Hysteresis filtering in the space of bounded measurable functions

Krejčí, Pavel ; Laurençot, Philippe

Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 3, pp. 755-772

Résumé

We define a mapping which with each function $u\in L^{\infty}(0, T)$ and an admissible value of $r > 0$ associates the function $\xi$ with a prescribed initial condition $\xi^{0}$ which minimizes the total variation in the $r$-neighborhood of $u$ in each subinterval $[0, t]$ of $[0, T]$. We show that this mapping is non-expansive with respect to $u$, $r$ and $\xi^{0}$, and coincides with the so-called play operator if $u$ is a regulated function.