A Review on Approximation Results for Integral Operators in the Space of Functions of Bounded Variation

Abstract

We present a review on recent approximation results in the space of functions of bounded variation for some classes of integral operators in the multidimensional setting. In particular, we present estimates and convergence in variation results for both convolution and Mellin integral operators with respect to the Tonelli variation. Results with respect to a multidimensional concept ofφ-variation in the sense of Tonelli are also presented.

Discrete operators of sampling type and approximation in ‐variation

“…In the main properties of the φ‐variation are presented, while for further papers about such concept we refer to, e.g., .…”

“…In [34] the main properties of the -variation are presented, while for further papers about such concept we refer to, e.g., [1][2][3][4][5][6][7][8][9]11,23,28,[31][32][33]36].…”

Convergence and rate of approximation in $BV^{\varphi}(\mathbb R^N_+)$ for a class of Mellin integral operators

“…Results about homothetictype operators in various settings can be found, for example, in [19,32,44,18,39,17,15,16,3,10,11,12], while for similar results about classical convolution operators see, e.g., [23,41,33,14,20,6,7,8,2,4].…”

“…Due to the homothetic structure of our operators, it seems that the most natural way to frame the theory is to work with the Haar measure in R N + , i.e., µ(A) := A t −1 dt, where A is a Borel subset of R N + . Results about homothetictype operators in various settings can be found, for example, in [19,32,44,18,39,17,15,16,3,10,11,12], while for similar results about classical convolution operators see, e.g., [23,41,33,14,20,6,7,8,2,4].…”

On a special class of modified integral operators preserving some exponential functions

“…is a sequence of kernel functions. Following notations and conditions in [6] (see also [20]), the functions K n satisfy the following conditions:…”