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On statistical \(\mathfrak{A}\)-Cauchy and statistical \(\mathfrak{A}\)-summability via ideal

Abstract

The notion of statistical convergence was extended to \(\mathfrak{I}\)-convergence by (Kostyrko et al. in Real Anal. Exch. 26(2):669–686, 2000). In this paper we use such technique and introduce the notion of statistically \(\mathfrak{A}^{\mathfrak{I}}\)-Cauchy and statistically \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-Cauchy summability via the notion of ideal. We obtain some relations between them and prove that under certain conditions statistical \(\mathfrak{A}^{\mathfrak{I}}\)-Cauchy and statistical \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-Cauchy summability are equivalent. Moreover, we give some Tauberian theorems for statistical \(\mathfrak{A}^{\mathfrak{I}}\)-summability.

Introduction and preliminaries

Fast [10], introduced the notion of statistical convergence, which is an extension of convergence. A sequence \(\eta =(\eta _{k})\) in \(\mathbb{R} \) is statistically convergent to the number \(\mathfrak{s}\) if the set \(K(\epsilon )=\{k\leq n:|\eta _{k}-\mathfrak{s}|\geq \epsilon , \forall \epsilon >0\}\) has natural density 0; \(\delta (K(\epsilon ))=\lim_{n} \frac{ \vert K(\epsilon ) \vert }{n}=0\), where \(\vert \cdot \vert \) indicates the number of elements in the set. We write st-\(\lim \eta =\mathfrak{s}\). More generalization and application on this work can be found in ([1, 5, 8, 12, 14, 16, 23, 27]). One of such generalizations is the ideal (or \(\mathfrak{I}\))-convergence [18] which generalizes the usual convergence as well as the statistical convergence.

A non-empty class \(\mathfrak{I}\ (\mathcal{F}, resp.)\subseteq \mathfrak{P}(\mathfrak{X})\) of subsets of \(\mathfrak{X}\neq\varnothing\) is called ideal (filter, resp.) if

(i) \(\emptyset \in \mathfrak{I}\) (\(\emptyset \notin \mathcal{F}, \text{resp.}\)), (ii) \((\mathcal{D}_{1}\cup \mathcal{D}_{2}\text{ for }\mathcal{D}_{1},\mathcal{D}_{2}\in \mathfrak{I})\ (\mathcal{D}_{1}\cap \mathcal{D}_{2}\text{ for } \mathcal{D}_{1},\mathcal{D}_{2}\in \mathcal{F},resp.)\in \mathfrak{I}\) (\(\in \mathcal{F}, resp.\)), (iii) \(\mathcal{D}_{1}\in \mathfrak{I}\), \(\mathcal{D}_{2}\subseteq \mathcal{D}_{1}\ (\mathcal{D}_{1}\in \mathcal{F},\mathcal{D}_{2}\supseteq \mathcal{D}_{1},resp.)\Longrightarrow \mathcal{D}_{2}\in \mathfrak{I}\) (\(\mathcal{D}_{2}\in \mathcal{F},resp.\)). An ideal \(\mathfrak{I}\) is called non-trivial if \(\mathfrak{I}\neq \varnothing \), \(\mathfrak{X}\notin \mathfrak{I}\), and is called admissible if \(\{ \mathfrak{a} \} \in \mathfrak{I}\), for each \(\mathfrak{a}\in \mathfrak{X}\).

Let \(\mathfrak{I}\) be a non-trivial ideal in \(\mathfrak{X}\), the filter \(\mathcal{F}_{\mathfrak{I}}= \{ M=\mathfrak{X}\setminus \mathcal{A}:\mathcal{A}\in \mathfrak{I} \} \) is called the filter associated with the ideal \(\mathfrak{I}\). Recall that a real sequence \(\eta =(\eta _{k})\) is said to be \(\mathfrak{I}\)-convergent to \(\mathfrak{s}\in \mathbb{R} \) if \(\{ k: \vert \eta _{k}-\mathfrak{s} \vert \geq \epsilon,\text{ for every }\epsilon >0 \} \in \mathfrak{I}\), and we write \(\mathfrak{I}\)-\(\lim_{k}\eta _{k}=\mathfrak{s}\), [18]. More generalization and recent work can be found in ([3, 15, 17, 21, 22, 24, 25, 28, 29]).

Let \(\mathfrak{A}= ( \mathfrak{a}_{nk} ) \) be an infinite matrix and \(\eta =(\eta _{k})\) be a number sequence. By \(\mathfrak{A}\eta = ( \mathfrak{A}_{n} ( \eta ) ) \), we denote the \(\mathfrak{A}\)-transform of the sequence \(\eta = ( \eta _{k} ) \), where \(\mathfrak{A}_{n} ( \eta ) =\sum_{k=1}^{\infty }\mathfrak{a}_{nk} \eta _{k}\). A matrix \(\mathfrak{A}\) is regular if \(\mathfrak{A}\)-transforms c into c and \(\lim_{n}\mathfrak{A}_{n}(\eta )=\lim_{k}\eta _{k}\) for all \(\eta \in c\); the space of all convergent sequences. Let Ω denote the class of all nonnegative regular matrices. In [29], Savas et al. introduced the following definition. Let \(\mathfrak{A}= ( \mathfrak{a}_{nk} ) \in \Omega \). A real sequence \(\eta =(\eta _{k})\) is \(\mathfrak{A}^{\mathfrak{I}}\)-summable to \(\mathfrak{s}\in \mathbb{R} \) if the sequence \((\mathfrak{A}_{n}(\eta ))\) is \(\mathfrak{I}\)-convergent to \(\mathfrak{s}\), which we write \(\mathfrak{A}^{\mathfrak{I}}\)-\(\lim_{k}\eta _{k}=\mathfrak{s}\). Notice that, if \(\mathfrak{I}=\mathfrak{I}_{\delta }= \{ E\subseteq \mathbb{N}:\delta (E)=0 \} \), then \(\mathfrak{A}^{\mathfrak{I}}\)-summability becomes statistical \(\mathfrak{A}\)-summability due to [9].

Recently, Edely [6] introduced the notion of \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-summability and gave some relations with \(\mathfrak{A}^{\mathfrak{I}}\)-summability.

Definition 1.1

([6])

Let \(\mathfrak{I}\) be a non-trivial admissible ideal in \(\mathbb{N} \) and \(\mathfrak{A}= ( \mathfrak{a}_{nk} ) \in \Omega \). We say that a sequence \(\eta =(\eta _{k})\) is \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-summable to \(\mathfrak{s}\) if there is a set \(\mathfrak{H}\in \mathfrak{I}\) such that \(\mathfrak{M}=\mathbb{N} \setminus \mathfrak{H}= \{ m_{1},m_{2},\ldots\} \in \mathcal{F}_{\mathfrak{I}}\), and \(\lim_{i}\sum_{k}\mathfrak{a}_{m_{i}k}\eta _{k}=\lim_{i}y_{m_{i}}= \mathfrak{s}\). In this case we write \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-\(\lim \eta _{k}=\mathfrak{s}\).

Theorem 1.1

([6])

Let \(\mathfrak{I}\) be a non-trivial admissible ideal in \(\mathbb{N} \).

(a) If \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-\(\lim \eta _{k}= \mathfrak{s}\) then \(\mathfrak{A}^{\mathfrak{I}}\)-\(\lim \eta _{k}=\mathfrak{s}\).

(b) If \(\mathfrak{I}\) satisfies the condition \((AP)\) and \(\mathfrak{A}^{\mathfrak{I}}\)-\(\lim \eta _{k}=\mathfrak{s}\), then \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-\(\lim \eta _{k}=\mathfrak{s}\).

Definition 1.2

([28])

A real sequence \(\eta =(\eta _{k})\) is \(\mathfrak{I}\)-statistically convergent to \(\mathfrak{s}\in \mathbb{R} \) if \(\forall \epsilon >0\) and \(\nu >0\),

$$ \biggl\{ n:\frac{1}{n} \bigl\vert \bigl\{ k\leq n: \vert \eta _{k}-\mathfrak{s} \vert \geq \epsilon \bigr\} \bigr\vert \geq \nu \biggr\} \in \mathfrak{I} $$

then we write \(\mathfrak{I}\)-\(st\lim_{k}\eta _{k}=\mathfrak{s}\).

Remark 1.1

If \(\mathfrak{I}=\mathfrak{I}_{\mathrm{fin}}= \{ E\subseteq \mathbb{N}:E\text{ is finite} \} \), then \(\mathfrak{I}\)-statistical convergence coincides with the statistical convergence due to Fast [10].

Recently, Edely [7] also introduced the notion of statistically \(\mathfrak{A}^{\mathfrak{I}}\) and statistically \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-summable and gave some relations.

Definition 1.3

([7])

Let \(\mathfrak{A}= ( \mathfrak{a}_{jk} ) \in \Omega \). A sequence \(\eta =(\eta _{k})\) is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-summable to \(\mathfrak{s}\) if \(\forall \epsilon >0\) and every \(\nu >0\),

$$ \biggl\{ n\in \mathbb{N}:\frac{1}{n} \bigl\vert \bigl\{ j\leq n: \vert y_{j}- \mathfrak{s} \vert \geq \epsilon \bigr\} \bigr\vert \geq \nu \biggr\} \in \mathfrak{I}, $$

where \(y_{j}=\mathfrak{A}_{j}(\eta )\). Thus η is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-summable to \(\mathfrak{s}\) iff the sequence \((y_{j})\) is \(\mathfrak{I}\)-statistically convergent to \(\mathfrak{s}\), then we write \((\mathfrak{A}^{\mathfrak{I}})_{st}\)-\(\lim \eta =\mathfrak{I}\)-\(st\lim A\eta \).

Remark 1.2

(a) If \(\mathfrak{I}=\mathfrak{I}_{\mathrm{fin}}\), then statistical \(\mathfrak{A}^{\mathfrak{I}}\)-summable coincides with the statistical \(\mathfrak{A}\)-summable due to Edely and Mursaleen [9].

(b) If \(\mathfrak{A}=I\) the identity matrix, then statistical \(\mathfrak{A}^{\mathfrak{I}}\)-summable coincides with the \(\mathfrak{I}\)-statistical convergence due to Savas et al. [28]. If \(\mathfrak{I}=\mathfrak{I}_{\delta }\) and \(\mathfrak{A}=(C,1)\) the Cesàro matrix of order 1, then it reduces to statistical summability \((C,1)\) due to Móricz [20].

Definition 1.4

([7])

Let \(\mathfrak{A}=(\mathfrak{a}_{jk})\in \Omega \). A sequence \(\eta =(\eta _{k})\) is statistically \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-summable to \(\mathfrak{s}\) if there is a set \(M= \{ m_{i} \} \), where \(m_{1}< m_{2}<\cdots\) and \(M\in \mathcal{F}_{\mathfrak{I}}\), \(\delta ( M ) =1\), such that

$$ st-\lim_{i}\mathfrak{A}_{m_{i}}\eta =st-\lim _{i}y_{m_{i}}=\mathfrak{s}, $$

where \(y_{m_{i}}=\sum_{k}\mathfrak{a}_{m_{i}k}\eta _{k}\) i.e. \((\mathfrak{A}_{m_{i}}\eta )\) is statistically convergent to \(\mathfrak{s}\), and we write \((\mathfrak{A}^{\mathfrak{I}^{\ast }})_{st}\)-\(\lim \eta =\mathfrak{I}^{\ast }\)-\(st\lim \mathfrak{A}\eta =\mathfrak{s}\).

Remark 1.3

If \(\mathfrak{A}=I\), the identity matrix, then η is \(\mathfrak{I}^{\ast }\)-statistically convergent to the number \(\mathfrak{s}\), and we write \(\mathfrak{I}^{\ast }-st\lim \eta =\mathfrak{s}\).

Theorem 1.2

([7])

(a) If \((\mathfrak{A}^{\mathfrak{I}^{\ast }})_{st}\)-\(\lim \eta _{k}= \mathfrak{s}\) then \((\mathfrak{A}^{\mathfrak{I}})_{st}\)-\(\lim \eta _{k}=\mathfrak{s}\).

(b) If \(\mathfrak{I}\) satisfies the condition \((\mathit{APO})\), then whenever \(( \mathfrak{A}^{\mathfrak{I}} ) _{st}\)-\(\lim \eta _{k}= \mathfrak{s}\) we have \(( \mathfrak{A}^{\mathfrak{I}^{\ast }} ) _{st}\)-\(\lim_{k} \eta _{k}=\mathfrak{s}\).

Corollary 1.1

(a) If \(\mathfrak{I}^{\ast }\)-\(st\lim \eta _{k}=\mathfrak{s}\) then \(\mathfrak{I}\)-\(st\lim \eta _{k}=\mathfrak{s}\).

(b) If \(\mathfrak{I}\) satisfies the condition \((\mathit{APO})\), then whenever \(\mathfrak{I}\)-\(st\lim \eta _{k}=\mathfrak{s}\) we have \(\mathfrak{I}^{\ast }\)-\(st\lim \eta _{k}=\mathfrak{s}\).

Recall that \(\mathcal{I}\) satisfies the \((\mathit{APO})\) condition (cf. [2, 11]), if for every sequence \((\mathcal{C}_{n})\) of (pairwise disjoint) sets from \(\mathfrak{I}\) such that \(\delta (\mathcal{C}_{n})=0\) for each n, then there exist sets \(\mathcal{D}_{n}\in \mathfrak{I}\), \(n\in \mathbb{N} \) such that the symmetric difference \(\mathcal{C}_{n}\Delta \mathcal{D}_{n}\) is finite for every n, \(\bigcup_{n}\mathcal{D}_{n}\in \mathfrak{I}\), \(\delta (\bigcup_{n}\mathcal{D}_{n})=0\).

Remark 1.4

In what follows, \(\mathfrak{I}\) will be a non-trivial admissible ideal in \(\mathbb{N} \).

In this paper we use a technique and introduce the notion of statistically \(\mathfrak{A}^{\mathfrak{I}}\)-Cauchy and statistically \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-Cauchy summability via the notion of ideal. We obtain some relations between them and prove that under certain conditions statistical \(\mathfrak{A}^{\mathfrak{I}}\)-Cauchy and statistical \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-Cauchy summability are equivalent. Moreover, we give some Tauberian theorems for statistical \(\mathfrak{A}^{\mathfrak{I}}\)-summability.

Some related concepts

The concept of \(\mathfrak{I}\)-limit superior and inferior of a real sequence was given in [3], see also [17]. In this section we define and study some relations of statistically \(\mathfrak{A}^{\mathfrak{I}}\)-limit superior and statistically \(\mathfrak{A}^{\mathfrak{I}}\)-limit inferior of a real number sequence \(\eta =(\eta _{k})\).

Definition 2.1

Let \(\mathfrak{A}=(\mathfrak{a}_{jk})\in \Omega \) and \(\eta =(\eta _{k})\) be a real sequence. Let us write \(G_{\eta }\) and \(F_{\eta }\), for some \(\upsilon >0\), as

$$ G_{\eta }= \biggl\{ g\in \mathbb{R}: \biggl\{ n\in \mathbb{N}: \frac{1}{n} \bigl\vert \{ j\leq n:y_{j}>g \} \bigr\vert > \upsilon \biggr\} \notin \mathfrak{I} \biggr\} $$

and

$$ F_{\eta }= \biggl\{ f\in \mathbb{R}: \biggl\{ n\in \mathbb{N}: \frac{1}{n} \bigl\vert \{ j\leq n:y_{j}< f \} \bigr\vert > \upsilon \biggr\} \notin \mathfrak{I} \biggr\} . $$

Then we define

$$ \bigl( \mathfrak{A}^{\mathfrak{I}} \bigr) _{st}-\lim \sup \eta = \mathfrak{I}-st\lim \sup \mathfrak{A}\eta = \textstyle\begin{cases} \sup G_{\eta } & \text{if }G_{\eta }\neq \varnothing, \\ -\infty & \text{if }G_{\eta }=\varnothing,\end{cases} $$

and

$$ \bigl( \mathfrak{A}^{\mathfrak{I}} \bigr) _{st}-\lim \inf \eta = \mathfrak{I}-st\lim \inf \mathfrak{A}\eta =\textstyle\begin{cases} \inf F_{\eta } & \text{if }F_{\eta }\neq \varnothing, \\ \infty & \text{if }F_{\eta }=\varnothing.\end{cases} $$

Remark 2.1

If \(A=I\), then the statistical \(\mathfrak{A}^{\mathfrak{I}}\)-limit superior and statistical \(\mathfrak{A}^{\mathfrak{I}}\)-limit inferior of η reduced to \(\mathfrak{I}\)-statistical limit superior and inferior due to Mursaleen et al. [22]. Moreover if \(\mathfrak{I}=\mathfrak{I}_{\mathrm{fin}}\), then we have statistical limit superior and inferior cases due to [14].

The following result can be proved straightforward from Definition 2.1 and the least upper bound argument.

Theorem 2.1

(a) If \(( \mathfrak{A}^{\mathfrak{I}} ) _{st}\)-\(\lim \sup x=l_{1}\) is finite, then \(\forall \epsilon >0\),

$$ \biggl\{ n\in \mathbb{N}:\frac{1}{n} \bigl\vert \{ j\leq n:y_{j}>l_{1}- \epsilon \} \bigr\vert >\upsilon \biggr\} \notin \mathfrak{I} $$
(2.1)

for some \(\upsilon >0\), and

$$ \biggl\{ n\in \mathbb{N}:\frac{1}{n} \bigl\vert \{ j\leq n:y_{j}>l_{1}+ \epsilon \} \bigr\vert >\upsilon \biggr\} \in \mathfrak{I} , $$
(2.2)

for all \(\upsilon >0\). Conversely If (2.1) and (2.2) hold \(\forall \epsilon >0\), then \(( \mathfrak{A}^{\mathfrak{I}} ) _{st}\)-\(\lim \sup \eta = l_{1}\).

(b) If \(( \mathfrak{A}^{\mathfrak{I}} ) _{st}\)-\(\lim \inf \eta =l_{2}\) is finite, then \(\forall \epsilon >0\),

$$ \biggl\{ n\in \mathbb{N}:\frac{1}{n} \bigl\vert \{ j\leq n:y_{j}< l_{2}+ \epsilon \} \bigr\vert >\upsilon \biggr\} \notin \mathfrak{I} $$
(2.3)

for some \(\upsilon >0\), and

$$ \biggl\{ n\in \mathbb{N}:\frac{1}{n} \bigl\vert \{ j\leq n:y_{j}< l_{2}- \epsilon \} \bigr\vert >\upsilon \biggr\} \in \mathfrak{I} $$
(2.4)

for all \(\upsilon >0\). Conversely If (2.3) and (2.4) hold for every \(\epsilon >0\), then \(( \mathfrak{A}^{\mathfrak{I}} ) _{st}\)-\(\lim \inf \eta = l_{2}\).

Definition 2.2

Let \(\mathfrak{A}=(\mathfrak{a}_{jk})\in \Omega \). Then \(\eta =(\eta _{k})\) is said to be statistically \(\mathfrak{A}^{\mathfrak{I}}\)-bounded if there is a number \(t\in \mathbb{R} \) such that, for any \(\upsilon >0\),

$$ \biggl\{ n\in \mathbb{N}:\frac{1}{n} \bigl\vert \bigl\{ j\leq n: \vert y_{j} \vert >t \bigr\} \bigr\vert >\upsilon \biggr\} \in \mathfrak{I}. $$

Remark 2.2

(a) If \(\mathfrak{A}=I\), then the statistical \(\mathfrak{A}^{\mathfrak{I}}\)-boundedness reduces to \(\mathfrak{I}\)-statistical boundedness due to [22]. Moreover if \(\mathfrak{I}=\mathfrak{I}_{\mathrm{fin}}\), then we have the statistical bounded case of η due to [14].

(b) Statistical \(\mathfrak{A}^{\mathfrak{I}}\)-boundedness implies that \(( \mathfrak{A}^{\mathfrak{I}} ) _{st}\)-\(\lim \inf \eta \) and \(( \mathfrak{A}^{\mathfrak{I}} ) _{st}\)-\(\lim \sup \eta \) are finite.

(c) If \(\eta \in \ell _{\infty }\), then η is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-bounded.

(d) If η is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-summable then η is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-bounded.

The following theorems can be directly obtained from Theorem 3.2 and Theorem 3.4 of [22].

Theorem 2.2

Let \(\mathfrak{A}=(\mathfrak{a}_{jk})\in \Omega \). Then, for any real sequence \(\eta =(\eta _{k})\),

$$ \bigl( \mathfrak{A}^{\mathfrak{I}} \bigr) _{st}-\lim \inf \eta \leq \bigl( \mathfrak{A}^{\mathfrak{I}} \bigr) _{st}-\lim \sup \eta. $$

Remark 2.3

From Definition 2.1 and Theorem 2.2, we have, for any real sequence η,

$$ \lim \inf \eta \leq \bigl( \mathfrak{A}^{\mathfrak{I}} \bigr) _{st}- \lim \inf \eta \leq \bigl( \mathfrak{A}^{\mathfrak{I}} \bigr) _{st}- \lim \sup \eta \leq \lim \sup \eta. $$

Theorem 2.3

Let \(\mathfrak{A}=(\mathfrak{a}_{jk})\in \Omega \) and \(\eta =(\eta _{k})\) be statistically \(\mathfrak{A}^{\mathfrak{I}}\)-bounded. Then η is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-convergent iff \(( \mathfrak{A}^{\mathfrak{I}} ) _{st}\)-\(\lim \sup \eta = ( \mathfrak{A}^{\mathfrak{I}} ) _{st}\)-\(\lim \inf \eta \).

Example 2.1

Let \(B_{i}\) be mutually disjoint infinite sets such that \(\mathbb{N} =\bigcup_{i=1}^{\infty }B_{i}\). Let \(\mathfrak{I}\) be the class defined as

$$ \mathfrak{I}= \{ B\subset \mathbb{N}: B\text{ intersects only finite numbers of }B_{i} \}, $$

then \(\mathfrak{I}\) is a non-trivial admissible ideal in \(\mathbb{N} \). Define \(\eta =(\eta _{k})\) as

$$ \eta _{k}=\textstyle\begin{cases} 1 &\text{if } k\in B_{i}, k\text{ is odd,} \\ 0 &\text{otherwise} ,\end{cases} $$

and let \(\mathfrak{A}= ( \mathfrak{a}_{jk} ) \) be the identity matrix.

Since η is bounded, η is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-bounded. Since \(G_{\eta }= ( -\infty,1 ) \) and \(F_{\eta }= ( 0,\infty ) \), we have \(( \mathfrak{A}^{\mathfrak{I}} ) _{st}\)-\(\lim \inf \eta =0\), and \(( \mathfrak{A}^{\mathfrak{I}} ) _{st}\)-\(\lim \sup \eta =1\). Hence η is not statistically \(\mathfrak{A}^{\mathfrak{I}}\)-convergent.

Example 2.2

Let \(\mathfrak{I}\) and \(\mathfrak{A}\) be defined as in Example 2.1. Define \(\eta =(\eta _{k})\) as

$$ \eta _{k}=\textstyle\begin{cases} k &\text{if } k\in B_{1}, \\ 0 &\text{otherwise} .\end{cases} $$

Then, for any \(\upsilon >0\),

$$ \biggl\{ n\in \mathbb{N} :\frac{1}{n}\bigl\vert \bigl\{ j\leq n:\vert y_{j}\vert >1\bigr\} \bigr\vert >\upsilon \biggr\} \in \mathfrak{I}, $$

hence η is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-bounded. Since \(G_{\eta }= ( -\infty,0 ) \) and \(F_{\eta }= ( 0,\infty ) \), we have \(( \mathfrak{A}^{\mathfrak{I}} ) _{st}\)-\(\lim \inf \eta =0\), and \(( \mathfrak{A}^{\mathfrak{I}} ) _{st}\)-\(\lim \sup \eta =0\). Hence η is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-convergent to zero.

Statistical \(\mathfrak{A}^{\mathfrak{I}}\)-Cauchy and statistical \(\mathfrak{A}^{\mathfrak{I}^{\ast }}-\)Cauchy summability

Fridy [12], introduced the concept of Cauchy condition for statistical convergence for real sequences. In [4, 19] and [26] the notion of \(\mathfrak{I}\)-Cauchy sequence was studied which is a generalization of Cauchy condition for statistical convergence. Nabiev et al. [26] introduced the notion of a \(\mathfrak{I}^{\ast }\)-Cauchy sequence and proved that under certain conditions a \(\mathfrak{I}^{\ast }\)-Cauchy sequence is equivalent to a \(\mathfrak{I}\)-Cauchy sequence.

Definition 3.1

([4, 26])

A real sequence \(\eta =(\eta _{n})\) is a \(\mathfrak{I}\)-Cauchy sequence if \(\forall \epsilon >0\) there exists \(k=k(\epsilon )\in \mathbb{N} \) such that

$$ \bigl\{ n: \vert \eta _{n}-\eta _{k} \vert \geq \epsilon \bigr\} \in \mathfrak{I}. $$

Definition 3.2

([26])

A real sequence \(\eta =(\eta _{n})\) is called an \(\mathfrak{I}^{\ast }\)-Cauchy sequence if there exists a set \(M= \{ m_{1}< m_{2}<\cdots<m_{k}<\cdots \} \subset \mathbb{N} \), \(M\in \mathcal{F}_{\mathfrak{I}}\) such that the subsequence \((\eta _{m_{k}})\) is Cauchy in \(\mathbb{R} \).

We introduce the notion of statistically \(\mathfrak{A}^{\mathfrak{I}}\)-Cauchy and statistically \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-Cauchy summability.

Definition 3.3

Let \(\mathfrak{A}= ( \mathfrak{a}_{jk} ) \in \Omega \). A real sequence \(\eta =(\eta _{k})\) is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-Cauchy summable if for any \(\epsilon >0\) and \(\forall \nu >0\) there is \(N=N(\epsilon )\in \mathbb{N} \) such that

$$ \biggl\{ j\leq n:\frac{1}{n} \bigl\vert \bigl\{ \vert y_{j}-y_{N} \vert \geq \epsilon \bigr\} \bigr\vert \geq \nu \biggr\} \in \mathfrak{I}. $$

Definition 3.4

Let \(\mathfrak{A}= ( \mathfrak{a}_{jk} ) \in \Omega \). A real sequence \(\eta =(\eta _{k})\) is statistically \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-Cauchy summable if there is a set \(M= \{ m_{1},m_{2},\ldots \} \), where \(m_{1}< m_{2}<\cdots\) , and \(M\in \mathcal{F}(\mathfrak{I})\), \(\delta (M)=1\), such that the subsequence \((y_{m_{i}})\) is statistically Cauchy in \(\mathbb{R} \).

Now, we give some relations between statistical \(\mathfrak{A}^{\mathfrak{I}}\) (or statistical \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\))-summability and statistical \(\mathfrak{A}^{\mathfrak{I}}\) (or statistical \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\))-Cauchy summability.

Theorem 3.1

A real sequence η is statistically \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-summable to \(\mathfrak{s}\) if and only if η is statistically \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-Cauchy summable.

Proof

The proof follows from Definition 1.4 and Definition 3.4 and using Theorem 1 of [12]; statistical convergence is equivalent to the statistical Cauchy for \(\mathbb{R} \). □

Theorem 3.2

A real sequence \(\eta =(\eta _{k})\) is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-summable to \(\mathfrak{s}\) iff η is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-Cauchy summable.

Proof

Let \(( \mathfrak{A}^{\mathfrak{I}} ) _{st}\)-\(\lim \eta _{k}= \mathfrak{s}\), then, for any \(\epsilon >0\) and \(\forall \nu >0\), we have the set

$$ B ( \nu ) = \biggl\{ n:\frac{1}{n} \biggl\vert \biggl\{ j \leq n: \vert y_{j}-\mathfrak{s} \vert \geq \frac{\epsilon }{2} \biggr\} \biggr\vert \geq \nu \biggr\} \in \mathfrak{I}. $$

Let us define B and C by

$$ B= \biggl\{ j\leq n: \vert y_{j}-\mathfrak{s} \vert \geq \frac{\epsilon }{2} \biggr\} $$

and

$$ C= \bigl\{ j\leq n: \vert y_{j}-y_{N} \vert \geq \epsilon \bigr\} , $$

where \(N\notin B\), such N exists as \(\mathfrak{I}\) is an admissible ideal, otherwise the set \(B ( \frac{1}{2} ) =\mathbb{N} \notin \mathfrak{I}\). We need first to show that \(C\subseteq B\). Now for any \(c\in C\), since

$$ \vert y_{c}-y_{N} \vert \leq \vert y_{c}- \mathfrak{s} \vert + \vert y_{N}-\mathfrak{s} \vert , $$

we have

$$ \vert y_{c}-\mathfrak{s} \vert + \vert y_{N}- \mathfrak{s} \vert \geq \epsilon. $$

Since \(N\notin B\), we have

$$ \vert y_{N}-\mathfrak{s} \vert < \frac{\epsilon }{2}, $$

therefore

$$ \vert y_{c}-\mathfrak{s} \vert >\frac{\epsilon }{2}. $$

Hence \(c\in B\). So we have \(C\subseteq B\), therefore

$$ \frac{1}{n} \vert C \vert \leq \frac{1}{n} \vert B \vert . $$

Hence for any \(\nu >0\), we have

$$ \biggl\{ n:\frac{1}{n} \vert C \vert \geq \nu \biggr\} \subseteq \biggl\{ n:\frac{1}{n} \vert B \vert \geq \nu \biggr\} =B(\nu )\in \mathfrak{I}. $$

Therefore \(\{ n:\frac{1}{n} \vert \{ j\leq n: \vert y_{j}-y_{N} \vert \geq \epsilon \} \vert \geq \nu \} \in \mathfrak{I}\), hence η is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-Cauchy summable.

Conversely, let η be statistically \(\mathfrak{A}^{\mathfrak{I}}\)-Cauchy summable. Then, for any \(\epsilon >0\) and \(\forall \nu >0\), there exists \(N=N(\epsilon )\in \mathbb{N} \) such that

$$ F(\upsilon )= \biggl\{ n:\frac{1}{n} \biggl\vert \biggl\{ j\leq n: \vert y_{j}-y_{N} \vert \geq \frac{\epsilon }{2} \biggr\} \biggr\vert \geq \nu \biggr\} \in \mathfrak{I}, $$

therefore

$$ G(\nu )= \biggl\{ n:\frac{1}{n} \biggl\vert \biggl\{ j\leq n: \vert y_{j}-y_{N} \vert \geq \frac{\epsilon }{2} \biggr\} \biggr\vert < \nu \biggr\} \in \mathcal{F}_{\mathfrak{I}}. $$

First, let us show that η is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-bounded. Let us define F and G by

$$ F= \biggl\{ j: \vert y_{j}-y_{N} \vert < \frac{\epsilon }{2} \biggr\} $$

and

$$ G= \bigl\{ j: \vert y_{j} \vert < \epsilon + \vert y_{t} \vert \bigr\} , $$

where \(t\in \mathbb{N} \) satisfied \(\vert y_{t}-y_{N} \vert <\frac{\epsilon }{2}\), such t exists as I is an admissible ideal, otherwise, the set \(F ( \frac{1}{2} ) =\mathbb{N} \notin I\). We need first to show that \(F\subseteq G\). Now for any \(a\in F\), since

$$ \vert y_{a}-y_{t} \vert \leq \vert y_{a}-y_{N} \vert + \vert y_{N}-y_{t} \vert < \epsilon. $$

Therefore

$$ \vert y_{a} \vert \leq \vert y_{a}-y_{t} \vert + \vert y_{t} \vert < \epsilon + \vert y_{t} \vert , $$

hence \(a\in G\). So we have \(F\subseteq G\), therefore

$$ \frac{1}{n} \vert F \vert \leq \frac{1}{n} \vert G \vert . $$

Hence for any \(\nu >0\), we have

$$ \biggl\{ n:\frac{1}{n} \vert F \vert >\nu \biggr\} \subseteq \biggl\{ n:\frac{1}{n} \vert G \vert >\nu \biggr\} . $$

Since \(G(\nu )\in \mathcal{F}_{\mathfrak{I}}\), we have \(\{ n:\frac{1}{n} \vert F \vert >\nu \} \in \mathcal{F}_{ \mathfrak{I}}\), therefore \(\{ n:\frac{1}{n} \vert G \vert >\nu \} \in \mathcal{F}_{\mathfrak{I}}\), so the set

$$ \biggl\{ n:\frac{1}{n} \bigl\vert \bigl\{ j\leq n: \vert y_{j} \vert < \epsilon + \vert y_{t} \vert \bigr\} \bigr\vert >\nu \biggr\} \in \mathcal{F}_{\mathfrak{I}}, $$

i.e.

$$ \biggl\{ n:\frac{1}{n} \bigl\vert \bigl\{ j\leq n: \vert y_{j} \vert >\epsilon + \vert y_{t} \vert \bigr\} \bigr\vert < \nu \biggr\} \in \mathcal{F}_{\mathfrak{I}}, $$

hence, the set

$$ \biggl\{ n:\frac{1}{n} \bigl\vert \bigl\{ j\leq n: \vert y_{j} \vert >\epsilon + \vert y_{t} \vert \bigr\} \bigr\vert >\nu \biggr\} \in \mathfrak{I}, $$

so η is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-bounded. We use that statistical \(\mathfrak{A}^{\mathfrak{I}}\)-boundedness implies that \(( \mathfrak{A}^{\mathfrak{I}} ) _{st}\)-\(\lim \inf \eta \) and \(( \mathfrak{A}^{\mathfrak{I}} ) _{st}\)-\(\lim \sup \eta \) are finite. Using Theorem 2.2, we have \(\alpha = ( \mathfrak{A}^{\mathfrak{I}} ) _{st}\)-\(\lim \inf \eta \leq ( \mathfrak{A}^{\mathfrak{I}} ) _{st}\)-\(\lim \sup \eta =\beta \). Given that η is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-Cauchy summable, then, for any \(\epsilon >0\) and \(\forall \nu >0\), there exists \(N=N(\epsilon )\in \mathbb{N} \) such that

$$ \biggl\{ n:\frac{1}{n} \biggl\vert \biggl\{ j\leq n: \vert y_{j}-y_{N( \frac{\epsilon }{2})} \vert \geq \frac{\epsilon }{2} \biggr\} \biggr\vert \geq \nu \biggr\} \in \mathfrak{I}. $$

Therefore

$$ \biggl\{ n:\frac{1}{n} \biggl\vert \biggl\{ j\leq n:y_{j}>y_{N( \frac{\epsilon }{2})}+ \frac{\epsilon }{2} \biggr\} \biggr\vert >\nu \biggr\} \in \mathfrak{I}, $$

hence by Theorem 2.1(a), we have

$$ \beta < y_{N(\frac{\epsilon }{2})}+\frac{\epsilon }{2}. $$
(3.1)

Also we have

$$ \biggl\{ n:\frac{1}{n} \biggl\vert \biggl\{ j\leq n:y_{j}< y_{N( \frac{\epsilon }{2})}- \frac{\epsilon }{2} \biggr\} \biggr\vert >\nu \biggr\} \in \mathfrak{I}, $$

hence by Theorem 2.1(b), we have

$$ y_{N(\frac{\epsilon }{2})}< \alpha +\frac{\epsilon }{2}. $$
(3.2)

Using (3.1) and (3.2), we have

$$ \beta < \alpha +\epsilon. $$

Hence, for any \(\vartheta >0\), we always have \(\beta <\alpha +\vartheta \), therefore \(\beta \leq \alpha \). Hence \(\alpha = ( \mathfrak{A}^{\mathfrak{I}} ) _{st}\)-\(\lim \inf \eta = ( \mathfrak{A}^{ \mathfrak{I}} ) _{st}\)-\(\lim \sup \eta =\beta \). Now by Theorem 2.3, η is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-convergent. □

Theorem 3.3

(a) If \(\eta =(\eta _{k})\) is statistically \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-Cauchy summable then η is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-Cauchy summable.

(b) If \(\mathfrak{I}\) satisfies the condition \((\mathit{APO})\), then η is statistically \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-Cauchy summable whenever η is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-Cauchy summable.

Proof

(a) The proof follows from Theorem 3.1, Theorem 1.2(a) and Theorem 3.2.

(b) The proof follows from Theorem 3.2, Theorem 1.2(b) and Theorem 3.1. □

Remark 3.1

The converse of Theorem 3.3 (a) is not true in general.

Example 3.1

In [7] Example 2.9, the following example was given.

Let \(B_{i}= \{ 2^{i-1}(2k-1):k\in \mathbb{N} \} \) be mutually disjoint infinite sets such that \(\mathbb{N} =\bigcup_{i=1}^{\infty }B_{i}\). Let \(\mathfrak{I}\) be the class defined as

$$ \mathfrak{I}= \bigl\{ B\subset \mathbb{N}: B\text{ intersects only finite numbers of }B_{i}^{ \prime }s \bigr\} , $$

then \(\mathfrak{I}\) is a non-trivial admissible ideal in \(\mathbb{N} \). Define \(\eta =(\eta _{k})\) by

$$ \eta _{k}=\frac{1}{i},\quad k\in B_{i}, $$

and \(\mathfrak{A}= ( \mathfrak{a}_{jk} ) \) by

$$ \mathfrak{a}_{jk}=\textstyle\begin{cases} 1 &\text{if } k=j^{2}, \\ 0 & \text{otherwise}. \end{cases} $$

It is shown that η is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-summable to zero but η is not statistically \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-summable to any number. Hence from Theorem 3.1 and Theorem 3.2 we conclude that η is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-Cauchy summable but η is not statistically \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-Cauchy summable.

Some Tauberian theorems

In [12], a Tauberian theorem was given for statistical convergence. The next results are Tauberian theorems for statistical \(\mathfrak{A}^{\mathfrak{I}}\)-summability. Let τ denote the collection of lower triangular nonnegative summability matrices \(\mathfrak{A}\) with (i) \(\sum_{k=1}^{n}\mathfrak{a}_{nk}=1\) and (ii) if \(K\subseteq \mathbb{N} \) such that \(\delta (K)=0\), then \(\lim_{n}\sum_{k\in K}\mathfrak{a}_{nk}=0\), (cf. [13]). From these conditions any \(\mathfrak{A}\in \tau \) is regular. Let us denote \(\Delta \eta _{k}=\eta _{k}-\eta _{k+1}\).

Theorem 4.1

Let \(\mathfrak{I}\) be a non-trivial admissible ideal in \(\mathbb{N}\) which satisfies the condition \((\mathit{APO})\). Let \(\mathfrak{A}=(\mathfrak{a}_{jk})\in \tau \) and \(\eta = ( \eta _{k} ) \) be a bounded sequence. If η is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-summable to \(\mathfrak{s}\) and \(\Delta A_{m_{i}}(\eta )=O ( \frac{1}{m_{i}} ) \), where \(M= \{ m_{i} \} \in \mathcal{F}_{\mathfrak{I}}\), then η is \(\mathfrak{I}\)-statistically convergent to \(\mathfrak{s} \).

Proof

Let η be statistically \(\mathfrak{A}^{\mathfrak{I}}\)-summable to \(\mathfrak{s}\) and \(\mathfrak{I}\) satisfy the condition \((\mathit{APO})\). From Theorem 1.2(b), η is statistically \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-summable to \(\mathfrak{s}\). Since \(\Delta A_{m_{i}}(\eta )=O ( \frac{1}{m_{i}} ) \), so by Theorem 3 of [12], η is \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-summable to \(\mathfrak{s}\). Since \(\mathfrak{A}=(\mathfrak{a}_{jk})\in \tau \), we have \(\mathfrak{A}=(\mathfrak{a}_{m_{i}k})\in \tau \). Therefore by Theorem 1 of Fridy and Miller [13], η is \(\mathfrak{I}^{\ast }\)-statistically convergent to \(\mathfrak{s}\). Hence by Corollary 1.1(a), η is \(\mathfrak{I}\)-statistically convergent to \(\mathfrak{s}\). □

Corollary 4.1

Let \(\mathfrak{I}\) be a non-trivial admissible ideal in \(\mathbb{N} \) which satisfies the condition \((\mathit{APO})\). Let \(\mathfrak{A}=(\mathfrak{a}_{jk})\in \tau \) and \(\eta = ( \eta _{k} ) \) be a bounded sequence. If η is statistically \(\mathfrak{A}^{\mathfrak{I}}\)-summable to \(\mathfrak{s}\) and \(\Delta \mathfrak{A}_{m_{i}}(\eta )=O ( \frac{1}{m_{i}} ) \), where \(M= \{ m_{i} \} \in \mathcal{F}_{\mathfrak{I}}\), then η is \(\mathfrak{A}^{\mathfrak{I}}\)-summable to \(\mathfrak{s}\).

Theorem 4.2

Let \(\mathfrak{I}\) be a non-trivial admissible ideal in \(\mathbb{N}\) which satisfies the condition \((\mathit{APO})\). Let \(\eta = ( \eta _{k} ) \) be a bounded sequence. If η is \(\mathfrak{I}\)-statistically convergent to \(\mathfrak{s}\) and \(\Delta \eta _{m_{i}}=O ( \frac{1}{m_{i}} ) \), where \(M= \{ m_{i} \} \in \mathcal{F}(I)\), then η is \(\mathfrak{I}\)-convergent to \(\mathfrak{s}\).

Proof

Let η be \(\mathfrak{I}\)-statistically convergent to \(\mathfrak{s}\). Since \(\mathfrak{I}\) satisfies the condition \((\mathit{APO})\), from Corollary 1.1(b), η is \(\mathfrak{I}^{\ast }\)-statistically convergent to \(\mathfrak{s}\). Since \(\Delta \eta _{m_{i}}=O ( \frac{1}{m_{i}} ) \), by Theorem 3 of [12], η is \(\mathfrak{I}^{\ast }\)-convergent to \(\mathfrak{s}\). Now by Proposition 3.2 of [18], η is \(\mathfrak{I}\)-convergent to \(\mathfrak{s}\). □

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Edely, O.H.H., Mursaleen, M. On statistical \(\mathfrak{A}\)-Cauchy and statistical \(\mathfrak{A}\)-summability via ideal. J Inequal Appl 2021, 34 (2021). https://doi.org/10.1186/s13660-021-02564-4

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MSC

  • 40A35
  • 40G15
  • 40E05

Keywords

  • Statistical \(\mathfrak{A}^{\mathfrak{I}}\)-limit superior
  • Statistical \(\mathfrak{A}^{\mathfrak{I}}\)-limit inferior
  • Statistical \(\mathfrak{A}^{\mathfrak{I}}\)-bounded
  • Statistical \(\mathfrak{A}^{\mathfrak{I}}\)-Cauchy summability
  • Statistical \(\mathfrak{A}^{\mathfrak{I}^{\ast }}\)-Cauchy summability
  • Tauberian theorem