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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 $$L^ 1(-\infty,+\infty)$$.