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On locally bounded solutions of Schilling's problem

100%

Morawiec J.

Annales Polonici Mathematici

2001

tom 76

nr 3

169-188

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.

Probability distribution solutions of a general linear equation of infinite order

63%

Kochanek T. , Morawiec J.

Annales Polonici Mathematici

2009

tom 95

nr 2

103-114

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

Refinement type equations: sources and results

63%

Kapica R. , Morawiec J.

Banach Center Publications

2013

tom 99

nr 1

87-110

It has been proved recently that the two-direction refinement equation of the form $f(x) = ∑_{n∈ }c_{n,1}f(kx-n) + ∑_{n∈ℤ}c_{n,-1}f(-kx-n)$ 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 $f(x) = ∑_{n∈ℤ}cₙf(kx-n)$, which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation $f(x) = ∫_{ℝ}c(y)f(kx-y)dy$ has also various interesting applications. This equation is a special case of the continuous refinement type equation of the form $f(x) = ∫_{Ω} |K(ω)| f(K(ω)x-L(ω)) dP(ω)$, which has been studied recently in connection with probability theory. The purpose of this paper is to give a survey on the above refinement type equations. We begin with a brief introduction of types of refinement equations. In the first part we present several problems from different areas of mathematics which lead to the problem of the existence/nonexistence of integrable solutions of refinement type equations. In the second part we discuss and collect recent results on integrable solutions of refinement type equations, including some necessary and sufficient conditions for the existence/nonexistence of integrable solutions of the two-direction refinement equation. Finally, we say a few words on the existence of extremely non-measurable solutions of the two-direction refinement equation.

The set of probability distribution solutions of a linear functional equation

63%

Morawiec J. , Reich L.

Annales Polonici Mathematici

2008

tom 93

nr 3

253-261

Let (Ω,𝓐,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 $F(x) = ∫_{Ω} F(τ(x,ω))dP(ω)$ we determine the set of all its probability distribution solutions.

Probability distribution solutions of a general linear equation of infinite order, II

63%

Kochanek T. , Morawiec J.

Annales Polonici Mathematici

2010

tom 99

nr 3

215-224

Let (Ω,𝓐,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 𝓐-measurable with respect to the second variable. We discuss the problem of existence of probability distribution solutions of the general linear equation $F(x) = ∫_Ω F(τ(x,ω)) P(dω)$. We extend our uniqueness-type theorems obtained in Ann. Polon. Math. 95 (2009), 103-114.

On Probability Distribution Solutions of a Functional Equation

63%

Morawiec J. , Reich L.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

tom 53

nr 4

389-399

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(min{x/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, $φ|_{(-∞,0]} = 0$, $φ|_{[1,∞)} = 1$}, 𝒞 = {φ: ℝ → ℝ|φ is continuous, $φ|_{(-∞,0]} = 0$, $φ|_{[1,∞)} = 1$}. We prove that the above equation has at most one solution in 𝒞 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 between solutions in both classes.