Abstract. We set up a new general coorbit space theory for reproducing representations of a locally compact second countable group G that are not necessarily irreducible nor integrable. Our basic assumption is that the kernel associated with the voice transform belongs to a Fréchet space T of functions on G, which generalizes the classical choice T " L 1 w pGq. Our basic example is T " Ş pPp1,`8q L p pGq, or a weighted versions of it. By means of this choice it is possible to treat, for instance, Paley-Wiener spaces and coorbit spaces related to Shannon wavelets and Schrödingerlets.
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].
This article is a continuation of the recent paper [21] by the first author, where off-diagonal-decay properties (often referred to as 'localization' in the literature) of Moore-Penrose pseudoinverses of (bi-infinite) matrices are established, whenever the latter possess similar off-diagonal-decay properties. This problem is especially interesting if the matrix arises as a discretization of an operator with respect to a frame or basis. Previous work on this problem has been restricted to wavelet-or Gabor frames. In [21] we extended these results to frames of parabolic molecules, including curvelets or shearlets as special cases. The present paper extends and unifies these results by establishing analogous properties for frames of α-molecules as introduced in recent work [22]. Since wavelets, curvelets, shearlets, ridgelets and hybrid shearlets all constitute instances of α-molecules, our results establish localization properties for all these systems simultaneously.
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