arXiv subjects

Michael Jury

Publications and source records attributed to Michael Jury.

5 recordsLinked to original sources

arXiv2023

The kernel perspective on dynamic mode decomposition

This manuscript revisits theoretical assumptions concerning dynamic mode decomposition (DMD) of Koopman operators, including the existence of lattices of eigenfunctions, common eigenfunctions between Koopman operators, and boundedness and compactness of Koopman operators. Counterexamples that illustrate restrictiveness of the assumptions are provided for each of the assumptions. In particular, this manuscript proves that the native reproducing kernel Hilbert space (RKHS) of the Gaussian RBF kernel function only supports bounded Koopman operators if the dynamics are affine. In addition, a new framework for DMD, that requires only densely defined Koopman operators over RKHSs is introduced, and its effectiveness is demonstrated through numerical examples.

math.FA↗

arXiv2022

An Optimal Approximation Problem For Free Polynomials

Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial $f$ in $d$ freely noncommuting arguments, find a free polynomial $p_n$, of degree at most $n$, to minimize $c_n := \|p_nf-1\|^2$. (Here the norm is the $\ell^2$ norm on coefficients.) We show that $c_n\to 0$ if and only if $f$ is nonsingular in a certain nc domain (the row ball), and prove quantitative bounds. As an application, we obtain a new proof of the characterization of polynomials cyclic for the $d$-shift.

math.FA↗

arXiv2022

Noncommutative partially convex rational functions

Motivated by classical notions of bilinear matrix inequalities (BMIs) and partial convexity, this article investigates partial convexity for noncommutative functions. It is shown that noncommutative rational functions that are partially convex admit novel butterfly-type realizations that necessitate square roots. The notion of xy-convexity, a strengthening of partial convexity arising in connection with BMIs, is also considered. A characterization of xy-convex polynomials is given.

math.FA↗

arXiv2019

Noncommutative partial convexity via $Γ$-convexity

Motivated by classical notions of partial convexity, biconvexity, and bilinear matrix inequalities, we investigate the theory of free sets that are defined by (low degree) noncommutative matrix polynomials with constrained terms. Given a tuple of symmetric polynomials $Γ$, a free set is called $Γ$-convex if it closed under isometric conjugation by isometries intertwining $Γ$. We establish an Effros-Winkler Hahn-Banach separation theorem for $Γ$-convex sets; they are delineated by linear pencils in the coordinates of $Γ$ and the variables $x$.

math.FA↗

arXiv2009

Agler interpolation families of kernels

An abstract Pick interpolation theorem for a family of positive semi-definite kernels on a set $X$ is formulated. The result complements those in \cite{Ag} and \cite{AMbook} and will subsequently be applied to Pick interpolation on distinguished varieties \cite{JKM}.

math.FA↗