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The set of probability distribution solutions of a linear functional equation

Janusz Morawiec, Ludwig Reich (2008)

Annales Polonici Mathematici

Let (Ω,,P) be a probability space and let τ: ℝ×Ω → ℝ be a function which is strictly increasing and continuous with respect to the first variable, measurable with respect to the second variable. Given the set of all continuous probability distribution solutions of the equation F ( x ) = Ω F ( τ ( x , ω ) ) d P ( ω ) we determine the set of all its probability distribution solutions.

The smallest positive eigenvalue of a quasisymmetric automorphism of the unit circle

Dariusz Partyka (1995)

Banach Center Publications

This paper provides sufficient conditions on a quasisymmetric automorphism γ of the unit circle which guarantee the existence of the smallest positive eigenvalue of γ. They are expressed by means of a regular quasiconformal Teichmüller self-mapping φ of the unit disc Δ. In particular, the norm of the generalized harmonic conjugation operator A γ : is determined by the maximal dilatation of φ. A characterization of all eigenvalues of a quasisymmetric automorphism γ in terms of the smallest positive eigenvalue...

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