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Exotic approximate identities and Maass forms

Fernando Chamizo, Dulcinea Raboso, Serafín Ruiz-Cabello (2013)

Acta Arithmetica

We obtain some approximate identities whose accuracy depends on the bottom of the discrete spectrum of the Laplace-Beltrami operator in the automorphic setting and on the symmetries of the corresponding Maass wave forms. From the geometric point of view, the underlying Riemann surfaces are classical modular curves and Shimura curves.

Exponential expressivity of ReLU k neural networks on Gevrey classes with point singularities

Joost A. A. Opschoor, Christoph Schwab (2024)

Applications of Mathematics

We analyze deep Neural Network emulation rates of smooth functions with point singularities in bounded, polytopal domains D d , d = 2 , 3 . We prove exponential emulation rates in Sobolev spaces in terms of the number of neurons and in terms of the number of nonzero coefficients for Gevrey-regular solution classes defined in terms of weighted Sobolev scales in D , comprising the countably-normed spaces of I. M. Babuška and B. Q. Guo. As intermediate result, we prove that continuous, piecewise polynomial high...

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