### Bounded solutions of Schilling's problem.

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We prove that for some parameters q ∈ (0,1) every solution f:ℝ → ℝ of the functional equation f(qx) = 1/(4q) [f(x-1) + f(x+1) + 2f(x)] which vanishes outside the interval [-q/(1-q),q/(1-q)] and is bounded in a neighbourhood of a point of that interval vanishes everywhere.

Let 0 < β < α < 1 and let p ∈ (0,1). We consider the functional equation φ(x) = pφ (x-β)/(1-β) + (1-p)φ(minx/α, (x(α-β)+β(1-α))/α(1-β)) and its solutions in two classes of functions, namely ℐ = φ: ℝ → ℝ|φ is increasing, ${\phi |}_{(-\infty ,0]}=0$, ${\phi |}_{[1,\infty )}=1$, = φ: ℝ → ℝ|φ is continuous, ${\phi |}_{(-\infty ,0]}=0$, ${\phi |}_{[1,\infty )}=1$. We prove that the above equation has at most one solution in and that for some parameters α,β and p such a solution exists, and for some it does not. We also determine all solutions of the equation in ℐ and we show the exact connection...

It has been proved recently that the two-direction refinement equation of the form $f\left(x\right)={\sum}_{n\in}{c}_{n,1}f(kx-n)+{\sum}_{n\in \mathbb{Z}}{c}_{n,-1}f(-kx-n)$ can be used in wavelet theory for constructing two-direction wavelets, biorthogonal wavelets, wavelet packages, wavelet frames and others. The two-direction refinement equation generalizes the classical refinement equation $f\left(x\right)={\sum}_{n\in \mathbb{Z}}c\u2099f(kx-n)$, which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation $f\left(x\right)={\int}_{\mathbb{R}}c\left(y\right)f(kx-y)dy$ has also various interesting applications....

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 $F\left(x\right)={\int}_{\Omega}F\left(\tau (x,\omega )\right)dP\left(\omega \right)$ we determine the set of all its probability distribution solutions.

Let (Ω,,P) be a probability space and let τ: ℝ × Ω → ℝ be strictly increasing and continuous with respect to the first variable, and -measurable with respect to the second variable. We obtain a partial characterization and a uniqueness-type result for solutions of the general linear equation $F\left(x\right)={\int}_{\Omega}F\left(\tau (x,\omega )\right)P\left(d\omega \right)$ in the class of probability distribution functions.

Let (Ω,,P) be a probability space and let τ: ℝ × Ω → ℝ be a mapping strictly increasing and continuous with respect to the first variable, and -measurable with respect to the second variable. We discuss the problem of existence of probability distribution solutions of the general linear equation $F\left(x\right)={\int}_{\Omega}F\left(\tau (x,\omega )\right)P\left(d\omega \right)$. We extend our uniqueness-type theorems obtained in Ann. Polon. Math. 95 (2009), 103-114.

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