Large algebras of singular functions vanishing on prescribed sets

TL;DR

This paper characterizes when the family of infinitely differentiable functions that vanish on a set Z contains large algebraic structures, extending previous results in the field.

Contribution

It provides a characterization of the non-vacuousness of these function families and constructs large algebraic structures within them.

Findings

Characterization of when such function families are non-empty

Existence of large algebraic structures inside these families

Extension of previous results in the literature

Abstract

In this paper, the non-vacuousness of the family of all nowhere analytic infinitely differentiable functions on the real line vanishing on a prescribed set Z is characterized in terms of Z. In this case, large algebraic structures are found inside such family. The results obtained complete or extend a number of previous ones by several authors.