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We consider the Radon transform associated to dual pairs (X, Ξ) in the sense of Helgason, with X = G/K and Ξ = G/H, where G = R d ⋊ K, K is a closed subgroup of GL(d, R) and H is a closed subgroup of G. Under some technical assumptions, we prove that if the quasi regular representations of G acting on L 2 (X) and L 2 (Ξ) are irreducible, then the Radon transform admits a unitarization intertwining the two representations. If, in addition, the representations are square integrable, we provide an inversion formula for the Radon transform based on the voice transform associated to these representations.

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Characterizing the function spaces corresponding to neural networks can provide a way to understand their properties. In this paper we discuss how the theory of reproducing kernel Banach spaces can be used to tackle this challenge. In particular, we prove a representer theorem for a wide class of reproducing kernel Banach spaces that admit a suitable integral representation and include one hidden layer neural networks of possibly infinite width. Further, we show that, for a suitable class of ReLU activation functions, the norm in the corresponding reproducing kernel Banach space can be characterized in terms of the inverse Radon transform of a bounded real measure, with norm given by the total variation norm of the measure. Our analysis simplifies and extends recent results in [34,29,30].

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