Seminario Geometría, Mecánica y Control
Positive Bases and dynamical sampling
Ponente: Juan I. Giribet (Universidad de San Andrés y CONICET, Argentina)Fecha: jueves 12 de febrero de 2026 - 15:00Lugar: Aula Naranja, ICMAT
Resumen:
In this talk we will revisit Chandler Davis’ notion of positive bases and explain how it fits naturally into the operator-driven constructions that appear in dynamical sampling. In particular we focus on families generated by iterates of a linear operator:
\(\{w_{1}\),\(Aw_{1}\),\(\dots\),\(A^{k_{1}}w_{1}\);\(\dots\);\(\;w_{r}\),\(Aw_{r}\),\(\dots\),\(A^{k_{r}}w_{r}\}\),
and ask when such orbit systems are not only spanning, but positively spanning — that is, when every vector can be written as a nonnegative combination of the orbit vectors.
Using spectral tools from dynamical sampling, we provide verifiable conditions on the spectrum of A that characterize when these orbit families form positive frames or minimal positive bases. In particular, we will highlight a complete description of when consecutive iterates of a single seed vector yield a minimal positive basis.
We will also show that, within this framework, the canonical dual of a positive frame is again positive, so that positivity is preserved under canonical reconstruction. Finally, we will discuss redundancy and how it applies to some engineering problems.
This is a joint work with Ursula Molter and Cristian Cruz.
