Holomorphic Extensions Associated with Fourier–Legendre Series and the Inverse Scattering Problem

In this paper, we consider the inverse scattering problem and, in particular, the problem of reconstructing the spectral density associated with the Yukawian potentials from the sequence of the partial-waves fℓ of the Fourier–Legendre expansion of the scattering amplitude. We prove that if the partial-waves fℓ satisfy a suitable Hausdorff-type condition, then they can be uniquely interpolated by a function f˜(λ)∈C, analytic in a half-plane. Assuming also the Martin condition to hold, we can prove that the Fourier–Legendre expansion of the scattering amplitude converges uniformly to a function f(θ)∈C (θ being the complexified scattering angle), which is analytic in a strip contained in the θ-plane. This result is obtained mainly through geometrical methods by replacing the analysis on the complex cosθ-plane with the analysis on a suitable complex hyperboloid. The double analytic symmetry of the scattering amplitude is therefore made manifest by its analyticity properties in the λ- and θ-planes. The function f(θ) is shown to have a holomorphic extension to a cut-domain, and from the discontinuity across the cuts we can iteratively reconstruct the spectral density σ(μ) associated with the class of Yukawian potentials. A reconstruction algorithm which makes use of Pollaczeck and Laguerre polynomials is finally given.

On the Connection between Spherical Laplace Transform and Non-Euclidean Fourier Analysis

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. Next, we introduce a Laplace-type transform (the so-called Spherical Laplace Transform) of the jump function across the cut. The main result of this paper is to establish the connection between the Spherical Laplace Transform and the Non-Euclidean Fourier Transform in the sense of Helgason. In this way, we find a connection between the unitary representation of SO ( 3 ) and the principal series of the unitary representation of SU ( 1 , 1 ) .

Measurement of the angular distribution of neutrons scattered from deuterium below 3 MeV

The differential cross section of neutron scattering on deuterium was investigated in the energy range from 400 keV to 2.5 MeV using the recoil detection method, irradiating with monoenergetic neutrons a proportional counter filled with deuterated gases. Comparing simulations of the transport of neutrons and recoil nuclei in the detector to the experimental pulse-height distribution, it was possible to establish a procedure for the determination of the coefficients of the Legendre expansion of the n-d angular distribution.

A New Method for De-Noising of Well Test Pressure Data Base on Legendre Approximation

In this paper, noise removing of the well test data is considered. We use the Legendre expansion to approximate well test data and a truncated strategy has been employed to reduce noise. The parameter of the truncation will be chosen by a discrepancy principle and a corresponding convergence result has been obtained. The theoretical analysis shows that a well numerical approximation can be obtained by the new method. Moreover, we can directly obtain the stable numerical derivatives of the pressure data in this method. Finally, we give some numerical tests to show the effectiveness of the method.

MELEXIR: maximum entropy Legendre expanded image reconstruction. A fast and efficient method for the analysis of velocity map imaging or photoelectron imaging data

The MELEXIR program obtains a Legendre expansion of the 3D velocity distribution from 2D images of ions or photoelectrons. The maximum entropy algorithm avoids inverse Abel transforms, is fast and applicable to low-intensity images.