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.

1 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.

2 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.

3 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.

4 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 $\mathfrak{s}$. □