Using hyperbolic form convolution with doubly isometry-invariant kernels, the explicit expression of the inverse of the de Rham laplacian ∆ acting on m-forms in the Poincaré space H is found. Also, by means of some estimates for hyperbolic singular integrals, L-estimates for the Riesz transforms ∇i∆−1, i ≤ 2, in a range of p depending on m,n are obtained. Finally, using these, it is shown that ...

We present a simplified theory of generalized eigenfunction expansions for a commuting family of bounded operators and with finitely many unbounded operators. We also study the convergence of these expansions, giving an abstract type of uniform convergence result, and illustrate the theory by giving two examples: The Fourier transform on Hecke operators, and the Laplacian operators in hyperboli...

There is a remarkable relation between two kinds of phase space distributions associated to eigenfunctions of the Laplacian of a compact hyperbolic manifold: It was observed in [1] that for compact hyperbolic surfaces XΓ = Γ\H Wigner distributions R S∗XΓ a dWirj = 〈Op(a)φirj , φirj 〉L2(XΓ) and Patterson–Sullivan distributions PSirj are asymptotically equivalent as rj → ∞. We generalize the defi...

Hyperbolic tangent and Sigmoid functions are used as non-linear activation units in the artificial deep neural networks. Since, these networks computationally expensive, customized accelerators designed for achieving required performance at lower cost power. The function MAC key building blocks of A low complexity accurate hardware implementation is to meet area targets such network accelerator...