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1
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Theorems of Thron's type for random-valued vector functions and the Krein-Rutman theorem

100%
Kapica R.
Annales Polonici Mathematici
|
2005
|
tom 85
|
nr 1
13-24
EN
We propose stochastic versions of some theorems of W. J. Thron [14] on the speed of convergence of iterates for random-valued functions on cones in Banach spaces.
2
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Convergence of sequences of iterates of random-valued vector functions

100%
Kapica R.
Colloquium Mathematicum
|
2003
|
tom 97
|
nr 1
1-6
EN
Given a probability space (Ω,𝓐, P) and a closed subset X of a Banach lattice, we consider functions f: X × Ω → X and their iterates $fⁿ: X × Ω^{ℕ} → X$ defined by f¹(x,ω) = f(x,ω₁), $f^{n+1}(x,ω) = f(fⁿ(x,ω),ω_{n+1})$, and obtain theorems on the convergence (a.s. and in L¹) of the sequence (fⁿ(x,·)).
3
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On the convergence of sequences of iterates of random-valued vector functions

100%
Kapica R.
Annales Polonici Mathematici
|
2007
|
tom 90
|
nr 3
193-201
EN
Given a probability space (Ω,𝓐,P) and a subset X of a normed space we consider functions f:X × Ω → X and investigate the speed of convergence of the sequence (fⁿ(x,·)) of the iterates $fⁿ:X × Ω^{ℕ} → X$ defined by f¹(x,ω ) = f(x,ω₁), $f^{n+1}(x,ω) = f(fⁿ(x,ω),ω_{n+1})$.
4
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Refinement type equations: sources and results

63%
Kapica R., Morawiec J.
Banach Center Publications
|
2013
|
tom 99
|
nr 1
87-110
EN
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.
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