Convexity conditions and intersections with smooth functions

Abstract:

A continuous function that agrees with each member of a family $\mathcal {F}$ of smooth functions in a small set must itself possess certain desirable properties. We study situations that arise when $\mathcal {F}$ consists of the family of polynomials of degree at most $n$, as well as certain larger families and when the small sets of agreement are finite. The conclusions of our theorems involve convexity conditions. For example, if a continuous function $f$ agrees with each polynomial of degree at most $n$ in only a finite set, then $f$ is $(n + 1)$-convex or $(n + 1)$-concave on some interval. We consider also certain variants of this theorem, provide examples to show that certain improvements are not possible and present some applications of our results.