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Reconstruction of entire functions from irregularly spaced sample points. (English) Zbl 0860.30029

Let

$G\left(z\right)=\left(z-{\lambda }_{0}\right)\prod _{n=1}^{\infty }\left(1-\frac{z}{{\lambda }_{n}}\right)\left(1-\frac{z}{{\lambda }_{-n}}\right),$

where ${\lambda }_{n}\in ℝ$ and $|{\lambda }_{n}-n|\le 1/2$, $\forall n\in ℤ$. The authors prove that

$\underset{n}{sup}|{G}^{\left(k\right)}\left({\lambda }_{n}\right)/{G}^{\text{'}}\left({\lambda }_{n}\right)|<\infty ·$

They apply this result to a problem concerning Hermite interpolation of entire functions of exponential type belonging to ${L}^{p}\left(ℝ\right)$. The corresponding problem for functions of several variables is also considered.

MSC:
 30D10 Representations of entire functions by series and integrals 30D15 Special classes of entire functions; growth estimates 41A05 Interpolation (approximations and expansions) 94A05 Communication theory