Discrete irregular sampling with larger gaps (Q5961707)

Let \(\widehat s\in l^2 (\mathbb{Z}_N)\) denote the discrete Fourier transform of \(s\in l^2 (\mathbb{Z}_N)\) and let \(B_M: =\{s\in l^2 (\mathbb{Z}_N): \widehat s(k)= 0\), \(|k|>M\}\). The author describes a new method for discrete irregular sampling, i.e. for \(r\) \((r\geq M+1)\) given pairs of sampling points \(\{n_1,n_1+1, \dots, n_r,n_r+1\} \subset \{1, \dots, N-1\}\) and corresponding values \(\{s(n_j), s(n_j+1)\}^r_{j=1}\) of \(s\in B_M\), they determine \(\widehat s(k)\) \((|k|<M)\). The problem is reduced to the solution of a linear system of equations with symmetric, positive definite coefficient matrix \(T=C+D^* \widetilde CD\), where \(C\), \(\widetilde C\) are Toeplitz matrices and where \(D\) is a diagonal matrix. The authors estimate the condition number of \(T\) under some conditions on the sampling gaps which allows gaps larger than the Nyquist rate. Numerical tests compare their method for example with the adaptive weights Toeplitz method of \textit{H. G. Feichtinger}, \textit{K. Gröchenig} and \textit{T. Strohmer} [Numer. Math. 69, No. 4, 423-440 (1995; Zbl 0837.65148)].