Abstract
We consider the functional equation
where 0 < a < 1, b in ℕ {1} and −1 = β 0 ≤ β1 ≤ … ≤ βb− 1 =1 are given parameters, ƒ: ℝ → ℝ is the unknown. We show that there is a unique bounded function ƒ which solves (F) and satisfies ƒ(t) = 0 for t < ∼-1/(1 − a), ƒ(t) = 1 for t > 1/(1 − a). This solution can be interpreted as the distribution function of a certain random series. It is known to be either singular or absolutely continuous, but the problem for which parameters it is absolutely continuous is largely open. We collect some previously established partial answers and generalize them. We also point out an interesting connection to the so-called Schilling equation.
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Dedicated to Prof. János Aczél on the occasion of his 70th birthday.
Research supported by NSERC and the Shrum endowment of Simon Fraser University
Research supported by a DFG fellowship
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Borwein, J.M., Girgensohn, R. Functional Equations and Distribution Functions. Results. Math. 26, 229–237 (1994). https://doi.org/10.1007/BF03323043
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DOI: https://doi.org/10.1007/BF03323043