Sampling Theorems for Inverse Problems on Riemannian Manifolds
Giovanni S. Alberti, Ernesto De Vito, Bianca Gariboldi, Giacomo Gigante
Source abstract
Abstract. We consider inverse problems consisting of the reconstruction of an unknown signal [Formula: see text] from noisy measurements [Formula: see text], where [Formula: see text] is a function on a Riemannian manifold without boundary [Formula: see text]. We consider the case when only pointwise samples are available, namely, [Formula: see text], where [Formula: see text] is a Marcinkiewicz–Zygmund family. We derive sampling theorems providing explicit bounds on the reconstruction error depending on [Formula: see text], the smoothness of [Formula: see text], and the properties of [Formula: see text]. We study in detail the case when [Formula: see text] 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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