In this paper, we present a method for solving inverse Sturm–Liouville problems by generalizing a Rundell–Sacks algorithm. The method is extended to deal with a general reference potential which can be adapted, e.g., to estimations of the jump-discontinuity points of the unknown potential. Moreover, its convergence properties are investigated. Numerical examples show that this modification can achieve more precise results from a given data set than the earlier method in using only the null reference potential and, therefore, the L2- and L∞-error can be reduced significantly.

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For point processes we establish a link between integration-by-parts-and splitting-formulas which can also be considered as integration-by-parts-formulas of a new type. First we characterize finite Papangelou processes in terms of their splitting kernels. The main part then consists in extending these results to the case of infinitely extended Papangelou and, in particular, Pólya and Gibbs processes. and denoted also by P ε z, . Here ε ∈ {−1, +1}. This class of processes has its origin in [26]. Finally the class of Gibbs processes G(V, ) is of fundamental importance. Such processes are, in general not uniquely, specified by some reference measure ∈ M and some abstract Boltzmann factor V (x, μ) by which one understands a non-negative, measurable function having the property V (x, ν + κ) = V (x, ν)V (x, κ), x ∈ X, ν, κ ∈ M ·· .

In [16], Zessin constructed the so-called Pólya sum process via partial integration technique. This process shares some important properties with the Poisson process such as complete randomness and infinite divisibility. This work discusses H-sufficient statistics for the Pólya sum process as it was done for the Poisson process in [14].

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