Almost absolute weighted summability with index k

TL;DR

This paper introduces a generalized space of almost summable series using weighted means, explores its topological properties, relationships with classical spaces, and characterizes related matrix operators.

Contribution

It extends the space of almost summable series with weights and factors, analyzing its structure and operator behavior, which was not previously studied.

Findings

The generalized space has specific topological properties.

Relations between the new space and classical sequence spaces are established.

Matrix operators acting on the space are characterized.

Abstract

The space $\widehat{\ell }_k$ of absolutely almost convergent series was introduced and studied by Das et al [4]; which plays an important role in summability theory, approximation theory, Fourier analysis, etc. In the present paper we generalize the space making use of some factors and weighted mean transformations, investigate its topological structures and relations between classical sequence spaces. Also we characterize certain matrix operators on it.

Equations

$$\frac{1}{m+1}{\sum }_{v=0}^m{x}_{n+v}\to \gamma \text{ as }m\to \infty \text{ uniformly in }n$$

$${\sum }_{m=0}^{\infty }{\left|{\psi }_{m,n}\right|}^k<\infty ,k>0,$$

$$\psi_{m,n}=\left\{\begin{array}[]{c}a_{n},\text{ \ }m=0\\ \frac{1}{m(m+1)}\mathop{\textstyle\sum}_{v=1}^{m}va_{n+v},\text{ }m\geq 1.\end{array}\right.$$

$${\hat{\ell }}_k=\left\{a:{\sum }_{m=0}^{\infty }{\left|{\psi }_{m,n}\right|}^k<\infty ,\text{ uniformly in }n,k>0\right\}.$$

$$T_{-1,n}(s)=s_{n-1},T_{m,n}(s)=\frac{1}{P_m}{\sum }_{v=0}^m{p}_v{s}_{n+v},m\ge 0.$$

$$F_{m,n}(a)=T_{m,n}(s)-T_{m-1,n}(s)=\left\{\begin{array}[]{c}a_{n},\text{ \ }m=0\\ \frac{p_{m}}{P_{m}P_{m-1}}\mathop{\textstyle\sum}_{v=1}^{m}P_{v-1}a_{n+v},\text{ }m\geq 1,\end{array}\right.$$

$$P_n=p_0+p_1+...+p_n\to \infty asn\to \infty ,P_{-1}=p_{-1}=0.$$

$$\sum _{m=0}^{\infty }{u}_m^{k-1}{\left|F_{m,n}(a)\right|}^k<\infty$$

$$\left(\frac{1}{u_m}\right)\in {\ell }_{\infty },$$

$$\sum _{m=0}^{\infty }{u}_m^{k-1}{\left(\frac{p_m}{P_m}\right)}^k<\infty ,$$

$$\sum _{m=M}^{\infty }{u}_m^{k-1}{\left|F_{m,n}(a)\right|}^k\le 1.$$

$$a_{m+n}=\frac{P_m}{p_m}{F}_{m,n}(a)-\frac{P_{m-2}}{p_{m-1}}{F}_{m-1,n}(a),$$

$$\left|{\sum }_{v=1}^m{P}_{v-1}{a}_{n+v}\right|$$

$$\sum _{m=0}^{\infty }{u}_m^{k-1}{\left|F_{m,n}(a)\right|}^k$$

$${\Vert a\Vert }_{{|f({\overline{N}}_p^u)|}_k}=\sup {\left\{\sum _{m=0}^{\infty }{u}_m^{k-1}{\left|F_{m,n}(a)\right|}^k:n\in \mathbb{N}\right\}}^{1/k}.$$

$$\sum _{m=M}^{\infty }{u}_m^{k-1}{\left|F_{m,n}(a)\right|}^k\le 1.$$

$$\sup \left\{\sum _{m=0}^{\infty }{u}_m^{k-1}{\left|F_{m,n}(a)\right|}^k:n\in \mathbb{N}\right\}$$

$$\underset{n}{\sup }{\left\{u_0^{k-1}{\left|a_n^{m_1}-a_n^{m_2}\right|}^k+\sum _{m=1}^{\infty }{u}_m^{k-1}{\left|\frac{p_m}{P_m{P}_{m-1}}{\sum }_{v=1}^m{P}_{v-1}\left(a_{n+v}^{m_1}-a_{n+v}^{m_2}\right)\right|}^k\right\}}^{1/k}<\epsilon .$$

$${\Vert a\Vert }_{{|f({\overline{N}}_p^u)|}_k}\le {\Vert a^{m_1}-a\Vert }_{{|f({\overline{N}}_p^u)|}_k}+{\Vert a^{m_1}\Vert }_{{|f({\overline{N}}_p^u)|}_k}<\infty ,$$

$$A_n\left(x\right)=\sum _{v=0}^{\infty }{a}_{nv}{x}_v$$

$${\ell }_k=\left\{x=\left(x_v\right)\in w:\sum _{v=0}^{\infty }{\left|x_v\right|}^k<\infty \right\}$$

$${\Vert x\Vert }_{{\ell }_k}={\left(\sum _{v=0}^{\infty }{\left|x_v\right|}^k\right)}^{1/k}.$$

$$\frac{1}{4C^2}{U}_p\left(A\right)\le L_p(A)\le U_p\left(A\right),$$

$$U_p\left(A\right)=\sum _{v=0}^{\infty }{\left(\sum _{n=0}^{\infty }\left|a_{nv}\right|\right)}^{p_v}<\infty$$

$$L_p(A)=\sup \left\{\sum _{v=0}^{\infty }{\left|\sum _{n\in N}{a}_{nv}\right|}^{p_v}:{\text{N is a finite subset of N}}_0\right\}<\infty .$$

$${\sum }_{m=0}^{\infty }{u}_m^{k-1}{\left|b(m,n,j)\right|}^k<\infty \text{ uniformly in }n$$

$$\sup \left\{{\sum }_{m=0}^{\infty }{u}_m^{k-1}{\left|b(m,n,j)\right|}^k:n,j\in \mathbb{N}\right\}<\infty$$

$$b(m,n,j)=\left\{\begin{array}[]{c}a_{nj},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ m=0\\ \frac{p_{m}}{P_{m}P_{m-1}}\mathop{\textstyle\sum}_{v=1}^{m}P_{v-1}a_{n+v,,j},\ m\geq 1.\end{array}\right.$$

$$\sum _{m=0}^{\infty }{u}_m^{k-1}{\left|F_{m,n}(A(x))\right|}^k$$

$$q_{sn}(x)={\left(\sum _{m=0}^s{u}_m^{k-1}{\left|F_{m,n}(A(x))\right|}^k\right)}^{1/k}$$

Full text

1 Introduction

Any subspace of $w,$ the set of all sequences of complex numbers, is called a sequence space. A $BK$ space $X$ is a Banach sequence space with the property that the map $p_{n}:X\rightarrow\mathbb{C}$ defined by $p_{n}(x)=x_{n}$ is continuous for all $n\geq 0,$ where $\mathbb{C}$ denotes the complex field. Let $\ell_{\infty}$ be the subspace of all bounded sequences of $w.$ A sequence $\left(x_{n}\right)\in\ell_{\infty}$ is said to be almost convergent to $\gamma$ if all of its Banach limits $\left[1\right]$ are equal to $\gamma$. Lorentz $\left[8\right]$ characterized almost convergence that a sequence $\left(x_{n}\right)$ is almost convergent to $\gamma$ if and

*2010 AMS Subject Classification: *40C05, 40D25, 40F05, 46A45

*Keywords: Sequence space, Bk-space, Absolute and almost summability, Matrix transformations

*only if

[TABLE]

This notation plays an important role in summability theory, approximation theory and Fourier analysis and was investigated by several authors. For example, it was later used to define and study some concepts such as conservative and regular matrices, some sequence spaces and matrix transformations (see [2], [6], [7], [9], [10], [11], [15]).

Absolute almost convergence emerges naturally as absolute analogue of almost convergence just as absolute convergence emerged out of the concept of convergence. To introduce this concept, let $\Sigma a_{v}$ be a given infinite series with $s_{n}$ as its n-th partial sum. The series $\Sigma a_{v}$ is said to be absolutely almost convergent series if (see [3])

[TABLE]

uniformly in $n,$ where

[TABLE]

The space of all absolutely almost convergent series

[TABLE]

was first defined and studied in $\left[4\right]$. We note an important relation between $\widehat{\ell}$ and absolute Cesaro summability $\left|C,1\right|$ in Flett’s notation [5], $\widehat{\ell}\subset$ $\left|C,1\right|$ $\left[4\right].$

2 Main Results

The purpose of the present paper is to define an absolute almost weighted summability using some factors and weighted means, which extends the well known concept of absolute almost convergence of Das et al $\left[4\right],$ and to study its topological structures. Also we investigate relations between classical sequence spaces and characterize certain matrix operatos on it.

For any sequence $\left(s_{n}\right),$ we define $T_{m,n}$ by

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A straightforward calculation then shows that

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where $(p_{n})$ is a sequence of positive real numbers with

[TABLE]

So we give the following definition.

**Definition 2.1. **Let $\Sigma a_{v}$ be an infinite series with partial summations $s_{n}.$ Let $(p_{n})$ and $\left(u_{n}\right)$ be sequences of positive real numbers. The series $\Sigma a_{v}$ is said to be absolute almost weighted summable $\left|f(\overline{N}_{p}),u_{m}\right|_{k},$ $k\geq 1,$ if

[TABLE]

uniformly in $n.$

For $\left|f(\overline{N}_{p}^{u})\right|_{k},k\geq 1,$ we write the set of all series summable by the method $\left|f(\overline{N}_{p}),u_{m}\right|_{k}.$ Then, $\Sigma a_{v}$ is summable $\left|f(\overline{N}_{p}),u_{m}\right|_{k}$ iff the series $\Sigma a_{v}\in$ $\left|f(\overline{N}_{p}^{u})\right|_{k}.$ Note that, in the case$\ u_{m}=p_{m}=1$ for $m\geq 0,$ it reduces to the set of absolutely almost convergent series $\widehat{\ell}_{k}$ given by Das, Kuttner and Nanda $\left[3\right].$ Further, it is clear that the space $\left|\overline{N}_{p}^{u}\right|_{k}$ is derived from $\left|f(\overline{N}_{p}^{u})\right|_{k}$ by putting $n=0$ (see [9], [12], [13]), and also $\left|f(\overline{N}_{p}^{u})\right|_{k}\subset\left|\overline{N}_{p}^{u}\right|_{k},$ but the converse is not true.

First it would be appropriate to clarify some relations between the new method and classical sequence spaces such as $bs$ and $\ell_{\infty},$ where $bs$ and $\ell_{\infty}$ are the set of all bounded series and sequences, respectively.

**Theorem 2.2. **Let $(p_{m})$ and $(u_{m})$ be two sequences of positive numbers. If

$(i)$ If

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then $\left|f(\overline{N}_{p}^{u})\right|_{k}\subset\ell_{\infty},$ $k\geq 1.$

$\left(ii\right)$ If

[TABLE]

then $bs\subset\left|f(\overline{N}_{p}^{u})\right|_{k},$ $k>1.$

Proof. $\left(i\right)$ Given $a=(a_{v})\in$ $\left|f(\overline{N}_{p}^{u})\right|_{k}.$ Then, by the definition, there exists an integer $M$ such that, for all $n,$

[TABLE]

So, by $\left(2.3\right)$ and $\left(2.5\right),$ we have $\left|F_{m,n}(a)\right|\leq u_{m}^{-1+1/k}$ for $m\geq M$ and all $n.$ On the other hand, for $m\geq 1,$

[TABLE]

which gives that $a=(a_{v})\in\ell_{\infty}.$ This completes the proof.

$\left(ii\right)$ Let $a=(a_{v})\in bs.$ Say $M=\sup_{v}\left|\mathop{\textstyle\sum}_{j=0}^{v}a_{j}\right|.$ Then we have

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which implies that

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This completes the proof.

For the special case $u_{m}=p_{m}=1$ for all $m\geq 0,$ $\left|f(\overline{N}_{p}^{u})\right|_{k}=\widehat{\ell}_{k}$ and $\left(2.4\right)$ reduces to $\sum_{m=0}^{\infty}\left(m+1\right)^{-k}<\infty.$ So we have the following result in $\left[3\right].$

**Corollary 2.3. **For $k>1,$ $bs\subset\widehat{\ell}_{k}.$

**Theorem 2.4. **Let $(u_{m})$ be any sequence of positive numbers. Then, $\left|f(\overline{N}_{p}^{u})\right|_{k},$ $k\geq 1,$ is a $BK$ space with respect to the norm

[TABLE]

**Proof. ** It is routine to show that $\left(2.7\right)$ satisfies the norm conditions. We only note that $\left(2.7\right)$ is well defined. In fact, if $a\in\left|f(\overline{N}_{p}^{u})\right|_{k}$, then, as in the proof of part $\left(ii\right)$ of Theorem 2.2, there exists an integer $M$ such that, for all $n,$

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and  $\left(u_{m}^{1-1/k}\left|F_{m,n}(a)\right|\right)$ is bounded for all $m,n\geq 0.$ This gives

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To prove that it is a Banach space, let us take arbitrary Cauchy sequence $\left(a^{m}\right),$ where $a^{m}=\left(a_{v}^{m}\right)\in\left|f(\overline{N}_{p}^{u})\right|_{k}$ for $m\geq 0.$ Given $\varepsilon>0.$ Then there exists an integer $m_{0}$ such that $\left\|a^{m_{1}}-a^{m_{2}}\right\|_{\left|f(\overline{N}_{p}^{u})\right|_{k}}<\varepsilon$ for $m_{1},m_{2}>m_{0},$ or, equivalently,

[TABLE]

This also gives us $\left|a_{v}^{m_{1}}-a_{v}^{m_{2}}\right|^{k}<\varepsilon/u_{0}^{1-1/k}$ for $m_{1},m_{2}>m_{0},$and all $v,$ $i.e.,$ $\left(a_{v}^{m}\right)$ is a Cauchy sequence in the set of complex numbers $\mathbb{C}.$ So it converges to a number $a_{v}$ $(v=0,1,...),$ $i.e.,$ $\lim_{m\rightarrow\infty}a_{v}^{m}=a_{v}.$ Now letting $m_{2}\rightarrow\infty,$ by $\left(2.8\right)$ we have, for $\left\|a^{m_{1}}-a\right\|_{\left|f(\overline{N}_{p}^{u})\right|_{k}}<\varepsilon$ for $m_{1}>m_{0}.$ This means $\lim_{m\rightarrow\infty}a^{m}=a.$ Further, since

[TABLE]

then, $a\in\left|f(\overline{N}_{p}^{u})\right|_{k}.$ So, $\left|f(\overline{N}_{p}^{u})\right|_{k}$ is a Banach space. This completes the proof.

We note that if $E$ is a $BK$-space such that $bs\subset E\subset\ell_{\infty}$, then $E$ is not separable (and hence not reflexive)$\left(\text{see }\left[4\right]\right).$ Hence the following result at once follows from Theorem 2.2.

**Corollary 2.5. **If $(p_{m})$ and $(u_{m})$ are sequences of positive numbers satisfying $\left(2.3\right)$ and $\left(2.4\right)$, then $\left|f(\overline{N}_{p}^{u})\right|_{k}$ is not seperable for $k>1.$

3 Matrix Transformations on Space $\left|f(\overline{N}_{p})\right|_{k}$

In this section we characterize certain matrix transformations on the space $\left|f(\overline{N}_{p})\right|_{k}.$ First we recall some notations. Let $X,$ $Y$ be any subsets of $\omega$ and $A=(a_{nv})$ be an infinite matrix of complex numbers. By $A(x)=(A_{n}(x)),$ we indicate the $A$-transform of a sequence $x=\left(x_{v}\right)$, if the series

[TABLE]

are convergent for $n\geq 0.$ If $Ax\in Y,$ whenever $x\in X,$ then we say that $A$ defines a matrix mapping from $X$ into $Y$ and denote the class of all infinite matrices $A$ such that $A:X\rightarrow Y$ by $(X,Y).$ Also we denote the set of all $p$-absolutely convergent series by $\ell_{k},1\leq k<\infty,$ $i.e.,$

[TABLE]

which is a $BK$-space by respect to the norm

[TABLE]

Also we make use of the following lemma Sarıgöl $\left[14\right]$.

**Lemma 3.1. **Suppose that $A=\left(a_{nv}\right)$ is an infinite matrix with complex numbers and $p=\left(p_{v}\right)$ is a bounded sequence of positive numbers such that H=\sup_{v}p_{v}~{}and\ C=max\left\{1,2^{H-1}\right\}.\Then,

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provided that

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or

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Now we begin with first theorem given the characterization of the class $(\ell_{1},\left|f(\overline{N}_{p})\right|_{k}).$

Theorem 3.2. Let $u=(u_{n})$ be a sequence of positive numbers and let $A=(a_{vj})$ be an infinite matrix. Then, $A\in(\ell_{1},\left|f(\overline{N}_{p})\right|_{k}),1\leq k<\infty,$ if and only if

[TABLE]

and

[TABLE]

where

[TABLE]

**Proof. **Necessity. Suppose $A\in(\ell_{1},\left|f(\overline{N}_{p})\right|_{k}).$ Then, $A(x)\in\left|f(\overline{N}_{p})\right|_{k}$ for all $x\in\ell_{1},$ $i.e.,$

[TABLE]

uniformly in $n.$ If we put $x=e^{j}=\left(e_{v}^{j}\right)\in\ell_{1},$ $A(e^{j})\in\left|f(\overline{N}_{p})\right|_{k},$ where $e_{v}^{j}=1$ for $v=j,$ and zero otherwise, which gives that $\left(3.2\right)$ holds. Further,  since $\ell_{1}$ is Banach space, by the Banach-Steinhaus theorem, $A:\ell_{1}\rightarrow\mathbb{C}$ is a continuous linear map. So, for fixed $n$ and $s,$

[TABLE]

is a continuous seminorm on $\ell_{1},$ which implies that $\lim_{s\rightarrow\infty}q_{sn}(x)=q_{n}(x)$ is a continuous seminorm, or, equivalently, there exists a constant $K$ such that

[TABLE]

for every $x\in\ell_{1}.$ Applying $\left(3.3\right)$ with $x=e^{j}=\left(e_{v}^{j}\right)\in\ell_{1}$ we have, for all $j,n\geq 0,$

[TABLE]

which gives $\left(3.2\right).$

Sufficiency. Suppose $\left(3.1\right)$ and $\left(3.2\right)$ hold. Given $x\in\ell_{1}.$ Then, we should show $A(x)\in\left|f(\overline{N}_{p})\right|_{k}.$ For this, it is enough to prove that

[TABLE]

By applying Minkowski’s inequality we get

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where

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On the other hand, by $\left(3.2\right),$ since for all $l,n,j\geq 0,$

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it follows that

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Hence, for every $\varepsilon>0,$ there exists an integer $j_{0}$ such that, for all $l$ and $n,$

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Also, by $\left(3.1\right),$ since

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there exists integer $l_{0}$ so that, for $l\geq l_{0}$ and all $n,$

[TABLE]

So we have

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which implies, by $\left(3.4\right),$

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This completes the proof.

In the special case $p_{m}=u_{m}=1$ for all $m\geq 0,$ $\left|f(\overline{N}_{p})\right|=\widehat{\ell}_{k}$ and so the following result follows from Theorem 3.1.

Corollary 3.3. $A\in(\ell_{1},\widehat{\ell}_{k}),1\leq k<\infty,$ if and only if

[TABLE]

and

[TABLE]

where

[TABLE]

Theorem 3.4. Let $u=(u_{n})$ be a sequence of positive numbers and let $A=(a_{vj})$ be an infinite matrix . Then, $A\in(c,\left|f(\overline{N}_{p})\right|_{k}),1\leq k<\infty,$ if and only if

[TABLE]

[TABLE]

[TABLE]

Proof. Necessity. Let $A\in(c,\left|f(\overline{N}_{p})\right|_{k}).$ Then, $A(x)\in\left|f(\overline{N}_{p})\right|_{k}$ for every $x\in c.$ Now, let $e,e^{\left(j\right)}\in c.$ Then, we have $\left(3.7\right)$ and $\left(3.8\right),$ respectively, where $e=(1,1,...).$ Also, it follows as in the proof of Theorem 3.2 that

[TABLE]

Let $N$ be arbitrary finite set of natural numbers and define a sequence $x$ by

[TABLE]

then $x\in c$ and $\left\|x\right\|=1.$ Applying $\left(3.9\right)$ with this sequence $\left(3.10\right),$ we have

[TABLE]

Hence, it is seen from Lemma 3.1 together with $p_{v}=1$ for all $v$ that $\left(3.11\right)$ is equivalent to $\left(3.6\right).$

Sufficiency. Suppose that $\left(3.6\right),\left(3.7\right)$ and $\left(3.8\right)$ hold. Given $x\in c$ and say $\lim_{j}x_{j}=\beta.$ Then, by $\left(3.7\right),$ as in Theorem 3.2,

[TABLE]

Now it is enough to show that the tail of this series tends to zero uniformly in $n.$ To see that we write

[TABLE]

[TABLE]

It is clear from $\left(3.7\right)$ that $F_{M,n}^{1}\rightarrow 0$ as $M\rightarrow\infty$ uniformly in $n$. On the other hand, since $x_{j}\rightarrow\beta,$ for any $\varepsilon>0,$ there exists an integer $j_{0}$ such that

[TABLE]

which gives us, by $\left(3.6\right),$ for all $n\geq 0,$

[TABLE]

By $\left(3.8\right),$ the first term of the equality is smaller that $\varepsilon/2$ for suffiently large $M$ and all $n.$ This means $F_{M,n}^{2}\rightarrow 0$ as $M\rightarrow\infty$ uniformly in $n$. Hence, the theorem is established.

For $p_{m}=u_{m}=1,$ Theorem 3.4 is reduced to the following result.

Corollary 3.5. Let $A=(a_{vj})$ be an infinite matrix and $\left(b(m,n,j)\right)$ be as in $\left(3.5\right).$ Then, $A\in(c,\widehat{\ell}_{k}),1\leq k<\infty,$ if and only if

[TABLE]

[TABLE]

[TABLE]