Given a probability space (Ω,, P) and a closed subset X of a Banach lattice, we consider functions f: X × Ω → X and their iterates $f\u207f:X\times {\Omega}^{\mathbb{N}}\to X$ defined by f¹(x,ω) = f(x,ω₁), ${f}^{n+1}(x,\omega )=f(f\u207f(x,\omega ),{\omega}_{n+1})$, and obtain theorems on the convergence (a.s. and in L¹) of the sequence (fⁿ(x,·)).

Given a probability space (Ω,,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 $f\u207f:X\times {\Omega}^{\mathbb{N}}\to X$ defined by f¹(x,ω ) = f(x,ω₁), ${f}^{n+1}(x,\omega )=f(f\u207f(x,\omega ),{\omega}_{n+1})$.

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.

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....

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