Abstract
We present in this paper a quadrature formula for a certain Fourier-Bessel transform and, closely related to this, for the Hankel transform of order ν>−1. Such formulas originate in the context of a Galerkin-type projection of the weightedL 2(−∞, ∞; ωμ) space (ωμ is the weight function mentioned below) used to get a discrete representation of a certain physical problem in Quantum Mechanics. The generalized Hermitee polynomialsH μ0 (x),H μ1 (x),..., with weight function ωμ(x), are used as the basis on which such a projection takes place. It is shown that theN-dimensional vectors representing certain projected functions as well as the entries of theN×N matrix representing the kernel of that Fourier-Bessel transform, approach the exact functional values at the zeros of theNth generalized Hermitee polynomial whenN→∞.
These properties lead to propose this matrix as a finite representation of the kernel of the Fourier-Bessel transform involved in this problem and theN zeros of the generalized Hermitee polynomialH μN (x) as abscissas to yield certain quadrature formulae for this integral and for the related Hankel transform. The error function produced by this algorithm is estimated at theN nodes and its is shown to be of a smaller order than 1/N. This error estimate is valid for piecewise continuous functions satisfying certain integral conditions involving their absolute values. The algorithm is presented with some numerical examples.
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Communicated by C. Brezinski