Gunnar Forst

Potential theory on locally compact Abelian groups

I. Harmonic Analysis.- 1. Notation and Preliminaries.- 2. Some Basic Results From Harmonic Analysis.- 3. Positive Definite Functions.- 4. Fourier Transformation of Positive Definite Measures.- 5. Positive Definite Functions on ?.- 6. Periodicity.- II. Negative Definite Functions and Semigroups.- 7. Negative Definite Functions.- 8. Convolution Semigroups.- 9. Completely Monotone Functions and Bernstein Functions.- 10. Examples of Negative Definite Functions and Convolution Semigroups.- 11. Contraction Semigroups.- 12. Translation Invariant Contraction Semigroups.- III. Potential Theory for Transient Convolution Semigroups.- 13. Transient Convolution Semigroups.- 14. Transient Convolution Semigroups on the Half-Axis and Integrals of Convolution Semigroups.- 15. Convergence Lemmas and Potential Theoretic Principles.- 16. Excessive Measures.- 17. Fundamental Families Associated With Potential Kernels.- 18. The Levy Measure for a Convolution Semigroup.- Symbols.- General Index.

A characterization of self-decomposable probabilities on the half-line

SummaryIt is shown that a probability measure μ on ℝ+ is self-decomposable if and only if for s>0 the sequence

A characterization of potential kernels on the positive half-line

\left( {\frac{1}{{n!}}\mathop \smallint \limits_0^\infty e^{ - ts} (ts)^n d\mu (t)} \right)_{n \geqq 0} ,

Subordinates of the Poisson semigroup

determines a probability on ℕ0, that is self-decomposable in the sense of Steutel and van Harn.

A convolution equation relating the generalized Γ-convolutions and the Bondesson Class

SummarySelf-decomposable probability measures μ on ℝ+ are characterized in terms of minus the logarithm of the Laplace transform of μ, say f, by the requirement that s→sf′(s) is again minus the logarithm of the Laplace transform of an infinitely divisible probability on ℝ+. Iteration of this condition yields characterizations in the case of ℝ+ of Urbaniks classes Ln of multiply self-decomposable probabilities. The analogous characterization for discrete (multiply) self-decomposable probabilities on ℝ+ is discussed and used to give a representation of the generating functions for discrete completely self-decomposable probabilities on ℤ+. Classes of generalized Γ-convolutions analogous to the multiply self-decomposable probabilities on ℝ+ are studied as well as their discrete counterparts.

What Costs Are Minimized by Huffman trees

SummaryThe set P of potential kernels on the half-line [0, τ[ is characterized as the set of positive measures k on [0,τ[ for which the Laplace transform f=ℒк has the property that the sequence

Potential Theory for Transient Convolution Semigroups

\left( {S^n ( - 1)^n \frac{{D^n f(s)}}{{n!}}} \right)_{n \geqq 0}