Let , , be a nonnegative regular summability matrix. The sequence is called -statistically convergent to , notation , if for every ,
Let denote the space of -periodic functions on the real line with the usual sup-norm, and let be a sequence of positive linear operators mapping into itself. The author proves the following Korovkin type theorem
Theorem: Let and be as above. Then necessary and sufficient condition in order that
is that (1) holds for the three functions, 1, , and .
An application is given describing sufficient conditions on a matrix of coefficients , , ensuring that the operators
where the ’s and ’s are the Fourier coefficients of , converge to in the sup-norm.