On the calculation of the finite Hankel transform eigenfunctions

Abstract

In the recent years considerable attention has been focused on the numerical computation of the eigenvalues and eigenfunctions of the finite (truncated) Hankel transform, important for numerous applications. However, due to the very special behavior of the Hankel transform eigenfunctions, their direct numerical calculation often causes an essential loss of accuracy. Here, we discuss several simple, efficient and robust numerical techniques to compute Hankel transform eigenfunctions via the associated singular self-adjoint Sturm-Liouville operator. The properties of the proposed approaches are compared and illustrated by means of numerical experiments.

Gravitational Kaluza-Klein modes in the string-cigar braneworld

“…We solved the KK equations (34) and (55) by the matrix method [41] with second-order truncation error in order to attain the complete KK spectra of the GS and string-cigar scenarios. Since the angular number l leads to a degenerate spectrum, we are interested in the l = 0 solutions (which are referred to s waves).…”

Reprint of Variable-step finite difference schemes for the solution of Sturm–Liouville problems

“…We mention Initial Value Problems [4], Boundary Value Problems [5,6,11,12] and Sturm-Liouville problems [7][8][9][10]. In all these cases the proposed formulae show good stability properties and the developed codes turn out to be competitive with the existing software, in particular when the problem to be solved is stiff and the solution has a slope different from that of its derivatives.…”

“…Matrix M has a band structure (bandwidth depends on the order of the method) with some additional elements on the first and last rows, because of the initial and final methods (see [8,9] for more details).…”

“…The input parameter problem is a function which contains the data of the problem (5)- (9). A prototype of this function is the following.…”

“…The mainly used schemes are high order central finite differences, while other additional formulae are considered depending on the Sturm-Liouville classification of the endpoints of the interval. The proposed approach appears efficient and robust to solve with constant stepsize a particular singular Sturm-Liouville problem arising from the numerical computation of the eigenvalues and eigenfunctions of the finite (truncated) Hankel transform (see [9]), important for numerous applications in signal and image processing, and also for the solution of a two-parameter singular Sturm-Liouville problem derived from the numerical simulation of the so called 'whispering gallery' modes (WGMs) occurring inside a prolate spheroidal cavity, see [10].…”