Open Access

Sequence spaces \(M(\phi)\) and \(N(\phi)\) with application in clustering

  • Mohd Shoaib Khan1,
  • Badriah AS Alamri2,
  • M Mursaleen2, 3Email author and
  • QM Danish Lohani1
Journal of Inequalities and Applications20172017:63

https://doi.org/10.1186/s13660-017-1333-z

Received: 14 November 2016

Accepted: 14 February 2017

Published: 21 March 2017

Abstract

Distance measures play a central role in evolving the clustering technique. Due to the rich mathematical background and natural implementation of \(l_{p}\) distance measures, researchers were motivated to use them in almost every clustering process. Beside \(l_{p}\) distance measures, there exist several distance measures. Sargent introduced a special type of distance measures \(m(\phi)\) and \(n(\phi)\) which is closely related to \(l_{p}\). In this paper, we generalized the Sargent sequence spaces through introduction of \(M(\phi)\) and \(N(\phi)\) sequence spaces. Moreover, it is shown that both spaces are BK-spaces, and one is a dual of another. Further, we have clustered the two-moon dataset by using an induced \(M(\phi)\)-distance measure (induced by the Sargent sequence space \(M(\phi)\)) in the k-means clustering algorithm. The clustering result established the efficacy of replacing the Euclidean distance measure by the \(M(\phi)\)-distance measure in the k-means algorithm.

Keywords

clusteringdouble sequencek-means clusteringtwo-moon dataset

MSC

40H0546A45

1 Introduction

Clustering is a well-known procedure to deal with an unsupervised learning problem appearing in pattern recognition. Clustering is a process of organizing data into groups called clusters so that objects in the same cluster are similar to one another, but are dissimilar to objects in other clusters [1]. The main contribution in the field of clustering analysis was the pioneering work of MacQueen [1] and Bezdek [2]. They had introduced highly significant clustering algorithms such as k-means [1] and fuzzy c-means [2]. Among all clustering algorithms, k-means is the simplest unsupervised clustering algorithm that makes use of a minimum distance from the center, and it has many applications in scientific and industrial research [36] (for more information about the k-means clustering algorithm, see Section 5). K-means algorithm is distance dependent, so its outputs vary with changing distance measures. Among all distance measures, a clustering process was usually carried out through the Euclidean distance measure [7], but many times it failed to offer good results. In this paper, we define \(M(\phi)\)- and \(N(\phi)\)-distance measure. Further, \(M(\phi )\)-distance is used to cluster two-moon dataset. The output result is compared with the result of Euclidean distance measure to show the efficacy of \(M(\phi)\)-distance over the Euclidean distance measure. \(M(\phi )\) and \(N(\phi)\)-distance measures are the generalization of \(m(\phi)\)- and \(n(\phi)\)-distance measures introduced by Sargent [8] and further studied by Mursaleen [9, 10] (to know more about \(m(\phi )\) and \(n(\phi)\), refer to [810]). The \(M(\phi)\) and \(N(\phi)\) spaces are closely related to \(l_{p}\) distance measures. \(l_{p}\) measures and its variance are mostly used to solve the problems evolving in the fields of Market prediction [11], Machine Learning [12], Pattern Recognition [13], Clustering [20] etc.

Throughout the paper, by ω we denote the set of all real or complex sequences. Moreover, by \(l_{\infty}\), c and \(c_{0}\) we denote the Banach spaces of bounded, convergent and null sequences, respectively; and let \(l_{p}\) be the Banach space of absolutely p-summable sequences with p-norm \({\Vert \cdot \Vert } _{p}\). For the following notions, we refer to [14, 15]. A double sequence \(x = (x_{jk})\) of real or complex numbers is said to be bounded if \(\Vert x \Vert _{\infty} < \infty\), the space of all bounded double sequences is denoted by \(\mathcal{L}_{\infty}\). A double sequence \(x = (x_{jk})\) is said to converge to the limit L in Pringsheim’s sense (shortly, convergent to L) if for every \(\varepsilon> 0\), there exists an integer N such that \(\vert x_{jk} - L\vert < \varepsilon\) whenever \(j,k > N\). In this case L is called the p-limit of x. If in addition \(x \in\mathcal{L}_{\infty}\), then x is said to be boundedly convergent to L in Pringsheim’s sense (shortly, bp-convergent to L). A double sequence \(x = (x_{jk})\) is said to converge regularly to L (shortly, r-convergent to L) if x is p-convergent and the limits \(x_{j}: = \lim_{k}x_{jk}\) (\(j \in\mathbb {N} \)) and \(x^{k}: = \lim_{j}x_{jk}\) (\(k \in\mathbb{N} \)) exist. Note that in this case the limits \(\lim_{j}\lim_{k}x_{jk}\) and \(\lim_{k}\lim_{j}x_{jk}\) exist and are equal to the p-limit of x. In general, for any notion of convergence ν, the space of all ν-convergent double sequences will be denoted by \(\mathcal{C}_{\nu}\) and the limit of a ν-convergent double sequence x by \(\nu\textrm{-} \lim_{j,k}x_{jk}\), where \(\nu\in\{ p,\mathit{bp},r\}\).

Let Ω denote a vector space of all double sequences with the vector space operations defined coordinate-wise. Vector subspaces of Ω are called double sequence spaces. Let us consider a double sequence \(x = \{ x_{mn}\}\) and define the sequence \(s = \{ s_{mn}\}\) via x by
$$s_{mn}: = \sum_{i,j}^{m,n} x_{ij}\quad (m,n \in\mathbb{N} ). $$
Then the pair \((x,s)\) and the sequence \(s = \{ s_{mn}\}\) are called a double series and a sequence of partial sums of the double series, respectively. Let λ be the space of double sequences converging with respect to some linear convergence rule \(\mu\textrm{-} \lim:\lambda\to \mathbb{R}\). The sum of a double series \(\sum_{i,j = 1}^{\infty,\infty} x_{ij}\) with respect to this rule is defined by \(\mu\textrm{-} \sum_{i,j = 1}^{\infty,\infty} x_{ij}: = \mu\textrm{-} \lim s_{mn}\). Başar and Şever introduced the space \(L_{p}\) in [16]
$$L_{p}: = \biggl\{ \{ x_{mn}\} \in\Omega:\sum _{m,n} \vert x_{mn}\vert ^{p} < \infty \biggr\} \quad (1 \le p < \infty) $$
corresponding to the space \(l_{p}\) for \(p \ge1\) and examined some of its properties. Altay and Başar [17] have generalized the spaces of double sequences \(L_{\infty}\), \(C_{p}\) and \(C_{\mathit{bp}}\) to
$$\begin{aligned}& L_{\infty} (t) = \Bigl\{ \{ x_{mn}\} \in\Omega:\sup _{m,n \in\mathbb{N}} \vert x_{mn}\vert ^{t_{mn}} < \infty \Bigr\} , \\& C_{p}(t) = \Bigl\{ \{ x_{mn}\} \in\Omega:p \textrm{-} \lim _{m,n \to \infty} \vert x_{mn} - \ell \vert ^{t_{mn}} = 0 \Bigr\} , \end{aligned}$$
and
$$C_{\mathit{bp}}(t) = C_{p}(t) \cap L_{\infty} (t), $$
respectively, where \(t = \{ t_{mn}\}\) is the sequence of strictly positive reals \(t_{mn}\). In the case \(t_{mn} = 1\), for all \(m,n \in\mathbb{N}\), \(L_{\infty} (t)\), \(C_{p}(t)\) and \(C_{\mathit{bp}}(t)\) reduce to the sets \(L_{\infty}\), \(C_{p}\) and \(C_{\mathit{bp}}\), respectively. Further, let C be the space whose elements are finite sets of distinct positive integers. Given any element σ of C, we denote by \(c(\sigma)\) the sequence \(\{ c_{n}(\sigma)\}\) which is such that \(c_{n}(\sigma) = 1\) if \(n \in \sigma\), \(c_{n}(\sigma) = 0\) otherwise. Further, let
$$C_{s} = \Biggl\{ \sigma\in C:\sum_{n = 1}^{\infty} c_{n}(\sigma) \le s \Biggr\} $$
be the set of those σ whose support has cardinality at most s, and
$$\Phi= \biggl\{ \phi= \{ \phi_{n}\} \in\omega: \phi_{1} > 0,\Delta\phi_{n} \ge0\mbox{ and }\Delta \biggl( \frac{\phi_{n}}{n} \biggr) \le0\ (n = 1,2, \ldots) \biggr\} , $$
where \(\Delta\phi_{n} = \phi_{n} - \phi_{n - 1}\) and \(\phi _{0} = 0\).
For \(\phi\in\Phi\), the following sequence spaces were introduced and studied in [8] by Sargent and further studied by Mursaleen in [9, 10]:
$$m(\phi) = \biggl\{ x = \{ x_{n}\} \in\omega:\sup _{s \ge 1} \sup_{\sigma\in C_{s}} \biggl( \frac{1}{\phi_{s}} \sum_{n \in\sigma} \vert x_{n}\vert \biggr) < \infty \biggr\} , $$
and
$$n(\phi) = \Biggl\{ x = \{ x_{n}\} \in\omega:\sup _{u \in S(x)} \Biggl( \sum_{m,n = 1,1}^{\infty,\infty} \vert u_{n}\vert \Delta\phi_{n} \Biggr) < \infty \Biggr\} . $$

Remark 1.1

  1. (i)

    The spaces \(m(\phi)\) and \(n(\phi)\) are BK-spaces with their usual norms.

     
  2. (ii)

    If \(\phi_{n} = 1\) (\(n = 1,2,3,\ldots\)), then \(m(\phi) = l_{1}\) [\(n(\phi) = l_{\infty} \)], and if \(\phi_{n} = n\) (\(n = 1,2,3,\ldots\)), then \(m(\phi) = l_{\infty}\) [\(n(\phi) = l_{1} \)].

     
  3. (iii)

    \(l_{1} \subseteq m(\phi) \subseteq l_{\infty} \) [\(l_{1} \subseteq n(\phi) \subseteq l_{\infty} \)] for all \(\phi\in\Phi\).

     
  4. (iv)

    For any \(\phi\in\Phi\), \(m(\phi)\neq l_{p}\) [\(n(\phi)\neq l_{q} \)], \(1 < p < \infty\).

     
In this paper, we define Sargent’s spaces for double sequences \(x = \{ x_{mn}\}\). For this we first suppose U to be the set whose elements are finite sets of distinct elements of \(\mathbb{N} \times\mathbb{N}\) obtained by \(\sigma\times\varsigma\), where \(\sigma\in C_{s}\) and \(\varsigma\in C_{t}\) for each \(s,t \ge1\). Therefore, any element ζ of U means \((m,n)\); \(m \in\sigma\) and \(n \in\varsigma\) having cardinality at most st, where s is the cardinality with respect to m and t is the cardinality with respect to n. Given any element ζ of U, we denote by \(c(\zeta)\) the sequence \(\{ c_{mn}(\zeta)\}\) such that
$$c_{mn}(\zeta) = \textstyle\begin{cases} 1; & \mbox{if } (m,n) \in\zeta,\\ 0; & \mbox{otherwise}. \end{cases} $$
Further, we write
$$U_{st} = \Biggl\{ \zeta\in U:\sum_{m,n = 1}^{\infty,\infty} c_{mn}(\zeta) \le st \Biggr\} $$
for the set of those ζ whose support has cardinality at most st; and
$$\Theta= \biggl\{ \phi= \{ \phi_{mn}\} \in\Omega: \phi_{11} > 0,\Delta_{11}\phi_{mn} \ge0\mbox{ and } \Delta_{11} \biggl( \frac{\phi_{mn}}{mn} \biggr) \le0\ (m,n = 1,2, \ldots) \biggr\} , $$
where \(\Delta_{11}\phi_{mn} = \phi_{mn} - \phi_{m - 1,n} - \phi_{m,n - 1} + \phi_{m - 1,n - 1}\) and \(\phi_{00}\), \(\phi_{0m}\), \(\phi_{n0} = 0\), \(\forall m,n \in\mathbb{I}^{ +}\). Throughout the paper, we write \(\sum_{m,n \in\zeta}\) for \(\sum_{m \in \sigma} \sum_{n \in\varsigma}\), and \(S(x)\) is used to denote the set of all double sequences that are rearrangements of \(x = \{ x_{mn}\} \in \Omega\). For \(\phi\in\Theta\), we define the following sequence spaces:
$$M(\phi) = \biggl\{ x = \{ x_{mn}\} \in\Omega: \Vert x \Vert _{M(\phi )} = \sup_{s,t \ge1}\sup_{\zeta\in U_{st}} \biggl( \frac{1}{\phi_{st}}\sum_{m,n \in\zeta} \vert x_{mn} \vert \biggr) < \infty \biggr\} $$
and
$$N(\phi) = \Biggl\{ x = \{ x_{mn}\} \in\Omega: \Vert x \Vert _{N(\phi)} = \sup_{u \in S(x)} \Biggl(\sum _{m,n = 1}^{\infty,\infty} \vert u_{mn}\vert \Delta_{11}\phi_{mn} \Biggr) < \infty \Biggr\} . $$
Then the distances between \(x = \{ x_{mn}\}\) and \(y = \{ y_{mn}\}\) induced by \(M(\phi)\) and \(N(\phi)\) can be expressed as
$$d_{M(\phi)} = \sup_{s,t \ge1}\sup_{\zeta \in U_{st}} \biggl( \frac{1}{\phi_{st}}\sum_{m,n \in\zeta} \vert x_{mn} - y_{mn}\vert \biggr) $$
and
$$d_{N(\phi)} = \sup_{u,v \in S(x)} \Biggl( \sum _{m,n = 1}^{\infty,\infty} \vert u_{mn} - v_{mn} \vert \Delta_{11}\phi_{mn} \Biggr). $$

Remark 1.2

If \(\phi_{st} = 1\) (\(s,t = 1,2,3,\ldots\)), then \(M(\phi) = L_{1}\) [\(N(\phi) = L_{\infty} \)], and if \(\phi_{st} = st\) (\(s,t = 1,2,3,\ldots\)), then \(M(\phi) = L_{\infty}\) [\(N(\phi) = L_{1}\)].

We now state the following known results of [18] for single sequences (series) which can also be proved easily for double sequences (series).

Lemma 1.1

If the series \(\sum u_{n}x_{n}\) is convergent for every x of a BK-space E, then the functional \(\sum_{n = 1}^{\infty} u_{n}x_{n}\) is linear and continuous in E.

Lemma 1.2

If E and F are BK-spaces, and if \(E \subseteq F\), then there is a real number K such that, for all x of E,
$$\Vert x \Vert _{F} \le K\Vert x \Vert _{E}. $$

2 Properties of the spaces \(M(\phi)\) and \(N(\phi)\)

Theorem 2.1

The space \(M(\phi)\) is a BK-space with the norm.
$$ \Vert x \Vert _{M(\phi)} = \sup_{s,t \ge 1}\sup _{\zeta\in U_{st}} \frac{1}{\phi_{st}} \biggl( \sum _{m,n \in\zeta} \vert x_{mn}\vert \biggr). $$
(2.1)

Proof

It is a routine verification to show that \(M(\phi)\) is a normed space with the given norm (2.1), and so we omit it. Now, we proceed to showing that \(M(\phi)\) is complete. Let \(\{ x^{l}\}\) be a Cauchy sequence in \(M(\phi)\), where \(x^{l} = \{ x_{mn}^{l}\}_{m,n = 1,1}^{\infty,\infty}\) for every fixed \(l \in\mathbb{N}\). Then, for a given \(\varepsilon> 0\), there exists a positive integer \(n_{0}(\varepsilon) > 0\) such that
$$\bigl\Vert x^{l} - x^{r} \bigr\Vert _{M(\phi)} = \sup_{s,t \ge 1}\sup_{\zeta\in U_{st}}\frac{1}{\phi_{st}} \biggl( \sum_{m,n \in\zeta} \bigl\vert x_{mn}^{l} - x_{mn}^{r} \bigr\vert \biggr) < \varepsilon $$
for all \(l,r > n_{0}(\varepsilon)\), which yields, for each fixed \(s,t \ge 1\) and \(\zeta\in U_{st}\),
$$ \sum_{m,n \in\zeta} \bigl\vert x_{mn}^{l} - x_{mn}^{r} \bigr\vert \le\varepsilon \phi_{11}\quad \mbox{for all }l,r > n_{0}(\varepsilon). $$
(2.2)
Therefore
$$ \biggl\vert \sum_{m,n \in\zeta} \bigl\vert x_{mn}^{l} \bigr\vert - \sum_{m,n \in\zeta} \bigl\vert x_{mn}^{r} \bigr\vert \biggr\vert < \varepsilon\phi_{11} \quad \mbox{for all }l,r > n_{ 0} ( \varepsilon). $$
(2.3)
This means that \(\{ \sum_{m,n \in\zeta} \vert x_{mn}^{l}\vert \} _{l \in\mathbb{N}}\) is a Cauchy sequence in \(\mathbb{R}\) for every fixed \(s,t \ge1\) and \(\zeta \in U_{st}\). Since \(\mathbb{R}\) is complete, it converges, say
$$\sum_{m,n \in\zeta} \bigl\vert x_{mn}^{l} \bigr\vert \to\sum_{m,n \in\zeta} \vert x_{mn} \vert \quad \mbox{as }l \to\infty. $$
Since absolute convergence implies convergence in \(\mathbb{R}\), hence
$$ \sum_{m,n \in\zeta} x_{mn}^{l} \to\sum _{m,n \in\zeta} x_{mn} \quad \mbox{as } l \to\infty. $$
(2.4)
Hence we have
$$ \lim_{l \to\infty} \biggl\Vert \sum_{m,n \in\zeta} x_{mn}^{l} - \sum_{m,n \in\zeta} x_{mn} \biggr\Vert _{M(\phi)} = 0. $$
(2.5)
Let \(y^{l} = \sum_{m,n \in\zeta} \vert x_{mn}^{l}\vert \). Then \(\{ y^{l}\} \in l_{\infty}\). Therefore
$$\sup_{l \in\mathbb{N}} \sum_{m,n \in\zeta} \bigl\vert x_{mn}^{l} \bigr\vert \le k. $$
Since \(\sum_{m,n \in\zeta} \vert x_{mn}\vert \le\sum_{m,n \in \zeta} \vert x_{mn} - x_{mn}^{l}\vert + \sum_{m,n \in\zeta} \vert x_{mn}^{l}\vert \le\varepsilon \phi_{11} + k\), it follows that \(x = \{ x_{mn}\} \in M(\phi)\). Since \(\{ x^{l}\}_{l \in\mathbb{N}}\) was an arbitrary Cauchy sequence, the space \(M(\phi)\) is complete. Now we prove that \(M(\phi)\) has continuous coordinate projections \(p_{mn}\), where \(p_{mn}:\Omega\to K\) and \(p_{mn}(x) = x_{mn}\). The coordinate projections \(p_{mn}\) are continuous since \(\vert x_{mn}\vert \le\sup_{s,t \ge1}\sup_{\zeta\in U_{st}}\phi_{st}\Vert x \Vert _{M(\phi)}\) for each \(m,n \in \mathbb{N}\). □

Remark 2.1

The space \(N(\phi)\) is a BK-space with the norm
$$\Vert x \Vert _{N(\phi)} = \sup_{u \in S(x)} \Biggl( \sum _{m,n = 1}^{\infty,\infty} \vert u_{mn}\vert \Delta_{11}\phi_{mn} \Biggr). $$

Lemma 2.1

  1. (i)

    If \(x \in M(\phi)\) [\(x \in N(\phi) \)] and \(u \in S(x)\), then \(u \in M(\phi)\) [\(u \in N(\phi) \)] and \(\Vert u\Vert = \Vert x\Vert \).

     
  2. (ii)

    If \(x \in M(\phi)\) [\(x \in N(\phi)\)] and \(\vert u_{mn}\vert \le \vert x_{mn}\vert \) for every positive integer m, n, then \(u \in M(\phi)\) [\(u \in N(\phi)\)] and \(\Vert u\Vert \le \Vert x\Vert \).

     

Proof

(i) Let \(x \in M(\phi)\), then \(\sup_{s,t \ge1} \sup_{\zeta\in U_{st}}\frac{1}{\phi_{st}}\sum_{m,n \in\zeta} \vert x_{mn}\vert < \infty\). So, we have
$$\frac{1}{\phi_{st}}\sum_{m,n \in\zeta} \vert x_{mn} \vert < \infty \quad \mbox{for each }\zeta\in U_{st}\mbox{ and }s,t \ge1. $$
Since the sum of a finite number of terms remains the same for all the rearrangements,
$$\frac{1}{\phi_{st}}\sum_{m,n \in\zeta} \vert u_{mn} \vert = \frac{1}{\phi_{st}}\sum_{m,n \in\zeta} \vert x_{mn}\vert \quad \mbox{for each }u \in S(x)\mbox{ and }\zeta\in U_{st}, s,t \ge1. $$
Hence
$$\sup_{s,t \ge1}\sup_{\zeta\in U_{st}}\frac{1}{\phi_{st}}\sum _{m,n \in\zeta} \vert u_{mn}\vert = \sup _{s,t \ge1} \sup_{\zeta\in U_{st}}\frac{1}{\phi_{st}}\sum _{m,n \in\zeta} \vert x_{mn}\vert < \infty, $$
thus \(u \in M(\phi)\) and \(\Vert u\Vert = \Vert x\Vert \).

(ii) By using the definition, easy to prove. □

Theorem 2.2

For arbitrary \(\phi\in\Theta\), we have \(\Delta_{11}\phi\in M(\phi)\) and \(\Vert \Delta_{11}\phi \Vert _{M(\phi)} \le2\).

Proof

Let s and t be arbitrary positive integers, let \(\sigma,\varsigma\in U_{st}\), and let \(\tau_{1}\), \(\tau_{2}\) constitute the element of σ and ς exceed by s and t respectively, also from the definition we have \(\Delta_{11}\phi\ge0\) and \(\Delta_{11} ( \frac{\phi_{mn}}{mn} ) \le0\). Then
$$\begin{aligned} \sum_{n \in\sigma,m \in\varsigma} \vert \Delta_{11} \phi_{mn} \vert \le& \sum_{n = 1,m = 1}^{s,t} \Delta_{11}\phi_{mn} + \sum_{n \in \tau_{1},m \in\tau_{2}} \Delta_{11}\phi_{mn} \\ \leq& \phi_{st} + \sum_{n \in\tau_{1},m \in\tau_{2}} \biggl( \frac{\phi_{m - 1,n - 1}}{(m - 1)(n - 1)} \biggr) \\ \leq & \phi_{st} + \left . \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \frac{\phi_{st}}{st} & + \frac{\phi_{s,t + 1}}{s(t + 1)} & + \frac{\phi_{s,t + 2}}{s(t + 2)} & +. &. \\ \frac{\phi_{s + 1,t}}{(s + 1)t} & + \frac{\phi_{s + 1,t + 1}}{(s + 1)(t + 1)} & + \frac{\phi_{s + 1,t + 2}}{(s + 1)(t + 2)} & +. &. \\ + & + & + &. & \\ . &. &. &. & \end{array}\displaystyle \right \} \quad \max(st) \textrm{-terms} \\ \leq& \phi_{st} + st\frac{\phi_{st}}{st} = 2\phi_{st}. \end{aligned}$$
 □

Lemma 2.2

If \(x \in M(\phi)\) and \(\{ c_{11},c_{12}, \ldots,c_{1n},c_{21},c_{22}, \ldots,c_{2n}, \ldots,c_{m1},c_{m2}, \ldots,c_{mn}\}\) is a rearrangement of \(\{ b_{11},b_{12}, \ldots,b_{1n},b_{21},b_{22}, \ldots,b_{2n}, \ldots,b_{m1},b_{m2}, \ldots,b_{mn}\}\) such that \(\vert c_{11}\vert \ge \vert c_{12}\vert \ge \cdots\ge \vert c_{1n}\vert \), \(\vert c_{21}\vert \ge \vert c_{22}\vert \ge\cdots\ge \vert c_{2n}\vert ,\dots, \vert c_{m1}\vert \ge \vert c_{m2}\vert \ge\cdots\ge \vert c_{mn}\vert \) and \(\vert c_{11}\vert \ge \vert c_{21}\vert \ge\cdots\ge \vert c_{m1}\vert \), \(\vert c_{12}\vert \ge \vert c_{22}\vert \ge\cdots \ge \vert c_{m2}\vert ,\vert c_{n1}\vert \ge \vert c_{n2}\vert \ge\cdots\ge \vert c_{nm}\vert \), then
$$\sum_{i,j = 1,1}^{m,n} \vert b_{ij}x_{ij} \vert \le \Vert x\Vert _{M(\phi )}\sum_{i,j = 1,1}^{m,n} \vert c_{ij}\vert \Delta_{11}\phi_{ij}. $$

Proof

In view of Lemma 2.1(i), it is sufficient to consider the case when \(b_{ij} = c_{ij}\) (\(i = 1,2, \ldots,m\); \(j = 1,2, \ldots,n\)). Then writing \(X_{mn} = \sum_{i,j = 1}^{m,n} \vert x_{ij}\vert \), we get
$$\begin{aligned} \sum_{i,j = 1,1}^{m,n} \vert b_{ij}x_{ij} \vert =& \sum_{i = 1}^{m - 1} \sum _{j = 1}^{n - 1} \bigl( \vert c_{ij}\vert - \vert c_{i,j + 1}\vert - \vert c_{i + 1,j}\vert +\vert c_{i + 1,j + 1}\vert \bigr)X_{ij} +\vert c_{mn}\vert X_{mn} \\ \le& \Vert x\Vert _{M(\phi)}\sum_{i = 1}^{m - 1} \sum_{j = 1}^{n - 1} \bigl( \vert c_{ij}\vert - \vert c_{i,j + 1}\vert - \vert c_{i + 1,j}\vert + \vert c_{i + 1,j + 1}\vert \bigr) \phi_{ij} + \Vert x\Vert _{M(\phi)}\vert c_{mn} \vert X_{mn} \\ =& \Vert x\Vert _{M(\phi)}\sum_{i,j = 1,1}^{m,n} \vert c_{ij}\vert \Delta_{11}\phi_{ij}. \end{aligned}$$
Hence we have \(\sum_{i,j = 1,1}^{m,n} \vert b_{ij}x_{ij}\vert \le \Vert x\Vert _{M(\phi)}\sum_{i,j = 1,1}^{m,n} \vert c_{ij}\vert \Delta_{11}\phi_{ij}\). □

Theorem 2.3

In order that \(\sum u_{ij}x_{ij}\) be convergent [absolutely convergent] whenever \(x \in M(\phi)\), it is necessary and sufficient that \(u \in N(\phi)\). Further, if \(x \in M(\phi)\) and \(u \in N(\phi)\), then
$$ \sum_{i,j = 1,1}^{\infty,\infty} \vert u_{ij}x_{ij} \vert \le \Vert u\Vert _{N(\phi)} \Vert x\Vert _{M(\phi)}. $$
(2.6)

Proof

Necessity. We now suppose that \(\sum u_{ij}x_{ij}\) is convergent whenever \(x \in M(\phi)\), then from Lemma 1.1 we have
$$\Biggl\vert \sum_{i,j = 1,1}^{\infty,\infty} u_{ij}x_{ij} \Biggr\vert \le K \Vert x\Vert _{M(\phi)} $$
for some real number K and all x of \(M(\phi)\). In view of Lemma 2.1(ii), we may replace \(x_{ij}\) by \(x_{ij} \operatorname{sgn} \{ u_{ij}\}\), obtaining
$$ \sum_{i,j = 1,1}^{\infty,\infty} \vert u_{ij}x_{ij} \vert \le K \Vert x\Vert _{M(\phi)}. $$
(2.7)
Let \(v \in S(u)\). Then taking x to be a suitable rearrangement of \(\Delta_{11}\phi\), it follows from Eq. (2.7) and Theorem 2.2 and Lemma 2.1(i) that
$$\sum_{i,j = 1,1}^{\infty,\infty} \vert v_{ij} \vert \Delta_{11}\phi_{ij} \le4K, $$
and thus \(u \in N(\phi)\).
Sufficiency. If \(x \in M(\phi)\) and \(u \in N(\phi)\), it follows from Lemma 2.2 that for every positive integer m and n,
$$\sum_{i,j = 1,1}^{\infty,\infty} \vert u_{ij}x_{ij} \vert \le \Vert u\Vert _{N(\phi)} \Vert x\Vert _{M(\phi)}. $$
 □

Theorem 2.4

In order that \(\sum u_{mn}x_{mn}\) be convergent [absolutely convergent] whenever \(x \in N(\phi)\), it is necessary and sufficient that \(u \in M(\phi)\).

Proof

Since sufficiency is included in Theorem 2.3, we only consider necessity. We therefore suppose that \(\sum u_{mn}x_{mn}\) is convergent whenever \(x \in N(\phi)\). By arguments similar to those used in Theorem 2.3, we may therefore have that
$$ \sum_{m,n = 1,1}^{\infty,\infty} \vert u_{mn}x_{mn} \vert \le K \Vert x\Vert _{N(\phi)} $$
(2.8)
for some real number K and all x of \(N(\phi)\). Let \(x = c(\zeta)\), where \(\zeta\in U_{st}\). Then \(x \in N(\phi)\), and
$$\Vert x\Vert _{N(\phi)} = \sup_{\xi\in U_{st}}\sum _{m,n \in\xi} \Delta_{11}\phi_{mn} \le4 \phi_{st}, $$
from Theorem 2.2 and Eq. (2.8) we have
$$\sum_{m,n \in\zeta} \vert u_{mn}\vert \le4K \phi_{st}\quad (\zeta\in U_{st}; s,t = 1,2,3, \ldots), $$
and thus \(u \in M(\phi)\). □

3 Inclusion relations for \(M(\phi)\) and \(N(\phi)\)

Lemma 3.1

In order that \(M(\phi) \subseteq M(\psi)\) [\(N(\phi) \supseteq N(\psi)\)], it is necessary and sufficient that
$$\sup_{s,t \ge1} \biggl( \frac{\phi_{st}}{\psi_{st}} \biggr) < \infty. $$

Proof

Since each of the spaces \(M(\phi)\) and \(N(\phi )\) is the dual of the other, by Theorems 2.3 and 2.4, the second version is equivalent to the first. Moreover, sufficiency follows from the definition of an \(M(\phi)\) space. We therefore suppose that \(M(\phi) \subseteq M(\psi)\). Since \(\Delta\phi\in M(\phi)\), it follows that \(\Delta \psi\in M(\psi)\), and hence we find that, for every positive integer \(s,t \ge1\),
$$\phi_{st} = \sum_{i,j = 1,1}^{s,t} \Delta_{11}\phi_{ij} \le\psi_{st} \Vert \Delta \phi \Vert _{M(\psi)},\quad \mbox{where }\Delta= \Delta_{11}. $$
 □

Theorem 3.1

  1. (i)

    \(L_{1} \subseteq M(\phi) \subseteq L_{\infty}\) [\(L_{1} \subseteq N(\phi) \subseteq L_{\infty} \)] for all ϕ of Θ.

     
  2. (ii)

    \(M(\phi) = L_{1}\) [\(N(\phi) = L_{\infty} \)] if and only if \(\mathit{bp}\textit{-}\lim_{s,t}\phi_{st} < \infty\).

     
  3. (iii)

    \(M(\phi) = L_{\infty}\) [\(N(\phi) = L_{1}\)] if and only if \(\mathit{bp}\textit{-}\lim_{s,t}(\phi_{st}/st) > 0\).

     

Proof

We prove here the first version, while the second version follows by Theorems 2.3 and 2.4. Since \(\phi_{11} \le\phi_{mn} \le mn\phi_{mn}\) for all ϕ of Θ, we have by Lemma 3.1 that (i) is satisfied. Further, from Lemma 3.1, it follows that \(M(\phi) \subseteq L_{1}\) if and only if \(\sup_{s,t \ge 1}\phi_{st} < \infty\), while \(L_{\infty} \subseteq M(\phi)\) if and only if \(\sup_{s,t \ge1}(\phi_{st}/st) < \infty\); since the sequences \(\{ \phi_{st}\}\) and \(\{ st/\phi_{st}\}\) are monotonic, (ii) and (iii) are also satisfied. □

Theorem 3.2

Suppose that \(1 < p < \infty\) and \(\frac{1}{p} + \frac{1}{q} = 1\). Then
  1. (i)

    Given any ϕ of Θ, \(M(\phi)\neq L_{p}\) [\(N(\phi )\neq L_{q}\)].

     
  2. (ii)

    In order that \(L_{p} \subset M(\phi)\) [\(N(\phi) \subset L_{q}\)], it is necessary and sufficient that \(\sup_{s,t \ge1} ( \frac{(st)^{1/q}}{\phi_{st}} ) < \infty\).

     
  3. (iii)

    In order that \(M(\phi) \subset L_{p}\) [\(N(\phi) \supset L_{q}\)], it is necessary and sufficient that \(\Delta\phi\in L_{p}\).

     
  4. (iv)

    \(\bigcup_{\Delta\phi\in L_{p}} M(\phi) = L_{p}\) [\(\bigcap_{\Delta\phi\in L_{p}} N(\phi) = L_{q} \)].

     

Proof

(i) Let us suppose that \(M(\phi) = L_{p}\).

Then, by Lemma 1.2, there exist real numbers \(r_{1}\) and \(r_{2}\) (\(r_{1} > 0\), \(r_{2} > 0\)) such that, for all x of \(M(\phi)\),
$$r_{1} \Vert x\Vert _{L_{p}} \le \Vert x\Vert _{M(\phi)} \le r_{2} \Vert x\Vert _{L_{p}}. $$
Taking \(x = c(\zeta)\), where \(\zeta\in U_{st}\), we have that
$$r_{1}(st)^{\frac{1}{p}} \le\frac{st}{\phi_{st}} \le r_{2}(st)^{\frac{1}{p}}\quad (s,t = 1,2,3, \ldots), $$
and hence that
$$r_{1} \le\frac{(st)^{\frac{1}{q}}}{\phi_{st}} \le r_{2} \quad (s,t = 1,2,3, \ldots). $$
In view of Lemma 3.1, this implies that \(M(\phi) = M(\psi)\), where \(\psi= \{ (mn)^{\frac{1}{q}}\}\). Since \(\Delta\psi\in M(\psi)\) by Theorem 2.2, but \(\Delta\psi\notin L_{q}\), this leads to a contradiction. Hence (i) follows.
(ii) If \(L_{q} \subset M(\phi)\), arguments similar to those used in the proof of (i) show that
$$ (st)^{1/q} \le K\phi_{st} \quad (s,t = 1,2,3, \ldots). $$
(3.1)
For sufficiency, we suppose that (3.1) is satisfied. Then, whenever \(x \in L_{p}\) and \(\zeta\in U_{st}\),
$$\sum_{m,n \in\zeta} \vert x_{mn}\vert \le \biggl( \sum_{m,n \in\zeta} \vert x_{mn}\vert ^{p} \biggr)^{\frac{1}{p}} \biggl( \sum_{m,n \in\zeta} 1 \biggr)^{\frac{1}{q}} \le \Vert x\Vert _{L_{q}}(st)^{\frac{1}{q}} < K\phi_{st}\Vert x\Vert _{L_{q}}, $$
and hence \(x \in M(\phi)\). In view of (i), it follows that \(L_{q} \subset M(\phi)\).

(iii) By Theorem 2.2, we have \(\Delta\phi\in M(\phi)\). For sufficiency, we suppose that \(\Delta\phi\in L_{p}\) and that \(x \in M(\phi)\). Then \(\{ u_{mn}\Delta_{11}\phi_{mn}\} \in L_{1}\) whenever \(u \in L_{q}\), and it therefore follows from Lemma 2.2 that \(\{ u_{mn}x_{mn}\} \in L_{1}\) whenever \(u \in L_{q}\). Since \(L_{p}\) is the dual of \(L_{q}\) and since \(M(\phi)\neq L_{p}\), it follows that \(M(\phi) \subset L_{q}\).

(iv) By using (iii) we have \(\bigcup_{\Delta\phi\in L_{p}} M_{\phi} \subseteq L_{p}\). Now, for obtaining the complementary relation \(L_{p} \subseteq\bigcup_{\Delta\phi\in L_{p}} M_{\phi}\), let us suppose that \(x \in L_{p}\). Then \(\lim_{m,n \to\infty} x_{mn} = 0\), and hence there is an element u of \(S(x)\) such that \(\{ \vert u_{mn}\vert \}\) is a non-increasing sequence. If we take \(\psi= \{ \sum_{i,j = 1,1}^{m,n} \vert u_{ij}\vert \}\), then it is easy to verify that \(\psi\in \Theta\) and that \(x \in M(\phi)\). Since \(\Delta\psi\in L_{p}\), the complementary relation is satisfied. □

4 Application of \(M(\phi)\) and \(N(\phi)\) in clustering

In this section, we implement a k-means clustering algorithm by using \(M(\phi)\)-distance measure. Further, we apply the k-means algorithm into clustering to cluster two-moon data. The clustering result obtained by the \(M(\phi)\)-distance measure is compared with the results derived by the existing Euclidean distance measures (\(l_{2}\)).

4.1 Algorithm to compute \(M(\phi)\) distance

Let \(x = [x_{1},x_{2},x_{3}, \ldots,x_{n}]_{1 \times n}\) and \(y = [y_{1},y_{2},y_{3}, \ldots,y_{n}]_{1 \times n}\) be two matrices of size \(1 \times n\), and let \(\phi_{m,n} = \phi_{1,n} = n\).
  1. (1)

    Calculate \(a_{i} = \frac{1}{\phi_{1,i}}\vert x_{i} - y_{i}\vert \), \(i = 1,2,3, \ldots,n\).

     
  2. (2)
    The \(M(\phi)\)-distance between x and y is d, where
    $$d = \max\{ a_{1},a_{1} + a_{2}, \ldots,a_{1} + a_{2} + \cdots+ a_{n}\}. $$
     

4.2 K-means clustering algorithm for \(M(\phi)\)-distance measure

Let \(X = [x_{1},x_{2},x_{3},\ldots,x_{n}]\) be the data set.
  1. (1)

    Randomly/judiciously select k cluster centers (in this paper we choose first k data points as the cluster center \(y = [x_{1},x_{2},\ldots,x_{k}]\)).

     
  2. (2)

    By using \(M(\phi)\) or \(N(\phi)\) distance measure (since both are dual of each other, in application point of view, we only consider \(M(\phi)\)), compute the distance between each data points and cluster centers.

     
  3. (3)

    Put data points into the cluster whose \(M(\phi)\)-distance with its center is minimum.

     
  4. (4)

    Define cluster centers for the new clusters evolved due to steps 1-3, the new cluster centers are computed as follows: \(c_{i} = \frac{1}{k_{i}}\sum_{j = 1}^{k_{i}} x_{i}\), where \(k_{i}\) denotes the number of points in the ith cluster.

     
  5. (5)

    Repeat the above process until the difference between two consecutive cluster centers reaches less than a small number ε.

     

4.3 Two-moon dataset clustering by using \(M(\phi)\)-distance measure in k-means algorithm

Two-moon dataset is a well-known nonconvex data set. It is an artificially designed two dimensional dataset consisting of 373 data points [19]. Two-moon dataset is visualized as moon-shaped clusters (see Figure 1).
Figure 1

Original shape of two-moon dataset.

By using \(M(\phi)\)-distance measure in the k-means clustering algorithm, the obtained result is represented in Figure 2. In Figure 3, we represent the result obtained by using the Euclidean distance measure in the k-means algorithm (we measure the accuracy of the cluster by using the formula, accuracy = (number of data points in the right cluster/total number of data points)). The experimental result shows that cluster accuracy of \(M(\phi)\)-distance measure is 84.72% while \(l_{2}\)-distance measure’s clustering accuracy is 78.55%. Thus, \(M(\phi)\)-distance measure substantially improves the clustering accuracy.
Figure 2

Clustering induced by \(\pmb{M(\phi)}\) -distance measure.

Figure 3

Clustering induced by Euclidean distance \(\pmb{l_{2}}\) .

5 Conclusions

In this paper, we defined Banach spaces \(M(\phi)\) and \(N(\phi)\) with discussion of their mathematical properties. Further, we proved some of their inclusion relation. Furthermore, we applied the distance measure induced by the Banach space \(M(\phi)\) into clustering to cluster the two-moon data by using the k-means clustering algorithm; the result of the experiment shows that the \(M(\phi)\)-distance measure extensively improves the clustering accuracy.

Declarations

Acknowledgements

The second and third authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, South Asian University
(2)
Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University
(3)
Department of Mathematics, Aligarh Muslim University

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