Sampling Theorems for Inverse Problems on Riemannian Manifolds

We consider inverse problems consisting of the reconstruction of an unknown signal 𝑓 from noisy measurements 𝑦 =𝐹⁡𝑓 +noise, where 𝐹⁡𝑓 is a function on a Riemannian manifold without boundary M. We consider the case when only pointwise samples are available, namely, 𝑦𝑗 =(𝐹⁡𝑓)⁢(𝑥𝑗) +𝜂𝑗, where {𝑥𝑗}𝑛 𝑗=1 ⊆M is a Marcinkiewicz–Zygmund family. We derive sampling theorems providing explicit bounds on the reconstruction error depending on 𝑛, the smoothness of 𝑓, and the properties of 𝐹. We study in detail the case when 𝐹 is a convolution on a compact two-point homogeneous space. As a corollary, we state a sampling theorem for convolutions on the two-dimensional sphere and discuss four relevant examples related to terrestrial and celestial measurements.