Nonperiodic sampling and the local three squares theorem
The local three squares problem concerns the question whether functions defined on a cube in \(\mathbb{R}^d\) are determined from their averages on certain smaller cubes. The authors prove a quite general result, namely: fix the dimension \(d\), consider positive numbers \(r_1, \dots, r_{d+1}\), and take any \(R\geq \sum_{i=1}^{d+1}r_i\). Then any function \(f\in L^2([-R,R]^d)\) is determined by its averages over cubes of side \(r_i\) that are completely contained in \([-R,R]^d\) and have edges parallel to the coordinate axes, if and only if \(r_i/r_j\) is irrational for \(i\neq j\). The main technical tool is the sampling theory for bandlimited functions.
scientific article; zbMATH DE number 1887456