The sinc interpolation obtained in the Shannon-Nyquist sampling theory [20] for finite energy signals is fundamental in signal processing. However, it has some limitations. These limitations are (i) that the signal is sampled uniformly at , for some sampling period , and (ii) that is band-limited, that is, is the maximum frequency present in the signal. Under these conditions the signal can be reconstructed from the uniform samples according to the following sinc interpolation:

where is the sinc function. Several issues of practical interest arise when numerically implementing this interpolation. Besides the infinite dimension of the problem, uniform sampling is not realistic. Moreover, band-limitedness is just an approximation to reality.

If Nonuniform samples are available, the signal reconstruction suffers from dimensionality and ill-conditioning [12]. Indeed, if the set of time-shifted sinc functions is considered a frame for , a Gramian matrix equation represents the following interpolation:

where is a matrix containing shifted sinc functions, is a vector with entries the samples of the signal, and are the expansion coefficients of the projection in terms of the sinc functions. The infinite dimension of this problem makes it unsolvable, and when the dimension is reduced the problem becomes ill-posed [12].

The problem in part is due to using shifted sinc functions as basis: these functions are not appropriate for the interpolation given that time-limited signals are of infinite frequency support according to the uncertainty principle. A more appropriate basis is given by the Prolate Spheroidal Wave functions (PSWFs) [15].

The PSWFs are real-valued functions, with finite time support , that maximize their energy in a given bandwidth. These functions are the eigenfunctions of the sinc-based integral equation

where is the eigenvalue corresponding to the eigenfunction . Given the orthogonality of the sinc functions, the above definition leads to the orthogonality of the PSWF functions, so that they, like the sinc functions, are basis for finite energy signals. Thus the sinc function , which belongs to the space of band-limited signals, can be expanded in terms of the PSWFs as

Replacing this equation in the sinc interpolation gives an interpolation in terms of the PSWFs:

with expansion coefficients

If the signal has a finite support in time and approximately finite support in frequency, the above sums become finite. In that case, the upper limit of the sum in (5) depends on the approximately maximum frequency of , and in (6) the upper limit of the sum depends on the finite support of . A sampled-version of the signal could then be

where relates to the time support, and to the frequency support of .

Thus, if is time-limited and essentially band-limited, (5) and (6) provide a reconstruction of the signal from Nonuniformly sampled signals. In [14], it is shown that in the case of jittering sampling, that is, a subset of the uniform samples is available, the signal can be reconstructed.

A discrete version of the PSWF is used in these cases. The discrete PSW functions (or discrete PSW sequences or DPSS) are parameterized by the time bandwidth product where is their length and its normalized bandwidth. Like their analog counterparts, they are defined as the solution to the eigenvalue problem

where , and . The real eigenfunctions are the DPSS, and the corresponding eigenvalues relate to the their energy concentration. The DPSS are also orthonormal.