## On the localization principle for multi-dimensional Shannon and conjugate Shannon sampling series. (Über das Lokalisierungsprinzip bei mehrdimensionalen Shannonschen und konjugierten Shannonschen Abtastreihen.)(German)Zbl 0931.42025

Summary: In this paper the behaviour of multidimensional Shannon and conjugate Shannon sampling series is investigated. For the one-dimensional Shannon as well as the one-dimensional conjugate Shannon sampling series there hold some localization principles. These localization principles are useful for the investigation of the convergence behaviour of the one-dimensional Shannon and conjugate Shannon sampling series. In this paper it is shown that for more than one dimension these localization principles cannot hold. These results solve a problem proposed by P. L. Butzer.

### MSC:

 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 41A05 Interpolation in approximation theory 94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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### References:

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