Sampling theorems for inverse problems on Riemannian manifolds

2025/08/14 by Alberti, Giovanni S., De Vito, Ernesto, Gariboldi, Bianca +1
#43A85 #45Q05 #94A20 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2508.10810

Abstract

We consider inverse problems consisting of the reconstruction of an unknown signal f from noisy measurements y=Ff+noise, where Ff is a function on a Riemannian manifold without boundary \mathcal M. We consider the case when only pointwise samples are available, namely y_j = (Ff)(x_j)+η_j, where \x_j\_j=1ⁿ⊆\mathcal 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.

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Related

• Reconstruction and interpolation of manifolds II: Inverse problems with partial data for distances observations and for the heat kernel
• Reconstructing Functions from Random Samples
• Reconstruction of a Riemannian manifold from noisy intrinsic distances
• Sampling theorems on bounded domain
• Topological Inference of Manifolds with Boundary