On functions with zero integral moments over balls of fixed radius
For a locally integrable function \(f\) on \({\mathbb R}^n\), consider the integrals over all balls of a fixed radius \(r\): \[ \int_{|x|\leq r} f(x+u) \,dx, \quad u \in {\mathbb R}^n. \] It is known that if all these integrals are 0, this does not guarantee that \(f = 0\), unless further conditions are satisfied. The authors consider the case where \(f\) has support in a ball \(B_R\) of radius \(R > r\), and the integrals of the function and its first moments over all balls of radius \(r\) contained in \(B_R\) vanish: \[ \int_{|x|\leq r} f(x+u) \,dx = \int_{|x|\leq r} x_j f(x+u) \,dx = 0, \quad j=1,\ldots,n. \] The question is whether this is sufficient to imply \(f=0\) in \(B_R\). The authors state that for \(R > 2r\), this is true. For \(R \leq 2r\), it holds under extra conditions. If \(f \in C^\infty\), it holds for \(R = 2r\) without the extra conditions. For the case \(R > 2r\), they also provide a very complicated inversion formula for recovering \(f\) from its integrals, based on series expansions.
scientific article; zbMATH DE number 6935999