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 grand Lebesgue space is defined using shift operator. It is shown that the space of infinitely differentiable finite functions is dense 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 generated by the Kantorovich polynomials, to Fejer and Abel-Poisson convolution operators.