Sharp Unified Smoothness Theory for Cavalieri Estimation via Fourier Decay

Cavalieri estimation is a widely used technique in stereology (applied geometric sampling) for approximating the volume of a solid by sampling cross-sectional areas along a fixed axis. Classical theory shows that, under systematic equidistant sampling (the well-known Cavalieri estimator), the variance decay depends on the smoothness of the area function, which is essentially measured by the number of continuous derivatives. This paper focuses on the natural assumptions under which the theory holds. We first obtain sharp and explicit variance rates: when the Fourier decay is of order s>1/2, the variance of the Cavalieri estimator decays as t2s with a constant independent of t. Building on this, we show that the smoothness condition expressed in terms of the algebraic Fourier decay subsumes both integer- and fractional-order frameworks used to date. Finally, we establish a matching converse showing that, under general assumptions, no broader smoothness framework extends the theory; that is, any algebraic variance decay implies the corresponding Fourier decay.