The analogs of Korovkin theorems in grand-Lebesgue spaces are proved. The subspace G
(−π; π) of grand
Lebesgue space is defined using shift operator. It is shown that the space of infinitely differentiable finite functions is
dense in G
(−π; π). The analogs of Korovkin theorems are proved in G
(−π; π). These results are established in
(−π; π) in the sense of statistical convergence. The obtained results are applied to a sequence of operators generated
by the Kantorovich polynomials, to Fejer and Abel-Poisson convolution operators.