In the present paper, we characterize the classes of all triangular matrices, \documentclass{aastex}
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$(|\bar N_p |,|\bar N_q^\theta |_k )$
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$(|\bar N_p^\theta |_k ,|\bar N_q |)$
\end{document} for the case k ≧ 1, where \documentclass{aastex}
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$|\bar N_p^\theta |_k = \left\{ {a = (a_n ):\sum\limits_{n = 1}^\infty {\theta _n^{k - 1} } \left| {\frac{{p_n }}{{p_n p_{n - 1} }}\sum\limits_{v = 1}^n {P_{v - 1} a_v } } \right|^k < \infty } \right\},$
\end{document} i.e., the set of series summable by absolute weighted mean summability method, and so extend the some well known results.