arXiv subjects

Enrico De Micheli

Publications and source records attributed to Enrico De Micheli.

25 records · Page 2Linked to original sources

arXiv2005

Holomorphic extension associated with Fourier-Legendre expansions

In this article we prove that if the coefficients of a Fourier-Legendre expansion satisfy a suitable Hausdorff-type condition, then the series converges to a function which admits a holomorphic extension to a cut-plane. Furthermore, we prove that a Laplace-type (Laplace composed with Radon) transform of the function describing the jump across the cut is the unique Carlsonian interpolation of the Fourier coefficients of the expansion. We can thus reconstruct the discontinuity function from the coefficients of the Fourier-Legendre series by the use of the Pollaczek polynomials.

math.CA↗

arXiv2005

On the solution of a class of Cauchy integral equations

In a previous paper we have presented a new method for solving a class of Cauchy integral equations. In this work we discuss in detail how to manage this method numerically, when only a finite and noisy data set is available: particular attention is focused on the question of the numerical stability.

math.CA↗

arXiv2005

Hausdorff moments, Hardy spaces and power series

In this paper we consider power and trigonometric series whose coefficients are supposed to satisfy the Hausdorff conditions, which play a relevant role in the moment problem theory. We prove that these series converge to functions analytic in cut domains. We are then able to reconstruct the jump functions across the cuts from the coefficients of the series expansions by the use of the Pollaczek polynomials. We can thus furnish a solution for a class of Cauchy integral equations.

math.CA↗

arXiv2005

On the regularization of Fredholm integral equations of the first kind

In this paper the problem of recovering a regularized solution of the Fredholm integral equations of the first kind with Hermitian and square-integrable kernels, and with data corrupted by additive noise, is considered. Instead of using a variational regularization of Tikhonov type, based on a priori global bounds, we propose a method of truncation of eigenfunction expansions that can be proved to converge asymptotically, in the sense of the $L^2$--norm, in the limit of noise vanishing. Here we extend the probabilistic counterpart of this procedure by constructing a probabilistically regularized solution without assuming any structure of order on the sequence of the Fourier coefficients of the data. This probabilistic approach allows us to use the statistical tools proper of time-series analysis, and in this way we attain a new regularizing algorithm, which is illustrated by some numerical examples. Finally, a comparison with solutions obtained by the means of the variational regularization exhibits how some intrinsic limits of the variational-based techniques can be overcome.

math.CA↗

arXiv2002

Riemannian Geometrical Optics: Surface Waves in Diffractive Scattering

The geometrical diffraction theory, in the sense of Keller,is here reconsidered as an obstacle problem in the Riemannian geometry. The first result is the proof of the existence and the analysis of the main properties of the diffracted rays, which follow from the non-uniqueness of the Cauchy problem for geodesics in a Riemannian manifold with boundary. Then,the axial caustic is here regarded as a conjugate locus, in the sense of the Riemannian geometry, and the results of the Morse theory can be applied.The methods of the algebraic topology allow us to introduce the homotopy classes of diffracted rays. These geometrical results are related to the asymptotic approximations of a solution of a boundary value problem for the reduced wave equation. In particular, we connect the results of the Morse theory to the Maslov construction, which is used to obtain the uniformization of the asymptotic approximations. Then, the border of the diffracting body is the envelope of the diffracted rays and, instead of the standard saddle point method, use is made of the procedure of Chester, Friedman and Ursell to derive the damping factors associated with the rays which propagate along the boundary. Finally, the amplitude of the diffracted rays when the diffracting body is an opaque sphere is explicitly calculated.

math-ph↗

arXiv2002

Non-local Potentials and Rotational Bands of Resonances in Ion Collisions

Sequences of rotational resonances (rotational bands) and corresponding antiresonances are observed in ion collisions. In this paper we propose a description which combines collective and single-particle features of cluster collisions. It is shown how rotational bands emerge in many-body dynamics, when the degeneracies proper of the harmonic oscillator spectra are removed by adding interactions depending on the angular momentum. These interactions can be properly introduced in connection with the exchange-forces and the antisymmetrization, and give rise to a class of non-local potentials whose spectral properties are analyzed in detail. In particular, we give a classification of the singularities of the resolvent, which are associated with bound states and resonances. The latter are then studied using an appropriate type of collective coordinates, and a hydrodynamical model of the trapping, responsible for the resonances, is then proposed. Accordingly, we derive, from the uncertainty principle, a spin-width of the unstable states which can be related to their angular lifetime.

nucl-th↗

arXiv2002

Orbiting Resonances and Bound States in Molecular Scattering

A family of orbiting resonances in molecular scattering is globally described by using a single pole moving in the complex angular momentum plane. The extrapolation of this pole at negative energies gives the location of the bound states. Then a single pole trajectory, that connects a rotational band of bound states and orbiting resonances, is obtained. These complex angular momentum singularities are derived through a geometrical theory of the orbiting. The downward crossing of the phase-shifts through pi/2, due to the repulsive region of the molecular potential, is estimated by using a simple hard-core model. Some remarks about the difference between diffracted rays and orbiting are also given.

physics.atom-ph↗