Characterization of the reconstruction behavior of generalized sampling series for bandlimited Paley-Wiener functions

It is desirable to have a stable reconstruction, in the sense of a uniformly convergent reconstruction process, for a class of functions as large as possible. Therefore, in this paper general sampling series are analyzed for the frequently utilized Paley-Wiener space PW
¹
_π
, which is the largest space in the scale of Paley-Wiener spaces consisting of bounded and bandlimited functions. 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 stable reconstruction is not possible for the space PW
¹
_π
. Moreover, a universal function is given that shows the divergence behavior for all sampling series. Finally, a lower and an upper bound is derived and used to describe the asymptotic behavior of the peak value of the finite sampling series.