Algebraic genericity and the differentiability of the convolution

Abstract

Despite the convolution preserving most of the smooth properties of the functions that take part in it, there exist differentiable functions whose convolution is not differentiable. In the present result, we study the algebraic genericity of the set of those functions. In particular, it is proved that periodic continuous functions can be approximated by functions belonging to a vector space each of whose nonzero members generates some convolution which is not differentiable.