Baire*1, Baire 1 and Zahorski properties of higher order derivatives (Q858132)

Let \(f:(a,b)\to \mathbb R\) be a continuous function; and, for each positive integer \(k\), let \(D_a^{2k}f(.)\) (respectively, \(D^{2k}f(.)\)) stand for the approximate (respectively, ordinary) de la Vallee Poussin derivative of \(f\) of order \(2k\). Define also \[ \theta_{2k+2}={h^{2k+2}\over (2k+2)!}\theta_{2k+2}(f,x,h){f(x+h)+f(x-h)\over 2}-\sum_{i=0}^{k}{h^{2i}\over (2i)!}D_a^{2i}f(x). \] Finally, take some part \(E\) of \((a,b)\). Under certain regularity conditions involving these objects over \(E\), it is established that (i) if \(D_a^{n-2}f\) (respectively, \(D^{n-2}f\)) exists finitely on \(E\), then \(D_a^{l}f\in {\mathcal B}_1^*(E)\) (respectively, \(D^{l}f\in {\mathcal B}_1^*(E)\)), for \(l=n-2,n-4,\dots\), and (ii) if \(D^n f\) exists (possibly infinite) on \(E\), then \(D^nf\in {\mathcal B}^1(E)\). In addition, some Zahorski and Denjoy properties for the ordinary symmetric de la Vallée Poussin derivatives \(D^nf\) are given.