Convex-cyclic matrices, convex-polynomial interpolation and invariant convex sets
Nathan S. Feldman, Paul McGuire · Operators and Matrices · 2017
We define a convex-polynomial to be one that is a convex combination of the monomials {1,z,z 2 ,...} . This paper explores the intimate connection between peaking convexpolynomials, interpolating convex-polynomials, invariant convex sets, and the dynamics of matrices. In particular, we use these intertwined relations to both prove which matrices are convexcyclic while at the same time proving that we can prescribe the values and a finite number of the derivatives of a convex-polynomial subject to certain natural constraints. These properties are also equivalent to determining those matrices whose nonempty invariant closed convex sets are all invariant subspaces.