On approximation properties of certain multidimensional nonlinear integrals

Abstract

We prove theorems on convergence of multidimensional nonlinear integrals in Lebesgue points of generated function, and show that the main results are applicable to a wide class of exponentially nonlinear integral operators, which may be constructed by using well known positive kernels in approximation theory.

On Certain Multidimensional Nonlinear Integrals

“…The de…nition of the space L 1 (R n ) is analogous. Our results generalize and improve Theorem 2.2 in [3] in two di¤erent directions in view of the usages of generalized domain of integration and generalized characteristic point, respectively. Now, we consider the kernel function of the operators of type (1.2).…”

“…: R n ! R: The conditions on the kernel function to be given below are revised versions of the conditions used by Almali and Gadjiev [3].…”

On pointwise convergence of the family of Urysohn‐type integral operators

“…Corollary 1. Let F(t) ≤ 1 be a nonnegative function such that monotonically increasing in (−∞, 0] and monotonically decreasing in [0, ∞) and satisfies (10). Then at each Lebesgue point x 0 of bounded function u ∈ L 1 (R)…”

“…Angeloni investigated the properties of the following nonlinear convolution integral operators: where is a family of kernels of the form K w ( s ; u ) = L w ( s ) H w ( u ), in which f is an element of the space of functions of bounded variation in R n , s ∈ R n and u ∈ R . Recently, in 2016, Esen Almali and Gadjiev showed approximation properties of certain multidimensional nonlinear integrals. In Bardaro et al, the monograph contains results related to nonlinear integral operators and applications obtained by the authors.…”

On approximation properties for non-linear integral operators

“…For general analysis on non-linear integral operators in different spaces and settings the book [1] is recommended. Also, for some recent works, we refer the reader to see [2,5] and [7]. Recently, Esen Almali investigated the problem of pointwise convergence at lebesgue points of functions for the family of singular integrals involving infinitive sum in [8].…”