Matérn B-Splines and the Optimal Reconstruction of Signals

Abstract—Starting from the power spectral density of Matérn stochastic processes, we introduce a new family of splines that is defined in terms of the whitening operator of such processes. We show that these Matérn splines admit a stable representation in a B-spline-like basis. We specify the Matérn B-splines (causal and symmetric) and identify their key properties; in particular, we prove that these generate a Riesz basis and that they can be written as a product of an exponential with a fractional polynomial B-spline. We also indicate how these new functions bridge the gap between the fractional polynomial splines and the cardinal exponential ones. We then show that these splines provide the optimal reconstruction space for the minimum mean-squared error estimation of Matérn signals from their noisy samples. We also propose a digital Wiener-filter-like algorithm for the efficient determination of the optimal B-spline coefficients. Index Terms—Fractional splines, Matérn processes, Matérn splines, minimum mean-squared error (MMSE) estimation, power spectral density (PSD), Riesz bases. I