Definition
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[2CS]Suppose that either \(I=ℝ^+\) or \(I=ℝ\) in the following, for simplicity.
Let \(\varepsilon {\gt}0\); given a bounded function \(f:I→ℝ\) 1 , we define the ”sup transform” as the function \(g:I→ℝ\) given by
\begin{equation} \label{eq:trasf_ sup} g(x)=\sup _{y∈ (x,x+\varepsilon )} f(y)~ ~ . \end{equation}
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We summarize this transformation with the notation \(g=F(\varepsilon ,f)\).