The number rigidity of a point process P entails that for a bounded set A the knowledge of P on Ac a.s. determines P(A); the k-order rigidity means we can recover the moments up to order k. We show that there is k-rigidity if the continuous component s of P’s structure factor has a zero of order k in 0, by exploiting a connection with Schwartz’ Paley-Wiener theorem for analytic functions of exponential type; these results apply to any random L2 wide sense stationary measure on Rd or Zd . In the continuous setting, these local conditions are also necessary if s has finitely many zeros, or is isotropic, or at the opposite separable. This explains why no model seems to exhibit rigidity in dimension d ⩾ 3, and allows to efficiently recover many recent rigidity results about point processes. In the discrete setting, these results hold provided #A > 2k. We derive new results about models of cluster lattices and give the first example of a stationary point process P ⊂ Rd exhibiting arbitrary low decay of the structure factor in 0, hence arbitrary high order of rigidity. For a continuous Determinantal point process with kernel K, k-rigidity is equivalent to 1/(1 − K^2) having a zero of order 2k in 0, which answers questions on completeness and number rigidity. We also explore the consequences of these statements in the less tractable realm of Riesz gases.