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BIT Numerical Mathematics

, Volume 41, Issue 3, pp 490–503 | Cite as

Computing the Hilbert Transform of the Generalized Laguerre and Hermite Weight Functions

  • Walter Gautschi
  • Jörg Waldvogel
Article

Abstract

Explicit formulae are given for the Hilbert transform \(f_\mathbb{R} \)w(t)dt/(tx), where w is either the generalized Laguerre weight function w(t) = 0 if t ≤ 0, w(t) = tαet if 0 <#60; t <#60; ∞, and α > −1, x > 0, or the Hermite weight function w(t) = et2, −∞ <#60; t <#60; ∞, and −∞ <#60; x <#60; ∞. Furthermore, numerical methods of evaluation are discussed based on recursion, contour integration and saddle-point asymptotics, and series expansions. We also study the numerical stability of the three-term recurrence relation satisfied by the integrals \(f_\mathbb{R} \) π n (t;w)w(t)dt/(tx), n = 0 ,1 ,2 ,..., where π n (⋅w) is the generalized Laguerre, resp. the Hermite, polynomial of degree n.

Hilbert transform classical weight functions computational methods 

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Copyright information

© Swets & Zeitlinger 2001

Authors and Affiliations

  • Walter Gautschi
    • 1
  • Jörg Waldvogel
    • 2
  1. 1.Department of Computer SciencesPurdue UniversityWest LafayetteUSA
  2. 2.Seminar fürAngewandte MathematikZürichSwitzerland

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