General convergence theorem with respect to Cesari variation and applications

Abstract

LET fi c IR” be an open set and let BV(Q) be the space of those L’(Q) functions with bounded variation in Q. The aim of this paper is to present a unitary approach to the problem of convergence of certain operators in BP’(Q) with respect to Cesari variation. As is well known, convolution integral operators with suitable kernels, moment type operators and other type of linear integral operators defined on BV(/(n), have the property that a sequence of them converges in variation, in length or in area (see, e.g. [I-S]). All these results are proved by two fundamental steps: (1) application of suitable semicontinuity properties; (2) use of an appropriate “variation diminishing” property. In general, step 2 is more difficult to prove. Here, we show that for a sequence of operators T,,: BV(fi) + BV’(/(n), which satisfy some continuity assumptions, it is sufficient to assume an “almost variation diminishing property” for the restriction of T, to the subspace C”(n) fl BV(Q). This simplifies many of those proofs, because the variation, the length and the area, have an integral representation for regular functions. A simple extension of the Anzellotti-Giaquinta density theorem [9] enables us to prove the convergence theorem for T, , with respect to F-variation on BV(i2) which is defined by using the Goffman-Serrin approach [lo] (here 5 is a continuous sublinear functional). This abstract approach is very useful because it enables us to state directly analogous approximation theorems for the Serrin integral of the calculus of variation [II]. Among the examples, we quote some discrete operators as the (modified) generalized sampling series [12-141 and as a corollary, we state a convergence in variation for these operators. Theorems 1 and 4 below extend and unify many previous results on convergence in variation, length and area (see the examples). Moreover, they also improve some results of [8].