Summability process by singular operators

The aim of this paper is to obtain some approximation theorems for a sequence of singular operators that do not have to be positive in general. In the approximation, we mainly use a general matrix summability process introduced by Bell [Order summability and almost convergence, Proc. Amer. Math. Soc. 38 (1973) 548-552], which includes many well-known convergence methods, such as, the ordinary convergence, almost convergence, the Cesaro mean, the order summability, and so on. An application presented at the end of the paper shows that our approximation result is more applicable than the classical aspects