Large Structures Seminar: Dima Pasechnik

The problem of reconstructing a measure in $\mathbb{R}^d$ from a (truncated) multi-sequence of its moments has important applications, and is in general very hard to solve. We concentrate on a natural case of a measure mu with piecewise-polynomial density supported on a compact polyhedron $P$, and show that such problems can be solved exactly, due to existence of a natural integral transform of the measure (known as Fantappie transformation), which is a rational function $F_\mu(u)$.

The denominator of $F_\mu(u)$ is the product of linear functions of the form $1-\langle u,v\rangle$, with $v$ belonging to certain finite multiset $V(P)$. $F_\mu(u)$ is closely related to a more well-known Laplace transform $L_\mu(u)$ of a related “conified” measure arising in the theory of hyperplane arrangements.

There are interesting applications of $L_\mu$ to compact (not necessarily convex) polyhedra $P$. Let $I(P)$ be the indicator function of $P$. Then $I(P)$ can be decomposed (up to a measure $0$ subset) as a sum, with $+1$ or $-1$ coefficients, of $I(D)$, where $D$ runs through simplices with vertices in $V(P)$. This can be viewed as a non-convex generalisation of triangulations of convex polytopes. Further refinements and applications will be discussed.