Generalised smooth functions. II (Q1813366)

This paper is concerned with real-valued functions defined on an interval. The key concepts are as follows: symmetric de la Vallée- Poussin derivatives \(D^ mf\) of even and odd order, two related smoothness conditions \(S_ m^*\), \(s_ m^*\) defined differently for even and odd orders, the Peano derivative \(f_{(m)}\) of order \(m\). Examples are given to show that there is no containment between the classes \(S_ m^*\) and \(s_ m^*\) for given order \(m\). Continuous functions which are \(S_ m^*\) on a closed set \(E\) are shown to have a Baire \(^*1\) de la Vallée-Poussin derivative of order \(m-2,m-4,\dots\) on \(E\). In the case that a continuous function belongs to \(S_ m^*\) on an interval \(I\), the \((m-2)\)nd derivative is shown to exist and to be continuous on \(I\); if \(D^{m-1}f\) exists on a closed set \(E\subset I\), \(D^{m-1}f\) is shown to be Baire 1 on \(E\) and the upper and lower \((m- 1)\)st Peano derivatives are shown to be finite on a set having the power of the continuum in each subinterval of \(I\). A final result involving the smoothness of periodic, integrable functions is given. (Part I has been reviewed in Zbl 0511.26006.).