Generalized Kernels for Reconstructing Irregularly Sampled Bandlimited Signals

Theoretically, irregularly sampled bandlimited signals can be reconstructed even if very few demands of the structure of the sampling set are fullfilled. In [1] the authors proposed an algorithm which renders a unique reconstruction without supposing Nyquist-density or any separation of the sampling instants. The proofs for the convergence of the obtained function sequence and the uniqueness of the solution can also be found in [1]. The main advantage of the algorithm in comparison with known solutions is that it demands little of the structure of the sampling set and makes it possible to implement it in a direct way, without the need of further modelling. The main disadvantage is that it may lead to systems of linear equations with very poorly conditioned matrices, ones which depend on the structure of the given set of sampling instants. This paper investigates alternative kernels to be used with this algorithm and gives the proof for the availability of them, i. e. the corresponding matrices are in any case regular. The new kernels turn out to be much more compatible with the requirements of numerical implementation because they lead to matrices with a significantly reduced condition number.