolimits_{i=1}^{n} x_i = \sum\nolimits_{y\in\mathbb{R}}\delta_g(y)\cdot y ,$ which is an extended and much more general form of the ``natural density version of the Osikiewicz theorem''.
Note that in [Linear Algebra and Appl. -- 2015. -- {\bf 487}. -- P. 22--42],
the regularity of the matrix was used in the entire investigation, whereas in the present paper the investigation is actually performed with respect to a special type of matrix, which is not necessarily regular. |
| first_indexed | 2026-03-24T02:07:12Z |
| format | Article |
| fulltext |
UDC 517.5
K. Bose, P. Das, S. Sengupta (Jadavpur Univ., Kolkata, West Bengal, India)
ON SPLICED SEQUENCES AND THE DENSITY OF POINTS WITH RESPECT
TO A MATRIX CONSTRUCTED BY USING A WEIGHT FUNCTION*
ПРО СПЛЕТЕНI ПОСЛIДОВНОСТI ТА ГУСТИНУ ТОЧОК ВIДНОСНО
МАТРИЦI, ЩО СКОНСТРУЙОВАНА ЗА ДОПОМОГОЮ ВАГОВОЇ ФУНКЦIЇ
Following the line of investigation in [Linear Algebra and Appl. – 2015. – 487. – P. 22 – 42], for y \in \BbbR and a sequence
x = (xn) \in \ell \infty we define а new notion of density \delta g with respect to a weight function g of indices of the elements xn
close to y, where g : \BbbN \rightarrow [0,\infty ) is such that g(n) \rightarrow \infty and n/g (n) \nrightarrow 0. We present the relationships between the
densities \delta g of indices of (xn) and the variation of the Cesàro-limit of (xn). Our main result states that if the set of limit
points of (xn) is countable and \delta g(y) exists for any y \in \BbbR , then \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
1
g(n)
\sum n
i=1
xi =
\sum
y\in \BbbR
\delta g(y) \cdot y, which is
an extended and much more general form of the “natural density version of the Osikiewicz theorem”. Note that in [Linear
Algebra and Appl. – 2015. – 487. – P. 22 – 42], the regularity of the matrix was used in the entire investigation, whereas in
the present paper the investigation is actually performed with respect to a special type of matrix, which is not necessarily
regular.
У цьому викладi ми слiдуємо роботi [Linear Algebra and Appl. – 2015. – 487. – P. 22–42]. Так, для y \in \BbbR i послi-
довностi x = (xn) \in \ell \infty ми вводимо нове поняття густини \delta g вiдносно вагової функцiї g вiд iндексiв елементiв
xn, близьких до y, де функцiя g : \BbbN \rightarrow [0,\infty ) така, що g(n) \rightarrow \infty i n/g (n) \nrightarrow 0. Наведено спiввiдношення
мiж густинами \delta g iндексiв елементiв (xn) i варiацiями границi Чезаро для (xn). В основному результатi ствер-
джується, що у випадку, коли множина граничних значень для (xn) є злiченною, а \delta g(y) iснує для всiх y \in \BbbR ,
\mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
1
g(n)
\sum n
i=1
xi =
\sum
y\in \BbbR
\delta g(y) \cdot y, що є розширеною та набагато бiльш загальною формою „природної
густинної версiї теореми Осiкевича”. Вiдмiтимо, що в [Linear Algebra and Appl. – 2015. – 487. – P. 22 – 42] регу-
лярнiсть матрицi використовувалась протягом усього дослiдження. Водночас у нашiй роботi дослiдження насправдi
виконується для спецiального типу матрицi, що необов’язково є регулярною.
1. Introduction. For n,m \in \BbbN with n < m, let [n,m] denote the set \{ n, n+1, n+2, . . . ,m\} . Let
A \subset \BbbN . Define
d(A) = \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
| A \cap [1, n]|
n
and d(A) = \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
| A \cap [1, n]|
n
.
The numbers d(A) and d(A) are called the upper natural density and the lower natural density of
A, respectively. If d(A) = d(A), then this common value is called the natural density of A and we
denote it by d(A). Let \scrI d be the family of all subsets of \BbbN which have natural density 0. Then \scrI d is
a proper nontrivial admissible ideal of subsets of \BbbN . The notion of natural density was used by Fast
[8] and Scoenberg [23] to define the notion of statistical convergence.
In [4] a natural extension of the notions of natural density and statistical convergence were
introduced, by replacing n with a non linear term n\alpha , 0 < \alpha < 1, in the definition of asymptotic
density. The motivation came from the urge to investigation different kinds of densities and the
problem of comparing them with the natural density. Very recently in has been shown in [2] that one
can Further, extend the concept of natural density by considering natural density of weight g where
* The second author is thankful to SERB, DST, New Delhi for granting a research project No. SR/S4/MS:813/13 during
the tenure of which this work was done. The first author is thankful to UGC for granting Junior Research Fellowships
during the tenure of which this work was done.
c\bigcirc K. BOSE, P. DAS, S. SENGUPTA, 2019
1192 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9
ON SPLICED SEQUENCES AND THE DENSITY OF POINTS WITH RESPECT TO A MATRIX . . . 1193
g : \BbbN \rightarrow [0,\infty ) is a function with \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty g (n) = \infty and
n
g (n)
\nrightarrow 0. It has been observed in [2]
that one can construct uncountably many noncomparable P -ideals corresponding to different choices
of the weight function g, all different from the ideal \scrI d.
In another direction Osikiewicz had developed the ideas of finite and infinite splices in [20]. Let
E1, E2, E3, . . . , Ek, . . . be a partition of \BbbN into countable number of sequences. Let y1, y2, y3, . . .
. . . , yk, . . . be distinct real numbers. Let (xn) be such that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty ,n\in Ei
xn = yi.
Then (xn) is called an infinite-splice. (In the same way Osikiewicz defined a finite splice taking
finite number of sequences and finite number of distinct real numbers.) He proved the following.
Theorem 1 (Natural density (or Cesàro) version of Osikiewicz theorem [20]). Assume that (xn)
is a splice over a partition \{ Ei\} . Let yi = \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty ,n\in Ei xn. Assume that d(Ei) exists for each i and\sum
i
d(Ei) = 1.
Then
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
n
n\sum
k=1
xk =
\sum
i
yi \cdot d(Ei).
In fact Osikiewicz considered a more general case, namely matrix summability method and
A-density with the use of regular infinite matrices A the details of which are presented in the pre-
liminaries. Very recently in [3] a new approach was made to study the general version of Osikiewicz
theorem by defining the notion of the A-density of a point and an alternative version of the same
result was established. In fact it was shown that the assumptions of Osikiewicz theorem imply those
of the following theorem.
Theorem 2. Suppose that x = (xn) is a bounded sequence, \delta A(y) exists for every y \in \BbbR and\sum
y\in D
\delta A(y) = 1. Then
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
(Ax)n =
\sum
y\in D
\delta A(y) \cdot y.
Consequently, the Osikiewicz result follows from Theorem 2.
As a natural consequence in this paper we extend the ”natural density version of Osikiewicz
theorem” with the help of a weighted density function. But instead of considering the original
approach of Osikiewicz, we follow the more natural line of investigation of [3]. In order to do that
we define the notion of the density of a point with respect to a weight function and prove some of
its consequences. Note that the corresponding results do not follow from the results of [3] as the
redefined matrix with respect to a weight function is not necessarily a regular matrix. This shows
that results similar to [20] or [3] can be obtained for special kinds of nonregular matrices also. For
simplicity we do not use the matrix notation inside the body of the paper.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9
1194 K. BOSE, P. DAS, S. SENGUPTA
2. Preliminaries. We first present the necessary definitions and notations which will form the
background of this article.
If x = (xn) is a sequence and A = (an,k) is a summability matrix, then by Ax we denote the
sequence ((Ax)1, (Ax)2, (Ax)3, . . . ) where (Ax)n =
\sum \infty
k=1
an,kxk. The matrix A is called regular
if \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty xn = L implies \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty (Ax)n = L. The well-known Silverman – Toeplitz theorem
characterizes regular matrices in the following way. A matrix A is regular if and only if
(i) \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty an,k = 0,
(ii) \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
\sum \infty
k=1
an,k = 1,
(iii) \mathrm{s}\mathrm{u}\mathrm{p}n\in \BbbN
\sum \infty
k=1
| an,k| < \infty .
We say that a nonnegative matrix A = (aij) is nonregular if it fails to satisfy any of the three
conditions (i), (ii) and (iii) prescribed above.
For a nonnegative regular matrix A and E \subset \BbbN , following Freedman and Sember [12], the
A-density of E, denoted by \delta A(E), is defined as follows:
\delta A(E) = \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
\sum
k\in E
an,k = \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
\infty \sum
k=1
an,k1E(k) = \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
(A1E)n,
where 1E is a 0-1 sequence such that 1E(k) = 1 \Leftarrow \Rightarrow k \in E. If \delta A(E) = \delta A(E) then we say that
the A-density of E exists and it is denoted by \delta A(E). Clearly, if A is the Cesàro matrix, i.e.,
an,k =
\left\{
1
n
if n \geq k,
0 otherwise,
then \delta A coincides with the natural density.
Throughout by \ell \infty we denote the set of all bounded sequences of reals.
We first recall the original Osikiewicz theorem.
Theorem 3 (Osikiewicz [20]). Assume that A is a nonnegative regular summability matrix.
Assume that (xn) \in \ell \infty is a splice over a partition \{ Ei\} . Let yi = \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty ,n\in Ei xn. Assume
that \delta A(Ei) exists for each i and \sum
i
\delta A(Ei) = 1.
Then
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\infty \sum
k=1
an,kxk =
\sum
i
yi \cdot \delta A(Ei).
In [3] in a new approach, the authors had defined for a sequence (xn) the density \delta A(y) of
indices of those xn which are close to y which was not dealt with till then in the literature. This was
a more general approach than that of Osikiewicz.
Fix (xn) \in \ell \infty . For y \in \BbbR let
\delta A(y) = \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \rightarrow 0+
\delta A(\{ n : | xn - y| \leq \varepsilon \} )
and
\delta A(y) = \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \rightarrow 0+
\delta A(\{ n : | xn - y| \leq \varepsilon \} ).
If \delta A(y) = \delta A(y), then the common value is denoted by \delta A(y).
The main result of [3] was the following.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9
ON SPLICED SEQUENCES AND THE DENSITY OF POINTS WITH RESPECT TO A MATRIX . . . 1195
Theorem 4. Let x = (xn) \in \ell \infty . Suppose that the set of limit points of (xn) is countable and
\delta A(y) exists for any y \in \BbbR where A is a nonnegative regular matrix. Then
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
(Ax)n =
\sum
y\in \BbbR
\delta A(y) \cdot y.
Now recall that a nonempty family \scrI of subsets of \BbbN is an ideal in \BbbN if for A, B \subset \BbbN :
(i) A, B \in \scrI implies A \cup B \in \scrI ; (ii) A \in \scrI , B \subset A imply B \in \scrI . Further, if
\bigcup
A\in \scrI A = \BbbN ,
i.e., \{ n\} \in \scrI \forall n \in \BbbN , then \scrI is called admissible or free. An ideal \scrI is called a P -ideal if for any
sequence of sets (Dn) from \scrI , there is another sequence of sets (Cn) in \scrI such that Dn \vartriangle Cn is
finite for all n and
\bigcup
nCn \in \scrI . Equivalently, if for each sequence (An) of sets from \scrI there exists
A\infty \in \scrI such that An \setminus A\infty is finite for all n \in \BbbN , then \scrI becomes a P -ideal.
For a bounded sequence (xn), we now recall the following definitions (see [17]):
(i) (xn) is \scrI -convergent to y if for any \varepsilon > 0, \{ n : | xn - y| \geq \varepsilon \} \in \scrI .
(ii) A point y is called an \scrI -cluster point of (xn) if \{ n : | xn - y| \leq \varepsilon \} /\in \scrI for any \varepsilon > 0.
(iii) y is called an \scrI -limit point of (xn) if there is a set B \subset \BbbN , B /\in \scrI , such that \mathrm{l}\mathrm{i}\mathrm{m}n\in B xn = y.
We now start our main discussions. Let g : \BbbN \rightarrow [0,\infty ) be a function with \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty g (n) = \infty .
The upper density of weight g was defined in [2] by the formula
dg(A) = \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
| A \cap [1, n]|
g (n)
for A \subset \BbbN . The lower density of weight g, dg(A) is defined in a similar way. Then the family
\scrI g = \{ A \subset \BbbN : dg(A) = 0\}
forms an ideal. It has been observed in [2] that \BbbN \in \scrI g if and only if
n
g (n)
\rightarrow 0. Therefore, we
additionally assume that n/g (n) \nrightarrow 0 so that \BbbN /\in \scrI g and \scrI g becomes a proper admissible P -ideal
of \BbbN (see [2]). As a natural consequence we can consider the following definition.
Definition 1. A sequence (xn) of real numbers is said to converge dg -statistically to x if, for
any given \epsilon > 0, dg(A\epsilon ) = 0 where
A\epsilon = \{ n \in \BbbN : | xn - x| \geq \epsilon \} .
Further, one should observe that if we define A = (aij), i, j = 1, 2, . . . ,\infty , such that
aij =
\left\{
1
g(i)
if i \leq j,
0 otherwise,
where g : \BbbN \rightarrow (0,\infty ) is a weight function defined above then clearly A is not necessarily a regular
matrix (though for certain choices of g, for example, if g(n) = n + 1, the generated matrix would
be regular). In fact for appropriately chosen functions g the corresponding matrices may not satisfy
all the three conditions of a regular matrix. For example, if we take g(n) =
\surd
n, then for the
corresponding matrix
(i) \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty an,k = 0,
(ii) \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
\sum \infty
k=1
an,k = \infty .
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9
1196 K. BOSE, P. DAS, S. SENGUPTA
Our main result, Theorem 5, reveals that we can actually obtain Osikiewicz like theorems for matrices
which are not necessarily regular.
We now define the main concepts, namely, the notions of g-densities at a point where the upper
density of weight g is defined by
\delta g(y) = \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \rightarrow 0+
\delta g\{ n : | xn - y| \leq \varepsilon \}
and the lower density of weight g is defined by
\delta g(y) = \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \rightarrow 0+
\delta g\{ n : | xn - y| \leq \varepsilon \} .
If \delta g(y) = \delta g(y), then the common value is denoted by \delta g(y).
3. Main results. The main result which we are going to establish in this paper is the following.
Theorem 5. Let x = (xn) \in \ell \infty and the set of limit points of (xn) is countable. Let A be a
matrix generated by a weight function g for wich \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
n
g(n)
= M (say) finitely exists. Suppose
\delta g(y) exists for all y \in \BbbR . Then
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
(Ax)n =
\sum
y\in \BbbR
\delta g(y) \cdot y
or, equivalently,
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
g(n)
n\sum
i=1
xi =
\sum
y\in \BbbR
\delta g(y) \cdot y.
We start with the following observation.
Lemma 1. Let (xn) \in \ell \infty and \delta g(y) exists for all y \in \BbbR . Then D := \{ y \in \BbbR : \delta g(y) > 0\} is
countable and \sum
y\in D
\delta g(y) \leq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n
n
g(n)
.
Proof. If \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n
n
g(n)
= \infty , then there is nothing to prove. So let \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n
n
g(n)
= M < \infty
and let (rn) be a strictly decreasing sequence converging to M. For m \in \BbbN let
Dm :=
\biggl\{
y \in \BbbR
\bigm| \bigm| \bigm| \delta g(y) \geq 1
m
\biggr\}
.
Note that D1 \subset D2 \subset . . . \subset Dm \subset . . . and D =
\bigcup
mDm. Now if y1, y2, . . . , yl \in Dm be distinct,
let us choose \varepsilon = \mathrm{m}\mathrm{i}\mathrm{n}i \not =j
| yi - yj |
3
> 0. Consequently, the sets Ei = \{ n : xn \in (yi - \varepsilon , yi + \varepsilon )\} are
pairwise disjoint. Moreover,
\delta g(Ei) \geq \delta g(yi) \geq
1
m
\Rightarrow \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n
| Ei \cap [1, n]|
g(n)
\geq 1
m
.
Then, for any \tau > 0, there exists n1 \in \BbbN such that
| Ei \cap [1, n]|
g(n)
>
1
m
- \tau for all n \geq n1.
Again \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n
n
g(n)
< rn for every n \in \BbbN . So, for any fixed rp, we get n2 \in \BbbN such that
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9
ON SPLICED SEQUENCES AND THE DENSITY OF POINTS WITH RESPECT TO A MATRIX . . . 1197
n
g(n)
< M + \delta < rp \forall n \geq n2 and for a suitably chosen \delta . Let n0 = \mathrm{m}\mathrm{a}\mathrm{x}\{ n1, n2\} . As Ei’s are
disjoint, we have
| (
\bigcup l
i=1Ei) \cap [1, n]|
g(n)
=
l\sum
i=1
| Ei \cap [1, n]|
g(n)
\geq l
m
- l\tau \forall n \geq n0.
But \bigm| \bigm| \bigm| (\bigcup l
i=1Ei) \cap [1, n]
\bigm| \bigm| \bigm|
g(n)
\leq n
g(n)
< rp.
Now note that
l
m
- l\tau \leq rp evidently implies l \leq mrp
1 - \tau m
. Hence, choosing \tau so that 1 - \tau m > 0
we observe that l must be finite. Thus, Dm is finite for each m which implies that D =
\bigcup
mDm
can be at most countable.
Again
\sum
y\in Dm
\delta g(y) \leq
l\sum
i=1
\delta g(Ei) =
l\sum
i=1
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n
| Ei \cap [1, n]|
g(n)
\leq
\leq
l\sum
i=1
\biggl(
| Ei \cap [1, n]|
g(n)
+
\varepsilon 0
l
\biggr)
for all n \geq N (say)
where \varepsilon 0 is arbitrary. So
\sum
y\in Dm
\delta g(y) \leq
\bigm| \bigm| \bigm| (\bigcup l
i=1Ei) \cap [1, n]
\bigm| \bigm| \bigm|
g(n)
+ \varepsilon 0 \leq
n
g(n)
+ \varepsilon 0 \leq rp
for suitably chosen \varepsilon 0. Finally, in view of the fact that D =
\bigcup
mDm we get\sum
y\in D
\delta g(y) = \mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
\sum
y\in Dm
\delta g(y) \leq rp.
Since this is true for any rp, letting p \rightarrow \infty we get
\sum
y\in D
\delta g(y) \leq M.
Lemma 1 is proved.
Note that in general, one cannot prove that D =
\bigl\{
y \in \BbbR : \delta g(y) > 0
\bigr\}
is nonempty. Also the
above lemma would not remain true if one would change \delta g(y) to \delta g(y), that is D1 :=
\bigl\{
y \in \BbbR :
\delta g(y) > 0
\bigr\}
need not be countable. An example in this respect is given in [3] for g(n) = n.
The next result extends the natural density version of the Osikiewicz theorem. We will later show
that the condition
\sum
y\in D
\delta A(y) = M implies that the set of indices of (xn) can be divided into
appropriate splices. The method which we use in our proof is similar to that of Osikiewicz, but not
analogous as we use essentially new arguments.
Theorem 6. Let (xn) be a bounded sequence and g be a weight function such that \delta g(y) exists
for every y \in \BbbR and moreover
\sum
y\in D
\delta g(y) = M. Then
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
g(n)
n\sum
i=1
xi =
\sum
y\in D
\delta g(y) \cdot y.
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1198 K. BOSE, P. DAS, S. SENGUPTA
Proof. Since (xn) is bounded, there exists a K > 0 such that | xn| \leq K for every n \in \BbbN . Let
D = \{ yi : i = 1, 2, . . .\} where yi’s are distinct. Let \varepsilon > 0 be given and let r \in \BbbN be such that
r\sum
i=1
\delta g(yi) > M - \varepsilon and
\bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
i=r+1
yi \cdot \delta g(yi)
\bigm| \bigm| \bigm| \bigm| \bigm| < \varepsilon .
Let N \in \BbbN be such that
1
3
\mathrm{m}\mathrm{i}\mathrm{n}
i \not =j
| yi - yj | >
1
N
for all i, j \in 1, 2, . . . , r
and such that the sets Ei =
\Bigl\{
j : | xj - yi| <
1
N
\Bigr\}
have the following property:
\delta g(yi) -
\varepsilon
r(k + 1)
\leq \delta g(Ei) \leq \delta g(Ei) \leq \delta g(yi) +
\varepsilon
r(k + 1)
for i = 1, 2, . . . , r. Obviously E1, . . . , Er are pairwise disjoint. Now we can choose an m0 (\in \BbbN )
such that
\delta g(Ei) -
1
N
<
| Ei \cap [1, n]|
g(n)
< \delta g(Ei) +
1
N
for every n \geq m0 and i = 1, 2, . . . , r. Therefore,
\delta g(yi) -
1
N
- \varepsilon
r(k + 1)
<
| Ei \cap [1, n]|
g(n)
< \delta g(yi) +
1
N
+
\varepsilon
r(k + 1)
and, consequently, \bigm| \bigm| \bigm| \bigm| | Ei \cap [1, n]|
g(n)
- \delta g(yi)
\bigm| \bigm| \bigm| \bigm| < 1
N
+
\varepsilon
r(k + 1)
(1)
for every n \geq m0 and i = 1, 2, . . . , r. Then, for n \geq m0, we have
1
g(n)
n\sum
i=1
xi \leq
| E1 \cap [1, n]|
g(n)
\biggl(
y1 +
1
N
\biggr)
+
| E2 \cap [1, n]|
g(n)
\biggl(
y2 +
1
N
\biggr)
+ . . .
. . .+
| Er \cap [1, n]|
g(n)
\biggl(
yr +
1
N
\biggr)
+K
| (E1 \cup . . . \cup Er)
c \cap [1, n]|
g(n)
.
Now we can choose m1 > m0 such that, for all n \geq m1,
n
g(n)
< M + \varepsilon .
Then
M + \varepsilon >
n
g(n)
=
|
\bigcup r
i=1Ei \cap [1, n]|
g(n)
+
| (
\bigcup r
i=1Ei)
c \cap [1, n]|
g(n)
and, consequently,
|
\bigcup r
i=1Ei \cap [1, n]|
g(n)
=
r\sum
i=1
| Ei \cap [1, n]|
g(n)
>
r\sum
i=1
\delta g(yi) -
r
N
- \varepsilon
K + 1
.
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ON SPLICED SEQUENCES AND THE DENSITY OF POINTS WITH RESPECT TO A MATRIX . . . 1199
Therefore, for n \geq m1, we obtain
| (
\bigcup r
i=1Ei)
c \cap [1, n]|
g(n)
< (M + \varepsilon ) -
\biggl(
M - r
N
-
\biggl(
1 +
1
K + 1
\biggr)
\varepsilon
\biggr)
=
r
N
+
\biggl(
2 +
1
K + 1
\biggr)
\varepsilon .
Subsequently we get, for n \geq m1,
1
g(n)
n\sum
i=1
xi \leq
| E1 \cap [1, n]|
g(n)
\biggl(
y1 +
1
N
\biggr)
+
| E2 \cap [1, n]|
g(n)
\biggl(
y2 +
1
N
\biggr)
+ . . .
. . .+
| Er \cap [1, n]|
g(n)
\biggl(
yr +
1
N
\biggr)
+
Kr
n
+
\biggl(
2 +
1
K + 1
\biggr)
\varepsilon K.
Analogously,
1
g(n)
n\sum
i=1
xi \geq
| E1 \cap [1, n]|
g(n)
\biggl(
y1 -
1
N
\biggr)
- | E2 \cap [1, n]|
g(n)
\biggl(
y2 -
1
N
\biggr)
+ . . .
. . .+
| Er \cap [1, n]|
g(n)
\biggl(
yr -
1
N
\biggr)
- Kr
n
-
\biggl(
2 +
1
K + 1
\biggr)
\varepsilon K.
Thus,
1
g(n)
n\sum
i=1
xi -
r\sum
i=1
| Ei \cap [1, n]|
g(n)
\biggl(
yi +
1
N
\biggr)
\leq Kr
n
+
\biggl(
2 +
1
K + 1
\biggr)
\varepsilon K (2)
and
1
g(n)
n\sum
i=1
xi -
r\sum
i=1
| Ei \cap [1, n]|
g(n)
\biggl(
yi -
1
N
\biggr)
\geq - Kr
n
-
\biggl(
2 +
1
K + 1
\biggr)
\varepsilon K. (3)
Hence, by using (1) and (2), we have
1
g(n)
n\sum
i=1
xi -
\infty \sum
i=1
\delta g(yi) \cdot yi =
1
g(n)
n\sum
i=1
xi -
r\sum
i=1
\delta g(yi).yi -
\infty \sum
i=r+1
\delta g(yi) \cdot yi \leq
\leq 1
g(n)
n\sum
i=1
xi -
r\sum
i=1
\delta g(yi).yi +
\bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
i=r+1
\delta g(yi) \cdot yi
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 1
g(n)
n\sum
i=1
xi -
r\sum
i=1
\delta g(yi \cdot yi + \varepsilon =
=
\Biggl[
1
g(n)
n\sum
i=1
xi -
r\sum
i=1
| Ei \cap [1, n]|
g(n)
\biggl(
yi +
1
N
\biggr) \Biggr]
+
r\sum
i=1
\biggl[
| Ei \cap [1, n]|
g(n)
\biggl(
yi +
1
N
\biggr)
- \delta g(yi)\cdot yi
\biggr]
+\varepsilon \leq
\leq
r\sum
i=1
\biggl[ \biggl(
| Ei \cap [1, n]|
g(n)
- \delta g(yi)
\biggr) \biggl(
yi +
1
N
\biggr) \biggr]
+
1
N
r\sum
i+1
\delta g(yi) +
Kr
N
+
\biggl(
2K +
K
K + 1
+ 1
\biggr)
\varepsilon \leq
\leq
r\sum
i=1
\biggl[ \bigm| \bigm| \bigm| \bigm| \biggl( | Ei \cap [1, n]|
g(n)
- \delta g(yi)
\biggr) \bigm| \bigm| \bigm| \bigm| \biggl( | yi| + 1
N
\biggr) \biggr]
+
M
n
+
Kr
N
+
\biggl(
2K +
K
K + 1
+ 1
\biggr)
\varepsilon \leq
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9
1200 K. BOSE, P. DAS, S. SENGUPTA
\leq r
\biggl(
1
N
+
\varepsilon
r(K + 1)
\biggr) \biggl(
K + 1 +
1
N
\biggr)
+
M
n
+
Kr
N
+
\biggl(
2K +
K
K + 1
+ 1
\biggr)
\varepsilon .
Analogously, from (1) and (3), we get
1
g(n)
n\sum
i=1
xi -
\infty \sum
i=1
\delta g(yi) \cdot yi \geq
\geq - r
\biggl(
1
N
+
\varepsilon
r(K + 1)
\biggr) \biggl(
K + 1 +
1
N
\biggr)
- M
n
- Kr
N
-
\biggl(
2K +
K
K + 1
+ 1
\biggr)
\varepsilon .
Since N can be chosen arbitrarily large, we obtain\bigm| \bigm| \bigm| \bigm| \bigm| 1
g(n)
n\sum
i=1
xi -
\infty \sum
i=1
\delta g(yi) \cdot yi
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\biggl(
2K +
K
K + 1
+ 1
\biggr)
\varepsilon
for every \varepsilon > 0. Therefore, we can conclude that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
g(n)
n\sum
i=1
xi =
\infty \sum
i=1
\delta g(yi) \cdot yi.
Theorem 6 is proved.
Next we establish the following result.
Proposition 1. Let (xn) be a splice over a partition \{ Ei\} , yi = \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
n\in Ei
xn, \delta g(Ei) exists for
each i and
\sum \infty
i=1
\delta g(Ei) = M. Then \delta g(y) exists for every y \in \BbbR and \delta g(yi) = \delta g(Ei).
Proof. Let \varepsilon > 0 be given. We choose N \in \BbbN such that
\sum N
i=1
\delta g(Ei) > M - \varepsilon . Let \delta > 0 be
such that the intervals (yi - \delta , yi + \delta ) are pairwise disjoint for i = 1, 2, . . . , N. It is noted that the
sets of indices Ei \setminus \{ k : xk \in (yi - \delta , yi + \delta )\} are finite. So \delta g(yi) \geq \delta g(Ei). On the other hand, we
have
\delta g(yi) = \mathrm{l}\mathrm{i}\mathrm{m}
\eta \rightarrow 0+
\delta g \{ k : xk \in (yi - \eta , yi + \eta )\} =
= M - \mathrm{l}\mathrm{i}\mathrm{m}
\eta \rightarrow 0+
\delta g \{ k : xk /\in (yi - \eta , yi + \eta )\} \leq
\leq M - \delta g \{ k : xk /\in (yi - \delta , yi + \delta )\} \leq
\leq M -
N\sum
m=1
m \not =i
\delta g(Em) < \delta g(Ei) + \varepsilon .
Thus,
\delta g(yi) = \delta g(Ei).
Finally, let y be not in the set of limits \{ yi\} . As before for any \varepsilon > 0 we can find N such that\sum N
i=1
\delta g(Ei) > M - \varepsilon . Let \eta be the distance from y to the set \{ y1, . . . , yN\} . Then
\delta g
\biggl( \biggl\{
m : | xm - y| < \eta
2
\biggr\} \biggr)
< \varepsilon and, consequently, \delta g(y) = 0.
Proposition 1 is proved.
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ON SPLICED SEQUENCES AND THE DENSITY OF POINTS WITH RESPECT TO A MATRIX . . . 1201
Next we establish the following result which not only forms the basis of a necessary condition for
the existence of the limit \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
1
g(n)
\sum n
i=1
xi, but at the same time is an interesting observation.
Proposition 2. Suppose x = (xn) \in l\infty . If \delta g(y) = M, then My is a limit point of the
sequence \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
1
g(n)
\sum n
i=1
xi.
Proof. Since (xn) is bounded, there is K > 0 such that | xn| \leq K for all N \in \BbbN . Let y be
such that \delta g(y) = M. Also let N \in \BbbN and let EN =
\biggl\{
j \in \BbbN : | xj - y| < 1
N
\biggr\}
. Then there exists
kN \geq N such that
| EN \cap [1, kN ]|
g(kN )
> \delta g(EN ) - 1
N
= M - 1
N
.
Again as we have \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
n
g(n)
= M, we get
kN
g(kN )
< M +
1
N
.
Since y - 1
N
< xk < y +
1
N
for all xk \in EN and also - K \leq xk \leq K for each xk /\in EN , so we
have
| EN \cap [1, kN ]|
g(kN )
\biggl(
y - 1
N
\biggr)
-
| Ec
N \cap [1, kN ]|
g(kN )
K \leq 1
g(kN )
kN\sum
i=1
xi \leq
\leq | EN \cap [1, kN ]|
g(kN )
\biggl(
y +
1
N
\biggr)
+
| Ec
N \cap [1, kN ]|
g(kN )
K.
Thus,
y
\biggl(
kN
g(kN )
- M
\biggr)
- 1
N
| EN \cap [1, kN ]|
g(kN )
-
| Ec
N \cap [1, kN ]|
g(kN )
(K + y) \leq
\leq 1
g(kN )
kN\sum
i=1
xi - My \leq y
\biggl(
kN
g(kN )
- M
\biggr)
+
1
N
| EN \cap [1, kN ]|
g(kN )
+
| Ec
N \cap [1, kN ]|
g(kN )
(K - y)
and, consequently,\bigm| \bigm| \bigm| \bigm| \bigm| 1
g(kN )
kN\sum
i=1
xi - My
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\bigm| \bigm| \bigm| \bigm| | EN \cap [1, kN ]|
g(kN )
1
N
\bigm| \bigm| \bigm| \bigm| + \bigm| \bigm| \bigm| \bigm| | Ec
N \cap [1, kN ]|
g(kN )
(K + | y| )
\bigm| \bigm| \bigm| \bigm| + 1
N
| y| .
Since
| Ec
N \cap [1, kN ]|
g(kN )
=
kN
g(kN )
- | EN \cap [1, kN ]|
g(kN )
< M +
1
N
-
\biggl(
M - 1
N
\biggr)
=
2
N
,
we obtain \bigm| \bigm| \bigm| \bigm| \bigm| 1
g(kN )
kN\sum
i=1
xi - My
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\biggl(
M
N
+
1
N2
\biggr)
+
| Ec
N \cap [1, kN ]|
g(kN )
(K + | y| ) + 1
N
| y| \leq
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9
1202 K. BOSE, P. DAS, S. SENGUPTA
\leq
\biggl(
M
N
+
1
N2
\biggr)
+
2
N
(K + | y| ) + 1
N
| y| .
Therefore,
\mathrm{l}\mathrm{i}\mathrm{m}
N\rightarrow \infty
1
g(kN )
kN\sum
i=1
xi = My.
Proposition 2 is proved.
Corollary 1. Let (xn) be a bounded sequence. Suppose that there are y and z (y \not = z) with
\delta g(y) = \delta g(z) = M. Then the limit \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
1
g(n)
\sum n
i=1
xi does not exist.
One should note that Corollary 1 cannot be weakened by assuming \delta g(y), \delta g(z) > r for some
r \in (0,M). A counter example is given in Proposition 9 in [3] for g(n) = n.
Now we recall some important results from [3] which will be useful in the sequel.
Lemma 2 [3]. Let \scrI be an ideal of subsets of \BbbN . Assume that X := \{ n : xn \in [a, b]\} /\in \scrI .
Suppose that
\{ n : a \leq xn \leq t - \varepsilon \} \in \scrI or \{ n : t+ \varepsilon \leq xn \leq b\} \in \scrI
for any t \in (a, b) and any \varepsilon > 0 such that \varepsilon < \mathrm{m}\mathrm{i}\mathrm{n}\{ t - a, b - t\} . Then there is y \in [a, b] such that \{ n :
| xn - y| \geq \varepsilon \} \in \scrI for every \varepsilon > 0.
Proposition 3 [3]. Let \scrI be a P -ideal. Assume that (xn) \in \ell \infty does not have any \scrI -limit
points. Then the set of limit points of (xn), i.e., the set
\{ y \in \BbbR : xnk
\rightarrow y for some increasing sequence (nk) of natural numbers\} ,
is uncountable and closed.
Corollary 2 [3]. Let [a, b] be a fixed interval and \scrI be a P -ideal. Assume that \{ n : xn \in
\in [a, b]\} /\in \scrI and any point y \in (a, b) is not an \scrI -limit point of (xn). Then the set of limit points of
(xn) in [a, b], i.e., the set
\{ y \in (a, b) : xnk
\rightarrow y for some increasing sequence (nk) of natural numbers\} ,
is uncountable and closed.
Corollary 3 [3]. Let (xn) \in \ell \infty . Assume that the set of limit points of (xn) is countable. Then
the sequence (xn) has at least one \scrI -limit point for every P -ideal \scrI .
Now we prove certain results analogous to the results of [3] which will help us to reach our final
goal.
Lemma 3. Let r \in (0, 1), r1 \geq r2 \geq . . . , \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty rn = r and (En) be a decreasing sequence
of subsets of \BbbN .
(i) If \delta g(En) = rn, n \in \BbbN , then there is a subset E of \BbbN with \delta g(E) = r and such that En \setminus E
is finite for all n. Moreover, if \delta g(En) \rightarrow r, then \delta g(E) = r.
(ii) If \delta g(En) = rn, n \in \BbbN , then there is a subset E of \BbbN with \delta g(E) = r and such that En \setminus E
is finite for all n.
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ON SPLICED SEQUENCES AND THE DENSITY OF POINTS WITH RESPECT TO A MATRIX . . . 1203
Proof. (i) Let (pn) be an increasing sequence of natural numbers such that
| En \cap [1, j]|
g(j)
>
> rn - 1
3n
for every j \geq pn. For each n \in \BbbN now choose mn > pn such that\bigm| \bigm| (En \cap [1,mn]) \cap [1, j]
\bigm| \bigm|
g(j)
> rn - 1
3n
- 1
3n
> rn - 1
n
for all j, pn \leq j \leq pn+1. Thus, we have two increasing sequences of natural numbers (pn) and
(mn) such that, for all j \in [pn, pn+1],\bigm| \bigm| (En \cap [1,mn]) \cap [1, j]
\bigm| \bigm|
g(j)
> rn - 1
n
.
Put E =
\bigcup \infty
n=1En \cap [1,mn+1]. Take pn \leq j < pn+1. Then
| E \cap [1, j]|
g(j)
\geq
\bigm| \bigm| (En \cap [1,mn]) \cap [1, j]
\bigm| \bigm|
g(j)
> rn - 1
n
.
Thus, \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}n\rightarrow \infty
| E \cap [1, n]|
g(n)
\geq r, which means that \delta g(E) \geq r. Since E1 \supset E2 \supset . . . , so\bigcup \infty
n=1En \cap [1,mn+1] \subset Ej and Ej \setminus E \subset
\bigcup j - 1
n=1En \cap [1,mn+1]. Therefore, Ej \setminus E is finite, and,
consequently, \delta g(E) \leq \delta g(Ej) and \delta g(E) \leq \delta g(Ej). Hence, \delta g(E) = r and if \delta g(En) \rightarrow r, then
\delta g(E) = r.
(ii) As before we can choose two increasing sequences of natural numbers (pn) and (mn) such
that \bigm| \bigm| (En \cap [1,mn]) \cap [1, pn]
\bigm| \bigm|
g(pn)
> rn - 1
n
for every n. Put E =
\bigcup \infty
n=1En \cap [1,mn+1]. Then
| E \cap [1, pn+1]|
g(pn+1)
\geq
\bigm| \bigm| (En \cap [1,mn+1]) \cap [1, pn+1]
\bigm| \bigm|
g(pn+1)
\geq rn+1 -
1
n+ 1
.
Thus, \delta g(E) \geq r. Since En \setminus E is finite for all n, it now readily follows that \delta g(E) = r.
Lemma 3 is proved.
Theorem 7. As before let \scrI g = \{ A \subset \BbbN : \delta g(A) = 0\} . Let (xn) \in \ell \infty . A point y \in \BbbR is an
\scrI g -limit point of (xn) if and only if \delta g(y) > 0. Moreover, if \delta g(y) > 0, then there is E \subset \BbbN with
\delta g(E) = \delta g(y) and \mathrm{l}\mathrm{i}\mathrm{m}n\in E xn = y.
Proof. Assume that \delta g(y) = 0 and suppose y is an \scrI g -limit point of (xn). Then there is
E \subset \BbbN such that \delta g(E) > 0 and \mathrm{l}\mathrm{i}\mathrm{m}n\in E xn = y. Note that
\bigl\{
j : | xj - y| \leq \varepsilon
\bigr\}
\setminus E is finite for all
\varepsilon > 0. Hence, \delta g(E) \leq \delta g
\bigl( \bigl\{
j : | xj - y| \leq \varepsilon
\bigr\} \bigr)
for every \varepsilon > 0. Therefore, \delta g(E) = 0 which is a
contradiction.
Conversely, let \delta g(y) > 0. Let En =
\biggl\{
j : | xj - y| \leq 1
n
\biggr\}
. Then (En) is a decreasing sequence
with \delta g(En) \rightarrow \delta g(y). By Lemma 3 there is E such that En \setminus E is finite for all n with \delta g(E) =
= \delta g(y). Since almost all elements of E are contained in En, clearly \mathrm{l}\mathrm{i}\mathrm{m}j\rightarrow \infty
j\in E
xj = y. Hence, y is
an \scrI g -limit point of (xn).
The last part of the assertion follows in a similar way.
Theorem 7 is proved.
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1204 K. BOSE, P. DAS, S. SENGUPTA
Corollary 4. Let (xn) \in \ell \infty . A point y \in \BbbR is an \scrI g -cluster point of (xn) and it is not an
\scrI g -limit point if and only if :
(i) \delta g
\biggl( \biggl\{
j : | xj - y| \leq 1
n
\biggr\} \biggr)
> 0 for every n,
(ii) \delta g(y) = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\delta g
\biggl( \biggl\{
j : | xj - y| \leq 1
n
\biggr\} \biggr)
= 0.
Proposition 4. Let (xn) be a bounded sequence. Assume that y1, y2, . . . are the only distinct
real numbers such that \delta g(yi) > 0 for all i. Then there exists a partition E1, E2, . . . such that
\delta g(Ei) = \delta g(yi) for all i and \mathrm{l}\mathrm{i}\mathrm{m}n\in Ei xn = yi.
Proof. By Theorem 7 there are E\prime
1, E
\prime
2, . . . with \mathrm{l}\mathrm{i}\mathrm{m}n\in E\prime
i
xn = yi. Note that E\prime
i \cap E\prime
j is finite
if i \not = j. Define E1, E2, . . . in the following way. Let E\prime \prime
1 = E\prime
1, E
\prime \prime
m = E\prime
m \setminus
\bigcup m - 1
i=1 E\prime
i for m \geq 2.
Since E\prime
m \cap
\bigcup m - 1
i=1 E\prime
i is finite, so \delta g(E
\prime \prime
m) = \delta g(E
\prime
m) = \delta g(ym) for m \in \BbbN . Let E = \BbbN \setminus
\bigcup \infty
m=1E
\prime
m.
If E is finite, then put E1 = E \cup E\prime \prime
1 and Em = E \cup E\prime \prime
m for m \geq 2. If the set E is infinite, then
enumerate it as \{ n1, n2, . . .\} and put Em = E\prime \prime
m \cup \{ nm\} . Clearly \mathrm{l}\mathrm{i}\mathrm{m}n\in Em xn = ym.
Proposition 4 is proved.
Proposition 5. Let \{ En : n = 1, 2, . . .\} be a partition on \BbbN such that
\sum \infty
n=1
\delta g(En) <
< \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n
n
g(n)
= M (say). Then there is a partition \{ Fn : n = 0, 1, 2, . . .\} of \BbbN such that
(i) Fn \subset En,
(ii) \delta g(Fn) = \delta g(En) for all n,
(iii) \delta g(F0) = M -
\sum \infty
n=1
\delta g(En).
Proof. Let (\varepsilon n) be a strictly decreasing sequence of positive real numbers converging to 0. We
have \delta g(E1) = \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n
\biggl[
| E1 \cap [1, n]|
g(n)
\biggr]
. Furthermore, \delta g(Ec
1) = M - \delta g(E1). So
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n
| Ec
1 \cap [1, n]|
g(n)
= M - \delta g(E1) \Rightarrow
| Ec
1 \cap [1, n]|
g(n)
\geq M - \delta g(E1) + \varepsilon 1
for all n \geq N1(say). Since
1
g(n)
\rightarrow 0 as n \rightarrow \infty , we can choose N2 \in \BbbN large enough such that
| Ec
1 \cap \{ 1\} |
g(n)
< 2\varepsilon 1 for all n \geq N2. Let m1 = \mathrm{m}\mathrm{a}\mathrm{x}\{ N1, N2\} , and we set m0 = 1. Then
| Ec
1 \cap [1, n]|
g(n)
- | Ec
1 \cap \{ 1\} |
g(n)
\geq M - \delta g(E1) - \varepsilon 1
for each n \geq m1 and
| [m0, j] \setminus E1|
g(j)
\geq M - \delta g(E1) - \varepsilon 1
for all j \geq m1. Similarly, we have
\delta g
\bigl[
(E1 \cup E2)
c
\bigr]
= M - \delta g(E1 \cup E2) = M - \delta g(E1) - \delta g(E2),
i.e.,
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n
\Biggl[ \bigm| \bigm| (E1 \cup E2)
c \cap [1, n]
\bigm| \bigm|
g(n)
\Biggr]
= M - \delta g(E1) - \delta g(E2) \Rightarrow
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9
ON SPLICED SEQUENCES AND THE DENSITY OF POINTS WITH RESPECT TO A MATRIX . . . 1205
\Rightarrow
\bigm| \bigm| (E1 \cup E2)
c \cap [1, n]
\bigm| \bigm|
g(n)
\geq M - (\delta g(E1) + \delta g(E2)) + \varepsilon 2
for all n \geq K1(say). Now we choose a large K2 > N2 such that\bigm| \bigm| (E1 \cup E2)
c \cap [1,m1]
\bigm| \bigm|
g(n)
< 2\varepsilon 2
for all n \geq K2. Let m2 = \mathrm{m}\mathrm{a}\mathrm{x}\{ K1,K2\} . Then\bigm| \bigm| (E1 \cup E2)
c \cap [1, n]
\bigm| \bigm|
g(n)
-
\bigm| \bigm| (E1 \cup E2)
c \cap [1,m1]
\bigm| \bigm|
g(n)
\geq M - (\delta g(E1) + \delta g(E2)) - \varepsilon 2
whenever n \geq m2 and this implies that\bigm| \bigm| [m1, j] \setminus (E1 \cup E2)
\bigm| \bigm|
g(j)
\geq M - (\delta g(E1) + \delta g(E2)) - \varepsilon 2
for all j \geq m2.
Inductively we can define an increasing sequence (mn : n = 0, 1, 2, . . .) of natural numbers such
that, for all n = 1, 2, . . . ,\bigm| \bigm| [mn - 1, j] \setminus (E1 \cup E2 \cup . . . \cup En)
\bigm| \bigm|
g(j)
\geq M -
n\sum
i=1
\delta g(Ei) - \varepsilon n
whenever j \geq mn. Now let F0 =
\bigcup \infty
n=1([mn - 1,mn+1] \setminus
\bigcup n
i=1Ei). Then, for mn \leq j \leq mn+1, we
have
| F0 \cap [1, j]|
g(j)
\geq
\bigm| \bigm| [mn - 1, j] \setminus (E1 \cup E2 \cup . . . \cup En)
\bigm| \bigm|
g(j)
\geq M -
n\sum
i=1
\delta g(Ei) - \varepsilon n
for every n. So
\delta g(F0) = \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
j
| (F0 \cap [1, j]|
g(j)
\geq M -
\infty \sum
i=1
\delta g(Ei) (4)
as \varepsilon n \rightarrow 0.
Now let Fn = En \setminus F0 , n = 1, 2, . . . . Then F0 \cap En is finite for all n (note that F0 \cap E1 =
= \varphi , F0 \cap E2 \subset [1,m1], F0 \cap E3 \subset [1,m2], . . .). So \delta g(Fn) = \delta g(En) for all n and we obtain\bigcup \infty
n=1
Fn =
\bigcup \infty
n=1
(En \setminus F0) =
\bigcup \infty
n=1
En \setminus F0 = \BbbN \setminus F0 \Rightarrow F0 = \BbbN \setminus
\bigcup \infty
n=1
Fn.
Hence,
\delta g(F0) = M - \delta g(F
c
0 ) = M - \delta g
\Bigl( \bigcup \infty
n=1
Fn
\Bigr)
\leq M -
\infty \sum
n=1
\delta g(Fn) = M -
\infty \sum
n=1
\delta g(En). (5)
Combining (4) and (5), we finally observe that
\delta g(F0) = M -
\infty \sum
n=1
\delta g(En).
Proposition 5 is proved.
Finally, we prove a sufficient condition for a bounded sequence (xn) to have the property that\sum
y\in \BbbR
\delta g(y) = M.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9
1206 K. BOSE, P. DAS, S. SENGUPTA
Theorem 8. Let (xn) be a bounded sequence. Suppose that the set of limit points of (xn) is
countable and \delta g(y) exists for all y \in \BbbR . Then
\sum
y\in \BbbR
\delta g(y) = M.
Proof. Let D :=
\bigl\{
y \in \BbbR : \delta g(y) > 0
\bigr\}
. If possible let
\sum
y\in D
\delta g(y) < M. As the number of
limit points is countable, we are in a position to use Corollary 16 [3].
Let y be an \scrI g -limit point of (xn). Then there exists B \subset \BbbN , \delta g(B) > 0 such that \mathrm{l}\mathrm{i}\mathrm{m}n\in B xn =
= y. So for any \varepsilon > 0,
\bigl\{
n : | xn - y| \leq \varepsilon
\bigr\}
\supseteq B \setminus B0 where B0 \subset \BbbN is finite. Observe that
\delta g\{ n : | xn - y| \leq \varepsilon \} \geq \delta g(B) \Rightarrow \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \rightarrow 0+
\bigl[
\delta g\{ n : | xn - y| \leq \varepsilon \}
\bigr]
\geq \delta g(B) \Rightarrow \delta g(y) \geq \delta g(B) > 0.
So D \not = \varnothing . Now from Lemma 1 it follows that that D is countable. We enumerate D as \{ y1, y2, . . .\} .
By Proposition 4 there is a partition \{ E1, E2, . . .\} of \BbbN such that \delta g(Ek) = \delta g(yk) and \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
n\in Ek
=
= yk. Again applying Proposition 5 we get that there is a partition \{ F0, F1, F2, . . .\} of \BbbN such that
Fk \subset Ek, \delta g(Fk) = \delta g(Ek) for k = 1, 2, . . . ; F0 = \BbbN \setminus
\bigcup \infty
i=1 Fi so that \delta g(F0) = M -
\sum \infty
k=1
\delta g(Fk).
Obviously, \delta g(F0) > 0.
Now we consider the sequence (xn)n\in F0 and the ideal \scrI g| F0
= \{ E \subset F0 : E \in \scrI g\} . Since
\delta g(y) = 0 for all y /\in D, so by Theorem 7 y cannot be an \scrI g -limit of (xn)n\in F0 . Consequently, y
cannot be an \scrI g| F0
-limit point of (xn)n\in F0 . Now if any yi is an \scrI g| F0
-limit point of (xn)n\in F0 , then
there would be a set B \subset \BbbN , B \subset F0 such that B /\in \scrI g| F0
and \mathrm{l}\mathrm{i}\mathrm{m}n\in B xn = yi. Now B \subset F0
and B /\in \scrI g| F0
implies B /\in \scrI g. Again B \subset F0 implies B \cap Fi = \varnothing for all i = 1, 2, . . . . So
\mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty xn \in B \cup Fi = yi for all i. Consequently,
\delta g(yi) = \delta g(yi) = \mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \rightarrow 0+
\bigl[
\delta g\{ n : | xn - yi| \leq \varepsilon \}
\bigr]
\geq \delta g(B \cup Fi) > \delta g(Fi) = \delta g(yi)
which is a contradiction. So no yi is an \scrI g| F0
-limit point of (xn)n\in F0 , i.e.,(xn)n\in F0 has no
\scrI g| F0
-limit point.
Now to verify that the ideal \scrI g| F0
is a P -ideal let A1, A2, . . . \in \scrI g| F0
. Then A1, A2, . . . \in \scrI g. As
\scrI g is a P -ideal, we can get A\infty \in \scrI g such that An\setminus A\infty is finite for all n \in \BbbN . Now A\infty \cap F0 \in \scrI g| F0
and An \subset F0 for all n implies An \setminus (A\infty \cap F0) is finite for all n. So \scrI g| F0
is a P -ideal such that
(xn)n\in F0 has no \scrI g| F0
-limit point. Hence, the set of limit points of (xn)n\in F0 must be uncountable
(see Proposition 14 [3]), i.e., (xn) will have uncountably many limit points which contradicts the
assumption of the statement. Hence, it follows that
\sum
y\in \BbbR
\delta g(y) = M.
Theorem 8 is proved.
Finally, combining Theorem 6 with Theorem 8, we get the desired proof of our main result.
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Received 13.09.16,
after revision — 06.06.19
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 9
|
| id | umjimathkievua-article-1509 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:07:12Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/5e/1c5e56c61b8b2527eb975f2ea832335e.pdf |
| spelling | umjimathkievua-article-15092019-12-05T08:57:49Z On spliced sequences and the density of points with respect to a matrix constructed by using a weight function Про сплетенi послiдовностi та густину точок вiдносно матрицi, що сконструйована за допомогою вагової функцiї Bose, K. Das, P. Sengupta, S. Бозе, К. Дас, П. Сенгупта, С. UDC 517.5 Following the line of investigation in [Linear Algebra and Appl. -- 2015. -- {\bf 487}. -- P. 22--42], for $y\in\mathbb{R}$ and a sequence $x=(x_n)\in\ell^\infty$ we define а new notion of density $\delta_{g}$ with respect to a weight function $g$ of indices of the elements $x_n$ close to $y,$ where $ g\colon \mathbb{N}\to[ {0,\infty })$ is such that $ g(n) \to \infty $ and $ n / g(n) \nrightarrow 0.$ We present the relationships between the densities $\delta_{g}$ of indices of $(x_n)$ and the variation of the Ces\`aro-limit of $(x_n).$ Our main result states that if the set of limit points of $(x_n)$ is countable and $\delta_g(y)$ exists for any $y\in\mathbb{R},$ then $ \lim\nolimits_{n\to\infty} \dfrac{1}{g(n)}\displaystyle\sum\nolimits_{i=1}^{n} x_i = \sum\nolimits_{y\in\mathbb{R}}\delta_g(y)\cdot y ,$ which is an extended and much more general form of the ``natural density version of the Osikiewicz theorem''. Note that in [Linear Algebra and Appl. -- 2015. -- {\bf 487}. -- P. 22--42], the regularity of the matrix was used in the entire investigation, whereas in the present paper the investigation is actually performed with respect to a special type of matrix, which is not necessarily regular. УДК 517.5 У цьому викладі ми слідуємо роботі [Linear Algebra and Appl. -- 2015. -- {\bf{ 487}}. -- P. 22--42]. Так, для $y \in \mathbb{R}$ і послідовності $x = (x_n) \in \ell^{\infty}$ ми вводимо нове поняття густини $\delta_{g}$ відносно вагової функції $g$ від індексів елементів $x_n,$ близьких до $y,$ де функція $ n / g(n) \nrightarrow 0.$ Наведено співвідношення між густинами $\delta_{g}$ індексів елементів $(x_n)$ i варіаціями границі Чезаро для $(x_n).$ В основному результаті стверджується, що у випадку, коли множина граничних значень для $(x_n)$ є зліченною, а $\delta_g(y)$ існує для всіх $y\in\mathbb{R},$ $ \lim\nolimits_{n\to\infty} \dfrac{1}{g(n)}\displaystyle\sum\nolimits_{i=1}^{n} x_i = \sum\nolimits_{y\in\mathbb{R}}\delta_g(y)\cdot y ,$ що є розширеною та набагато більш загальною формою „природної густинної версії теореми Осікевича”. Відмітимо, що в [Linear Algebra and Appl. -- 2015. -- {\bf 487}. -- P. 22--42] регулярність матриці використовувалась протягом усього дослідження. Водночас у нашій роботі дослідження насправді виконується для спеціального типу матриці, що необов'язково є регулярною. Institute of Mathematics, NAS of Ukraine 2019-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1509 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 9 (2019); 1192-1207 Український математичний журнал; Том 71 № 9 (2019); 1192-1207 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1509/493 Copyright (c) 2019 Bose K.; Das P.; Sengupta S. |
| spellingShingle | Bose, K. Das, P. Sengupta, S. Бозе, К. Дас, П. Сенгупта, С. On spliced sequences and the density of points with respect to a matrix constructed by using a weight function |
| title | On spliced sequences and the density of points with respect
to a matrix constructed by using a weight function |
| title_alt | Про сплетенi послiдовностi та густину точок вiдносно
матрицi, що сконструйована за допомогою вагової функцiї |
| title_full | On spliced sequences and the density of points with respect
to a matrix constructed by using a weight function |
| title_fullStr | On spliced sequences and the density of points with respect
to a matrix constructed by using a weight function |
| title_full_unstemmed | On spliced sequences and the density of points with respect
to a matrix constructed by using a weight function |
| title_short | On spliced sequences and the density of points with respect
to a matrix constructed by using a weight function |
| title_sort | on spliced sequences and the density of points with respect
to a matrix constructed by using a weight function |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1509 |
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