Geodesic
The $\chi^{2}$ sequence spaces defined by a modulus
Kragujevac Journal of Mathematics, Tome 35 (2011) no. 1, p. 39 .

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In this paper we introduce the following sequence spaces \begin{center}$\left\{x\in \chi^{2}: P-im_{k,\ell} um_{m=0}^{nfty} um_{n=0}^{nfty}a_{k\ell}^{mn}feft(eft(eft(m+n\right)!eft|x_{mn}\right|\right)^{\frac{1}{m+n}}\right)=0\right\}$ \end{center} and $\left\{x\in \Lambda^{2}: \sup_{k,\ell}\sum_{m=0}^{\infty}\sum_{m=0}^{\infty}a_{k \ell}^{mn}f\left(\left|x_{mn}\right|^{\frac{1}{m+n}}\right) \infty\right\}$ where $f$ is a modulus function and $A$ is a nonnegative four dimensional matrix. We establish the inclusion theorems between these spaces and also general properties are discussed.
Classification : 40A05 40C05 40D05
Keywords: Gai sequence, Analytic sequence, Modulus function, Double sequences 
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Nagarajan Subramanian; Umakanta Misra; Vladimir Rakočević. The $\chi^{2}$ sequence spaces defined by a modulus. Kragujevac Journal of Mathematics, Tome 35 (2011) no. 1, p. 39 . http://geodesic.mathdoc.fr/item/KJM_2011_35_1_a3/