Discrete tomography through distribution theory (Q1011984)

The paper is concerned with the zero-sum arrays, that are the \(\mathbb{C}\)-valued functions on \(\mathbb{Z}^n\), \(a=(a_i)\in (\mathbb{C})^{\mathbb{Z}^n}\), \(i=(i_1,\dots,i_n)\in\mathbb{Z}^n\), \(a_i\in\mathbb{C}\), that verify the condition \(d_{t+p}(a)=\sum_{i\in\mathbb{Z}^n}t_{i-p}a_i=0\), for all \(p\in\mathbb{Z}^n\) and for a fixed array \(t=(t_i)\) with finite support, called window, which in addition have a polynomial growth \(a_k=O(|k|^N)\) as \(|k|\to\infty\). The author studies the annihilator of the Dirac delta function with its higher derivatives, and then he establishes a one-to-one correspondence between the space of zero-sum arrays of a window and the space of polynomial solutions of the associated partial differential equation (PDE). A dimension formula for the space of polynomial solutions of linear PDEs with constant coefficients is also presented. An inductive procedure for construction of a solution from that for PDE in a lesser number of variables and several examples of windows are finally addressed.