Estimation of an Entropy-based Functional

Given a function f from [0, 1] to the real line, we consider the (nonlinear) functional h obtained by evaluating the continuous entropy of the “density function” of f . Motivated by an application in signal processing, we wish to estimate h(f). Our main tool is a decomposition of h into two terms, which each have favorable scaling properties. We show that, if functions f and g satisfy a regularity condition, then the smallness of ∥f − g∥∞ and ∥f ′ − g∥∞, along with some basic control on derivatives of f and g, is sufficient to imply that h(f) and h(g) are close.