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The bandlimited interpolation of signals and the convergence behavior of the Shannon sampling series are discussed in order to show that it is desirable to have a uniformly convergent reconstruction, for as large a space of signals as possible. In this paper general sampling series are analyzed for the frequently utilized Paley-Wiener space PW pi 1, which is the largest space in the scale of Paley-Wiener spaces. The analysis is done not only for the Shannon sampling series, but for a whole class of axiomatically defined reconstruction processes. It is shown that for this very general class, which contains all common sampling series including the Shannon sampling series, a uniformly convergent reconstruction is not possible for the space PW pi 1. Moreover, a universal signal is identified that causes the divergence behavior for all sampling series. Finally, a lower and an upper bound are derived and used to describe the asymptotic behavior of the peak value of the finite sampling series.
Date of Publication: July 2008