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Error analysis associated with uniform Hermite interpolations of bandlimited functions. (English) Zbl 1208.41001

Let $\sigma >0$, $n\in ℤ$, and ${S}_{n}\left(t\right):=\frac{sin\left(\sigma t-n\pi \right)}{\sigma t-n\pi }$, for $t\ne \frac{n\pi }{\sigma }$; ${S}_{n}\left(t\right):=1$, for $t=\frac{n\pi }{\sigma }$. The authors consider the Hermite-type interpolation expansion

$f\left(t\right)=\sum _{n=-\infty }^{+\infty }\left\{f\left(\frac{n\pi }{\sigma }\right){S}_{n}^{2}\left(t\right)+{f}^{\text{'}}\left(\frac{n\pi }{\sigma }\right)\frac{sin\left(\sigma t-n\pi \right)}{\sigma }{S}_{n}\left(t\right)\right\},$

where $f\in P{W}_{\sigma }^{2}$, i.e., $f$ is an element of the Paley-Wiener space of entire ${L}^{1}\left(ℝ\right)$-functions of exponential type $\sigma$. They derive estimates (pointwise and uniform) for the truncation, amplitude and jitter type error from the expansion (1). In the last section they give some examples and comparisons, indicating that the use of formula (1) in approximation theory may be better than the use of the classical formula of Lagrange-type interpolation expansion.

##### MSC:
 41A05 Interpolation (approximations and expansions) 41A30 Approximation by other special function classes 94A20 Sampling theory