A quadrature formula for the Hankel transform
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 weightedL2(−∞, ∞; 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 polynomialsH0μ(x),H1μ(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 polynomialHNμ(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.
AMS(MOS) subject classification41A55 44A20 42A68 81A12
Unable to display preview. Download preview PDF.
- R.G. Campos, The para-Bose oscillator in a finite linear space, Il Nuovo Cimento 100 B (1987) 485–492.Google Scholar
- R. G. Campos and L. Juárez Z., A discretization of the continuous Fourier transform, Il Nuovo Cimento 107B (1992) 703–711.Google Scholar
- T.S. Chihara,An Introduction to Orthogonal Polynomials (Gordon and Breach, New York, 1978).Google Scholar
- A. Erdélyi (ed.),Higher Transcendental Functions, Vol. 2 (McGraw-Hill, New York, 1954), p. 194, andTables of Integral Transforms, Vol. 2.Google Scholar
- G. Szegö,Orthogonal Polynomials, 4th ed. (American Mathematical Society, Providence, RI, 1976) chapter 8.Google Scholar
- A. Erdélyi, Asymptotic forms for Laguerre polynomials. J. Indian Math. Soc. 24 (1960) 235–250.Google Scholar
- B. Muckenhoupt, Asymptotic forms for Laguerre polynomials, Proc. Amer. Math. Soc. 24 (1970) 288–292.Google Scholar
- F.G. Tricomi, Sul comportamento asintotico dei polinomi di Laguerre, Ann. Mat. Pura. Appl. 28 (1949) 263–289.Google Scholar