Signal Sampling and Recovery Under Long-Range Dependent Noise with Application to Lack-of-Fit Tests

The paper examines the impact of the additive correlated noise on the accuracy of a signal reconstruction algorithm originating from the Whittaker-Shannon sampling interpolation formula. The proposed reconstruction method is a smooth post-filtering correction of the classical Whittaker-Shannon interpolation series. We assess the global accuracy of the proposed reconstruction algorithm for long memory stationary errors being independent on the sampling rate. We also examine a class of long memory noise processes for which the correlation function depends on the sampling rate. Exact rates at which the reconstruction error tends to zero are evaluated. We apply our theory to the problem of designing non-parametric lack-of-fit tests for verifying a parametric assumption on a signal. The theory of the asymptotic behavior of quadratic forms of stationary sequences is utilized in this case