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On a Gaussian quadrature formula for entire functions of exponential type. (English) Zbl 0721.41039
Constructive theory of functions, Proc. Int. Conf., Varna/Bulg. 1987, 132-137 (1988).
[For the entire collection see Zbl 0695.00019.]
G. Frappier and Q. I. Rahman [Ann. Sci. Math. Q. 10, 17-26 (1986; Zbl 0589.30024)] proved the following theorem for functions \(\in B_{\tau}\), the linear space of all entire functions of exponential type \(\tau\) : For \(f(z)\in B_{\tau}\), \(\tau <2\sigma\), the quadrature formula \[ (1)\quad \int^{\infty}_{-\infty}f(x)dx=(\pi /v)\sum^{\infty}_{\nu =-\infty}f(v\pi /\sigma) \] holds if \(\int^{\infty}_{-\infty}f(x)dx\) and \(\sum^{\infty}_{k=- \infty}f(v\pi /\sigma)\) exist in the sense of Cauchy. If \(f(z)\in B_{2\sigma}\) and if \(f(x)=O(| x|^{-\delta}),\delta >1\) then (1) holds. R. P. Boas [TĂ´hoku Math. J., II. Ser. 24, 121-125 (1972; Zbl 0238.42009)] had shown that if \(f(z)\in B_{2\sigma}\cap L_ 1\), then formula (1) holds. Hence the author proves another result of this kind for \(f(z)\in B_{2\sigma}\).
MSC:
41A55 Approximate quadratures