Computing the Hilbert Transform of the
Generalized Laguerre and Hermite Weight Functions
The Hilbert transform W(x) of the function w(t) is defined as the
Cauchy principal-value integral of w(t)/(t-x) over
the real t-axis.
We give explicit formulae for the W(x) in the cases of the generalized
Laguerre weight function,
w(t) = t^a exp(-t) (1+sign(t))/2, and of the
Hermite weight function, w(t) = exp(-t^2). Furthermore, several numerical
evaluation schemes are discussed, based on various representations of the
objects under consideration. In this connection we study the numerical
stability of the three-term recurrence relation satisfied by certain
related integrals involving the Laguerre or the Hermite polynomials.
The complete paper appeared in BIT 41, 2001, 490-503.
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