The authors introduce a new trigonometric method of summation, denoted by \((R_{2+ \varepsilon})\). They say that a series with partial sums \(s_n\) is \((R_{2+ \varepsilon})\) summable to \(s\) if \[ \lim_{n\to 0} F(h) =s, \] where \[ F(h): ={h^{1 -\varepsilon} \over D(\varepsilon)} \sum^\infty_{k=1} A_k^{-\varepsilon} \left( {\sin(1/2)kh \over(1/2)k h} \right)^2, \quad 0-1. \] They prove some Abelian and Tauberian theorems for this method.