Webinar

Abstract:

Variation Diminishing Systems

Variation diminishment, i.e., the reducing of zero-crossing or local extrema is a elementary property that we can find in common tools such as low-pass filtering, probability theory, interpolation theory as well as approximation theory such as machine learning. The old and well-established theory behind this concept, however, has barely entered the respected communities and been mostly forgotten. Therefore, in the first part of this talk, I will review the basic concepts of this property and provide a brief historical excursion from its beginnings to its hopefully bright future. In the second part, I will discuss the variation diminishing property of $k$-positive systems, i.e., systems that map inputs with $k-1$ sign changes to outputs of at most the same variation. I will characterise this property for the Toeplitz and Hankel operators of finite-dimensional linear time invariant systems by introducing the novel concept of compound systems. Our main result is that these operators have a dominant approximation in the form of series or parallel interconnections of $k$ first order positive systems. In particular, our characterisation generalises the well-known properties of positive systems ($k=1$) and as well as the so-called relaxation systems ($k=\infty$).