Korovkin-type theorems and their statistical versions in grand Lebesgue spaces

Abstract

The analogs of Korovkin theorems in grand-Lebesgue spaces are proved. The subspace $G^{p)}(-pi;;pi)$ of grandLebesgue space is defined using shift operator. It is shown that the space of infinitely differentiable finite functions isdense in $G^{p)}(-pi;;pi)$ The analogs of Korovkin theorems are proved in $G^{p)}(-pi;;pi)$ These results are established in $G^{p)}(-pi;;pi)$ in the sense of statistical convergence. The obtained results are applied to a sequence of operators generatedby the Kantorovich polynomials, to Fejer and Abel-Poisson convolution operators.