Approximation by nonlinear integral operators via summability process

Abstract

In this paper, we study the approximation properties of nonlinear integral operators of convolution‐type by using summability process. In the approximation, we investigate the convergence with respect to both the variation semi‐norm and the classical supremum norm. We also compute the rate of approximation on some appropriate function classes. At the end of the paper, we construct a specific sequence of nonlinear operators, which verifies the summability process. Some graphical illustrations and numerical computations are also provided.

Characterization of Absolute and Uniform Continuity

“…Proof. The necessity is clear from Theorem 3.3 of [10]. The sufficiency part follows from Lemma 3.1 and the fact that C 2π is a closed subspace of B 2π .…”

“…is the length of the interval J and V J denotes the total variation on J, we proved in [10] that, for every f ∈ AC 2π ,…”

“…• L 1 2π , the space of all 2π-periodic and Lebesgue integrable functions. Then, we consider the nonlinear integral operators introduced in [10] (see also [2,3] in the case of A = {I}, the identity matrix):…”

“…Proof. The necessity immediately follows from Theorem 2.3 of [10]. For the sufficiency, assume that lim…”

“…They also play a crucial role in the characterization of absolutely continuous functions (see [5][6][7]). In our recent papers, we also consider regular summability methods instead of the usual convergence in the approximation by these operators (see [9][10][11]) in order to generalize and improve the known results in the literature. We should note that, in the approximation by these operators, we consider suitable functions spaces endowed with the variation semi-norm, the uniform (supremum) norm, the L p norm, and so on.…”