Best asymptotic bounds for truncation error in sampling expansions of band-limited signals (Corresp.)

For band-limited functions with finite energy, it is known that bounds on the truncation error incurred when the function is approximated by
2N + 1
terms in the cardinal expansion can be obtained that go to zero like
N^{-1/2}
. If the additional restriction is made that a guard band is present (that is, the function is sampled faster than the minimum rate), then bounds can be obtained that go to zero like
N^{-1}
, both for finite energy functions and for functions having absolutely integrable Fourier transforms. It is shown here that these bounds are all asymptotically the best possible. It is also shown that, in the absence of a guard band, bund-limited functions with absolutely integrable Fourier transforms exist for which the truncation error goes to zero arbitrarily slowly.