On the truncation error of the cardinal sampling expansion

Modern communication theory and practice are heavily dependent on the representation of continuous parameter signals by linear combinations, involving a denumerable set of random variables. Among the best known and most useful is the cardinal series
f_{n} (t) = \sum^{+n}_{-n} f(k) \frac{\sin \pi (t - k)}{ \pi ( t - k )}
for deterministic functions and wide-sense stationary stochastic processes bandlimited to
(-\pi, \pi)
. When, as invariably occurs in applications, samples
f(k)
are available only over a finite period, the resulting finite approximation is subject to a truncation error. For functions which are
L_{1}
Fourier transforms supported on
[-\pi + \delta, + \pi - \delta]
, uniform trunction error bounds of the form
O(n^{-1})
are known. We prove that analogous
O(n^{-1})
bounds remain valid without the guard band
\delta
and for Fourier-Stieltjes transforms; we require only a bounded variation condition in the vicinity of the endpoints
- \pi
and
+ \pi
of the basic interval. Our methods depend on a Dirichlet kernel representation for
f_{n}(t)
and on properties of functions of bounded variation; this contrasts with earlier approaches involving series or complex variable theory. Other integral kernels (such as the Fejer kernel) yield certain weighted truncated cardinal series whose errors can also be bounded. A mean-square trunction error bound is obtained for bandlimited wide-sense stationary stochastic processes. This error estimate requires a guard band, and leads to a uniform
O(n^{-2})
bound. The approach again employs the Dirichlet kernel and draws heavily on the arguments applied to deterministic functions.