Approximation by means of nonlinear integral operators in the space of functions with bounded \(\varphi \)-variation. (English) Zbl 1212.26016

Angeloni, Laura; Vinti, Gianluca

Differ. Integral Equ. 20, No. 3, 339-360 (2007); errata ibid. 23, No. 7-8, 795-799 (2010).

Summary: We study approximation problems by means of nonlinear convolution integral operators for functions belonging to \(BV_\varphi \)-spaces, i.e., functions with bounded \(\varphi \)-variation in the sense of Musielak-Orlicz. In particular, we obtain estimates and convergence results with respect to \(\varphi \)-variation. Introducing suitable Lipschitz classes that take into account the \(\varphi \)-variational functional, the problem of the rate of approximation is also considered.

MSC:

26A45	Functions of bounded variation, generalizations
26A46	Absolutely continuous real functions in one variable
41A25	Rate of convergence, degree of approximation
41A35	Approximation by operators (in particular, by integral operators)
47G10	Integral operators

Keywords:

approximation problem; nonlinear convolution integral operators; bounded \(\varphi \)-variation