Year: 2025
Type: preprint
Abstract: We consider inverse problems consisting of the reconstruction of an unknown signal $f$ from noisy measurements $y=Ff+\text{noise}$, 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 $\{x_j\}_{j=1}^n\subseteq\mathcal M$ 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.
Source: arXiv (Cornell University)
DOI: 10.1137/25m1788014
Authors: Alberti, Giovanni S., De Vito, Ernesto, Gariboldi, Bianca, Gigante, Giacomo
Institution: University of Genoa
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6979336464
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