Multidimensional Probabilistic Approximation in Wavelet Like Structure

Abstract

Let

$$ \varphi _{0}\left( x,y\right) :=\left\{ \begin{array}{l} 1,\text{ \ \ \ }x,y\geq 0 \\ 0,\text{ \ \ \ otherwise} \end{array} \right. $$

and $F\left( x,y\right)$ be a continuous probability distribution function on ℝ².

Then there exist linear wavelet type operators $L_{n}\left( F,x,y\right)$ which are also distribution functions and where the defining them wavelet function is $\varphi _{0}\left( x,y\right)$. These approximate $F\left( x,y\right)$ in the supnorm. The degree of this approximation is estimated by establishing a Jackson type inequality. Furthermore we give generalizations for the case of a wavelet function ≠ ϕ₀, which is just any distribution function on ℝ², also we extend these results in ℝ^r, r > 2. This chapter relies on [87].