Extrapolating a band-limited function from its samples taken in a finite interval

A band-limited signal of finite energy can be reconstructed from its samples taken at the Nyquist rate. Moreover, the reconstruction is stable, a feature crucial for implementation: a small error in the sample values generates only a correspondingly small error in the resulting signal. The Nyquist sample values are mutually independent, so that knowledge of them in a given interval
- T \leq t \leq 0
provides hardly any information about the behavior of the signal outside the interval. However, when the samples are taken at a greater rate--a case referred to as "over-sampling"--they are interrelated, and this redundancy can be exploited in various ways to improve the behavior of the reconstruction procedure. A natural question is whether it can also be used to form accurate estimates of the signal outside the interval of observation; this problem is relevant as well to prediction theory. With oversampling, when
T = \infty
, so that the samples are known on the entire half-line
t < 0
, they determine the signal everywhere, although the reconstruction is now no longer stable. Here we examine the case of finite
T;
of course, a finite amount of data can yield only limited accuracy. We prove that the samples can be used to form an approximation to the signal outside the sampling interval, with an error which, asymptotically as
T \rightarrow \infty
, decreases exponentially in
T
, over a range which grows linearly with
T
. However, as in the limiting case, these approximations are not useful in practice, since they require the sample values to be known exactly. In the presence of measurement error, the nature of the results changes: good approximations are now available for only a bounded distance outside the interval of observation, regardless of its length, and their accuracy and range of validity can be increased only by improving the precision of sample reading. Since physical measurements are never perfect, it is this conclusion which counts for applications. The same results hold for the extrapolation of bounded band-limited signals.