For a complex-valued deterministic signal of finite energy band-limited to the normalized frequency band|w| \leq \piexplicit coefficients\{a_{kn}\}are found such that for anyTsatisfying0 < T \leq 1/2,\left| f(t)-\sum^{2n}_{k=1}a_{kn}f(t - kT)\right| \leq E_{f}\cdot \beta^{n}whereE_{f}is the signal energy and\beta \doteq 0.6863. Thus the estimate off(t)in terms of2npast samples taken at a rate equal to or in excess of twice the Nyquist rate converges uniformly at a geometric rate tof(t)on(- \infty , \infty). The suboptimal coefficients\{a_{kn}\}have the desirable property of being pure numbers independent of both the particular band-limited signal and of the selected sampling rate1/T. It is also shown that these same coefficients can be used to estimate the value ofx(t)of a wide-sense stationary random process in terms of past samples.