×

zbMATH — the first resource for mathematics

Une formule de quadrature pour les fonctions entières de type exponentiel. (A quadrature formula for entire functions of exponential type). (French) Zbl 0589.30024
If \(0<\sigma <\infty\) and f is an entire function of exponential type\(<2\sigma\), or of type \(2\sigma\) and \(f(x)=O(\exp (| x|^{- \delta}))\), \(\delta >1\), then \[ \int^{\infty}_{-\infty}f(x)dx=\pi \sigma^{-1}\sum^{\infty}_{-\infty}f(n\pi /\sigma). \] These results could be deduced from the Poisson summation formula, but the authors’ proof is based on L. Hörmander’s approximation method [Math. Scand. 3, 21-27 (1955; Zbl 0065.303)].
Reviewer: R.P.Boas

MSC:
30D20 Entire functions of one complex variable, general theory
65D30 Numerical integration
65B15 Euler-Maclaurin formula in numerical analysis
PDF BibTeX XML Cite