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Sur une formule de quadrature pour des fonctions entières. (On a quadrature formula for entire functions). (French) Zbl 0617.41043
Let m be an odd integer and \(\sigma >0\). Further, let \(a_{0,0}=1\) whereas for \(m>1\) and \(0\leq \mu \leq m-1\) let \(a_{\mu,m-1}=(1/\mu !)\psi^{(\mu)}(0)\) whre \(\psi (z)=\prod^{(m-1)/2}_{\mu =1}(1+z^ 2/\mu^ 2).\) We prove that if f is an entire function of exponential type \(t<(m+1)\sigma\), then: \[ \int^{\to +\infty}_{\to - \infty}f(x)dx=\frac{\pi}{\sigma}\sum^{m-1}_{\mu =0,\mu even}\frac{1}{(2\sigma)^{\mu}}a_{\mu,m-1}\sum^{+\infty}_{v=- \infty}f^{(\mu)}(v\pi /\sigma) \] provided the integral on the left (taken in the sense of Cauchy) and the \((m+1)/2\) series on the right are convergent. The quadrature formula remains valid for entire functions of order 1 type \((m+1)\sigma\) belonging to \(L^ 1(-\infty,+\infty)\).

MSC:
41A55 Approximate quadratures
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