Janusz Morawiec

ON A REFINEMENT TYPE EQUATION

Abstract Let (Ω, A, P) be a complete probability space. We show that the trivial function is the unique L 1-solution of the following refinement type equation for a wide class of the given functions φ. This class contains functions of the form φ(x, ω) = α(ω)x – β(ω) with –∞ < ∫Ω log|α(ω)|dP(ω) < 0.

On the existence of irregular solutions of the two-coefficient dilation equation

Summary. We prove that for every real a and b such that

On a functional equation involving iterates and powers

ab \neq 0

Refinement equations and distributional fixed points

there exists a very irregular compactly supported solution

On a problem of Janusz Matkowski and Jacek Wesołowski

\varphi : {\Bbb R} \rightarrow {\Bbb R}

On local properties of compactly supported solutions of the two-coefficient dilation equation

of the functional equation¶

Attractor of Cantor Type with Positive Measure

\varphi(x)=a\varphi(2x)+b\varphi(2x-1)

Inhomogeneous refinement equations with random affine maps

.

Matrix refinement type equations

We present a complete list of all continuous solutions f:(0,+∞)→(0,+∞) of the equation f2(x)=γ[f(x)]αxβ, where α, β and γ>0 are given real numbers.MSC:39B22, 39B12, 26A18.

On continuous solutions of a problem of R.Schilling

Abstract Connections between L 1 -solutions of the refinement equation of the form f ( x ) = ∫ Ω | L | f ( Lx - M ) dP and distributional fixed points of a special random affine map are investigated. Using results on convergence of perpetuities the direct form of the Fourier transform of L 1 -solutions of the above refinement equation is obtained.