Approximate reconstruction of randomly sampled signals

We study the use of polynomial interpolation to approximate a function specified by samples taken at random moments satisfying a Poisson distribution with uniform mean sampling rate. Two different selection schemes are considered to determine which samples should be used in the construction of the polynomials, and detailed error estimates are derived for each case. The results are compared with the classical interpolation methods of convolution with a smoothing window. It is concluded that only low order polynomials are useful for interpolation in the presence of noise, but that they are comparable or superior to nonadaptive convolution in most cases, as well as computationally more efficient. Some simulation experiments are presented to support the theoretical estimates.