Abstract
The analogs of Korovkin theorems in grand-Lebesgue spaces are proved. The subspace G p −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 p −π; π . The analogs of Korovkin theorems are proved in G p −π; π . These results are established in G p −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.
Highlights
The concept of statistical convergence was first introduced by Fast ([20]) and Steinhaus in 1951 ([46])
This concept was treated as an almost everywhere convergence by Zygmund in the monograph [50], where it was introduced in the context of pointwise convergence of the Fourier series of summable function. This theory was further developed by Schoenberg [44], Peterson [39], Brown and Freedman [13], Connor [16], Erdös and Tenenbaum [19], Freedman and Sember [25], Fridy [26], Fridy and [27], Kuchukaslan et al [33], Maddox [35], Maharam [36], etc. Statistical convergence has important applications in different areas of mathematics, such as summation theory, number theory, probability theory, and approximation theory
Statistical convergence is related to the concept of statistical fundamentals, considered first by Fridy [26], who established the equivalence of these concepts for numerical sequences
Summary
The concept of statistical convergence was first introduced by Fast ([20]) and Steinhaus in 1951 ([46]). Let us state well-known Korovkin theorems which have important applications in the study of approximation problems in the spaces of continuous functions as well as in Lebesgue spaces ([2]). Theorem 2.1 (Korovkin’s first theorem) Let {Ln}n∈N be a sequence of positive operators from C([0; 1]) into F ([0; 1]) , satisfying the condition The following is valid: Theorem 2.2 Let 1 ≤ p < +∞ and {Kn}n∈N be a sequence of operators generated by the polynomials (2.1).
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