Luzin Type Approximation of Functions of Bounded Variation
Greg Francos · D-Scholarship@Pitt (University of Pittsburgh) · 2011
This paper is divided into two sections:(I) Consider the function space BV^m = {u ∈ W^{m-1,1}: D^α u is a measure for |α| = m}Such functions are called mth order functions of bounded variation. We show that a function in BV^m(R^n) possesses the so-called C^m-Luzin property; that is, it coincides with a C^m(R^n) function outside a set of arbitrarily small Lebesgue measure.(II) Consider a set of Lebesgue measureable functions f^α: R^N &rarr R indexed by themulti-indices in R^N of order |α| = m. We will prove that for any such collection, there isg &isin C^{m-1}(R^N) which is m-times differentiable almost everywhere, and such thatD^α g(x) = f^α(x) a.e. for all |α| = m.