On the Cesaro and Borel symmetric derivatives

Abstract

The paper discusses some properties of Cesaro and Borel derivatives. It was established that, according to Lebesgue, the concepts of Cesaro and Borel first-order symmetric (ordinary) derivatives are equivalent for a summable function. The dependence has been studied of the ordinary and symmetric Borel and Cesaro derivatives on the set of positive. In particular, if is a set of any positive measure, and at every point of this set there exists a symmetric Cesaro (respectively Borel) derivative, then almost everywhere on this set, there exists a Cesaro (respectively Borel) derivative equal to the symmetric derivative. It is also shown that a summable function can have an infinitely large Cesaro (respectively Borel) symmetric derivative on a set of zero measure.