How slow is Shannon׳s reconstruction for bandlimited signals?

Shannon׳s sampling theorem is fundamental in signal processing. It provides the exact reconstruction of a bandlimited signal from its samples at the Nyquist rate by the cardinal series. But this reconstruction formula is not applied widely in practice because of its slow convergence. It is of theoretical interests to find out how slow the convergence of this reconstruction could be. In this work, the convergence rate of Shannon׳s reconstruction is studied by the theory on the slow convergence of operator sequences. It is proved that the Shannon׳s reconstruction consists of a sequence of “arbitrarily slow  ” convergent operators. Specifically, for any positive sequence α(n)→0α(n)→0, there exists a bandlimited signal f such that the n  -th truncation error of its cardinal series is larger than α(n)α(n) for all n, where the truncation errors are measured in Lp norms, for 1