arXiv subjects

James Eldred Pascoe

Publications and source records attributed to James Eldred Pascoe.

16 recordsLinked to original sources

arXiv2026

Finite sum of squares, finite realization and noncommutative Carathéodory approximation

In the noncommutative polydisc, we first prove a positive sum of squares formula for a non-negative hereditary rational nc-function. The number of summands is finite. This result is used to derive a finite-dimensional realization formula for contractive nc-rational functions, where the colligation matrix is contractive. It is unitary if and only if the function is inner. Finally, we apply these results to generalize Carathéodory's classical theorem - approximating holomorphic self-maps of the unit disc by finite Blaschke products - to the setting of holomorphic functions on the noncommutative polydisc. This is in sharp contrast with the commutative situation where Carathéodory's approximation is known for Schur classes only in the unit disc and the unit bidisc.

math.FA↗

arXiv2026

On positive definite thresholding of correlation matrices

Standard thresholding techniques for correlation matrices often destroy positive semidefiniteness. We investigate the construction of positive definite functions that vanish on specific sets $K \subseteq [-1,1)$, ensuring that the thresholded matrix remains a valid correlation matrix. We establish existence results, define a criterion for faithfulness based on the linear coefficient of the normalized Gegenbauer expansion in analogy with Delsarte's method in coding theory, and provide bounds for thresholding at single points and pairs of points. We prove that for correlation matrices of rank $n$, any soft-thresholding operator that preserves positive semidefiniteness necessarily induces a geometric collapse of the feature space, as quantified by an $\mathcal{O}(1/n)$ bound on the faithfulness constant. Such demonstrates that geometrically unbiased soft-thresholding limits the recoverable signal.

math.ST↗

arXiv2026

Stable polynomials and bounded rational functions in the unit ball

We study polynomials with no zeros on the unit ball in complex Euclidean space with a view toward characterizing when a rational function is bounded on the ball. We give a complete local description of such polynomials in two variables near a boundary zero. In higher dimensions, we give a partial characterization of a simple boundary zero. Several applications are given including boundedness of rational functions with boundary singularities and constructions of examples with prescribed local properties.

math.CV↗

arXiv2026

Complete discrete Schoenberg-Delsarte theory for homogeneous spaces

We develop a theory of partially defined complete positivity preservers, extending Schoenberg's classical characterization to functions defined only on discrete subsets or constrained domains. We frame the extension problem through the theory of completely positive maps on operator systems -- we characterize general partially defined completely positive definite functions on general homogeneous spaces. We apply our interpolation to constrained packing problems and Delsarte theory, where one uses positive definite functions on homogeneous spaces to obtain bounds on various packing problems. We prove the specific positive definite function witnesses that a code is sharp for constrained angle codes must be from polynomials.

math.FA↗

arXiv2024

Stable polynomials and admissible numerators in product domains

Given a polynomial $p$ with no zeros in the polydisk, or equivalently the poly-upper half-plane, we study the problem of determining the ideal of polynomials $q$ with the property that the rational function $q/p$ is bounded near a boundary zero of $p$. We give a complete description of this ideal of numerators in the case where the zero set of $p$ is smooth and satisfies a non-degeneracy condition. We also give a description of the ideal in terms of an integral closure when $p$ has an isolated zero on the distinguished boundary. Constructions of multivariate stable polynomials are presented to illustrate sharpness of our results and necessity of our assumptions.

math.CV↗

arXiv2024

Local theory of stable polynomials and bounded rational functions of several variables

We provide detailed local descriptions of stable polynomials in terms of their homogeneous decompositions, Puiseux expansions, and transfer function realizations. We use this theory to first prove that bounded rational functions on the polydisk possess non-tangential limits at every boundary point. We relate higher non-tangential regularity and distinguished boundary behavior of bounded rational functions to geometric properties of the zero sets of stable polynomials via our local descriptions. For a fixed stable polynomial $p$, we analyze the ideal of numerators $q$ such that $q/p$ is bounded on the bi-upper half plane. We completely characterize this ideal in several geometrically interesting situations including smooth points, double points, and ordinary multiple points of $p$. Finally, we analyze integrability properties of bounded rational functions and their derivatives on the bidisk.

math.CV↗

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↗

arXiv2021

Positive univariate trace polynomials

A univariate trace polynomial is a polynomial in a variable x and formal trace symbols Tr(x^j). Such an expression can be naturally evaluated on matrices, where the trace symbols are evaluated as normalized traces. This paper addresses global and constrained positivity of univariate trace polynomials on symmetric matrices of all finite sizes. A tracial analog of Artin's solution to Hilbert's 17th problem is given: a positive semidefinite univariate trace polynomial is a quotient of sums of products of squares and traces of squares of trace polynomials.

math.RA↗

arXiv2020

Noncommutative rational functions invariant under the action of a finite solvable group

This paper describes the structure of invariant skew fields for linear actions of finite solvable groups on free skew fields in $d$ generators. These invariant skew fields are always finitely generated, which contrasts with the free algebra case. For abelian groups or solvable groups $G$ with a well-behaved representation theory it is shown that the invariant skew fields are free on $|G|(d-1)+1$ generators. Finally, positivity certificates for invariant rational functions in terms of sums of squares of invariants are presented.

math.RA↗

arXiv2020

Effective noncommutative Nevanlinna-Pick interpolation in the row ball, and applications

We provide an effective single-matrix criterion, in terms of what we call the elementary Pick matrix, for the solvability of the noncommutative Nevanlinna-Pick interpolation problem in the row ball, and provide some applications. In particular we show that the so-called "column-row property" fails for the free semigroup algebras, in stark contrast to the analogous commutative case. Additional applications of the elementary Pick matrix include a local dilation theorem for matrix row contractions and interpolating sequences in the noncommutative setting. Finally we present some numerical results related to the failure of the column-row property.

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↗

arXiv2019

Singularities of rational inner functions in higher dimensions

We study the boundary behavior of rational inner functions (RIFs) in dimensions three and higher from both analytic and geometric viewpoints. On the analytic side, we use the critical integrability of the derivative of a rational inner function of several variables to quantify the behavior of a RIF near its singularities, and on the geometric side we show that the unimodular level sets of a RIF convey information about its set of singularities. We then specialize to three-variable degree $(m,n,1)$ RIFs and conduct a detailed study of their derivative integrability, zero set and unimodular level set behavior, and non-tangential boundary values. Our results, coupled with constructions of non-trivial RIF examples, demonstrate that much of the nice behavior seen in the two-variable case is lost in higher dimensions.

math.CV↗

arXiv2018

Level curve portraits of rational inner functions

We analyze the behavior of rational inner functions on the unit bidisk near singularities on the distinguished boundary $\mathbb{T}^2$ using level sets. We show that the unimodular level sets of a rational inner function $ϕ$ can be parametrized with analytic curves and connect the behavior of these analytic curves to that of the zero set of $ϕ$. We apply these results to obtain a detailed description of the fine numerical stability of $ϕ$: for instance, we show that $\frac{\partial ϕ}{\partial z_1}$ and $\frac{\partial ϕ}{\partial z_2}$ always possess the same $L^{\mathfrak{p}}$-integrability on $\mathbb{T}^2$, and we obtain combinatorial relations between intersection multiplicities at singularities and vanishing orders for branches of level sets. We also present several new methods of constructing rational inner functions that allow us to prescribe properties of their zero sets, unimodular level sets, and singularities.

math.CV↗

arXiv2017

Derivatives of rational inner functions: geometry of singularities and integrability at the boundary

We analyze the singularities of rational inner functions on the unit bidisk and study both when these functions belong to Dirichlet-type spaces and when their partial derivatives belong to Hardy spaces. We characterize derivative $H^{\mathfrak{p}}$ membership purely in terms of contact order, a measure of the rate at which the zero set of a rational inner function approaches the distinguished boundary of the bidisk. We also show that derivatives of rational inner functions with singularities fail to be in $H^{\mathfrak{p}}$ for $\mathfrak{p}\ge\frac{3}{2}$ and that higher non-tangential regularity of a rational inner function paradoxically reduces the $H^{\mathfrak{p}}$ integrability of its derivative. We derive inclusion results for Dirichlet-type spaces from derivative inclusion for $H^{\mathfrak{p}}$. Using Agler decompositions and local Dirichlet integrals, we further prove that a restricted class of rational inner functions fails to belong to the unweighted Dirichlet space.

math.CV↗

arXiv2016

Regular and positive noncommutative rational functions

Call a noncommutative rational function $r$ regular if it has no singularities, i.e., $r(X)$ is defined for all tuples of self-adjoint matrices $X$. In this article regular noncommutative rational functions $r$ are characterized via the properties of their (minimal size) linear systems realizations $r=c^* L^{-1}b$. It is shown that $r$ is regular if and only if $L=A_0+\sum_jA_j x_j$ is privileged. Roughly speaking, a linear pencil $L$ is privileged if, after a finite sequence of basis changes and restrictions, the real part of $A_0$ is positive definite and the other $A_j$ are skew-adjoint. The second main result is a solution to a noncommutative version of Hilbert's 17th problem: a positive regular noncommutative rational function is a sum of squares.

math.RA↗

arXiv2016

Powerful numbers and the ABC-conjecture

The $abc$ conjecture is a very deep concept in number theory with wide application to many areas of number theory. In this article we introduce the conjecture and give examples of its applications. In particular we apply the $abc$ conjecture to the location of powerful numbers.

math.NT↗