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

Authors: YUSUF ZEREN, MIQDAD ISMAILOV, CEMİL KARAÇAM

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.

Keywords: Grand Lebesgue space, Korovkin theorems, shift operator, statistical convergence, positive linear operator, approximation process

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