Uniform linear prediction of bandlimited processes from past samples (Corresp.)

For
x(t)
either a deterministic or stochastic signal band-limited to the normalized frequency interval
\mid\omega\mid \leq \pi
, explicit coefficients
\{ a_{kn} \}
are exhibited that have the property that \begin{equation} lim_{n \rightarrow \infty} \parallel x(t) - \sum_{1}^n a_{kn} x(t - kT) \parallel = 0 \end{equation} in an appropriate norm and for any constant intersample spacing
T
satisfying
0 < T < \fac{1}{2}
; that is,
x(t)
may be approximated arbitrarily well by a linear combination of past samples taken at any constant rate that exceeds twice the associated Nyquist rate. Moreover, the approximation of
x(t)
is uniform in the sense that the coefficients
\{ a_{kn} \}
do not depend on the detailed structure of
x(t)
but are absolute constants for any choice of
T
. The coefficients that are obtained provide a sharpening of a previous result by Wainstein and Zubakov where a rate in excess of three times the Nyquist rate was required.