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Mihai Putinar

Publications and source records attributed to Mihai Putinar.

At least 19 recordsLinked to original sources
arXiv2026

Hyperbolic distance matrix completion

A completion theory for hyperbolic distance data is developed at the interface of matrix analysis, graph theory, and hyperbolic geometry. Krein's characterization of the metric space embeddability in Lobachevsky space leads to a natural anchoring procedure that transforms the indefinite data into a positive semidefinite kernel. In analogy with positive semidefinite and Euclidean distance matrix completion, chordality of the specification graph is shown to be the necessary and sufficient condition for local Lorentz-Gram data to admit global completion. Existence is complemented by explicit constructions. For trees, we obtain geodesic-rectification and product-distance completions; for chordal graphs, the latter extends to matrix-valued transfers along clique-trees. The resulting canonical completion is characterized by sparsity of its inverse and by a maximum-absolute-determinant principle. Its metric distortion exhibits a sharp dichotomy governed by clique separator size. Applications to exact recovery from sparse hyperbolic measurements and to hierarchical and phylogenetic data are developed.

math.MG↗
arXiv2026

Distance preservers for Lobachevsky space

We obtain a complete description of the class of entrywise preservers of Lorentz-Gram matrices. This resolves, for the case of constant negative curvature, the classification of entrywise preservers obtained by Schoenberg in the zero-curvature (Euclidean) and constant-positive-curvature (spherical) settings.

math.CA↗
arXiv2026

Holomorphic Interpolation of Multivariate Completely Monotone Functions

The integral representation of completely monotone functions of several real variables as Laplace or Stieltjes-Fantappi\'e transforms of positive measures opens a Hilbert space path toward their finite-point interpolation by simpler functions. We combine, within a non-commutative Radon transform framework, the matrix pencil realization of the positive semi-definite Hankel kernel associated with the sampling of a completely monotone function with Weyl's operational calculus and Fantappi\`e's analytic calculus. The interpolation is achieved by finitely determined entire or rational functions, respectively, which are directionally completely monotone. In our relaxation scheme, the original positive measure is approximated by a sequence of specific Wigner distributions, which can also be regarded as analytic functionals. Throughout the interpolation process, tight bounds are enforced on the modulus or the real part of the holomorphic extension to the underlying tube domain.

math.FA↗
arXiv2025

The multivariate Herglotz-Nevanlinna class: Rational approximation

We return to Takagi's variational principle, generalized after forty years to two complex variables by Pfister. Both isolating some extremal rational functions associated to a bounded holomorphic function in the unit disk, respectively the bidisk. The rational inner functions arising from the Takagi-Pfister skew eigenvectors lead to a Pade type approximation scheme. For these rational functions, we prove a Montessus de Ballore type convergence theorem, on the polydisk in any complex dimension. On the natural and more restrictive class of Agler holomorphic functions with non-negative real part, we show that Cayley rational inner functions match any finite section of the Taylor expansion at a prescribed point. We derive from the Hilbert space proof that the finite section coefficient set of Taylor series of the Agler functions in the Herglotz-Nevanlinna setting is semi-algbraic. The pole distribution of the Takagi-Pfister interpolation sequence is identified as a main open question on the subject.

math.CV↗
arXiv2025

The Multivariate Herglotz-Nevanlinna Class: Superresolution

Bounded holomorphic interpolation problems associated to finitely many data have, in general, distinct solutions. Uniqueness arises only in some convex extreme configurations. Rational inner functions in a polydisk are the best understood examples in this sense. We analyze the continuity of global solutions as functions of the finite interpolation data in neighbourhoods of elements distinguished by this uniqueness property. Our study covers rational inner or Cayley rational inner functions in the polydisk and automorphisms of the Euclidean ball. The proof of the main superresolution result is derived from optimization theory techniques and volume estimates of sublevel sets of real polynomials, both emerging from Markov's multivariable moment problem.

math.FA↗
arXiv2024

Quadrature domains packing

Given a finite family of compact subsets of the complex plane we propose a certificate of mutual non-overlapping with respect to area measure. The criterion is stated as a couple of positivity conditions imposed on a four argument analytic/anti-analytic kernel defined in a neighborhood of infinity. In case the compact sets are closures of quadrature domains the respective kernel is rational, enabling an effective matrix analysis algorithm for the non-overlapping decision. The simplest situation of two disks is presented in detail from a matrix model perspective as well as from a Riemann surface potential theoretic interpretation.

math.SP↗
arXiv2024

Carleman factorization of layer potentials on smooth domains

One of the unexplored benefits of studying layer potentials on smooth, closed hypersurfaces of Euclidean space is the factorization of the Neumann-Poincaré operator into a product of two self-adjoint transforms. Resurrecting some pertinent indications of Carleman and M. G. Krein, we exploit this grossly overlooked structure by confining the spectral analysis of the Neumann-Poincaré operator to the amenable $L^2$-space setting, rather than bouncing back and forth the computations between Sobolev spaces of negative or positive fractional order. An enhanced, fresh new look at symmetrizable linear transforms enters into the picture in the company of geometric-microlocal analysis techniques. The outcome is manyfold, complementing recent advances on the theory of layer potentials, in the smooth boundary setting.

math.AP↗
arXiv2023

Generic properties of the Neumann-Poincaré operator: simplicity of eigenvalues and cyclic vectors

Two generic properties of the Neumann--Poincaré operator are investigated. We prove that non-zero eigenvalues of the Neumann--Poincaré operator on smooth boundaries in three dimensions and higher are generically simple in the sense of Baire category. We also prove that the functions defined by the fundamental solutions to the Laplace operator located at points outside the surface are generically cyclic vectors in the sense that the collection of those points where the functions are not cyclic vectors is of measure zero.

math.SP↗
arXiv2023

Matrix positivity preservers in fixed dimension. II: positive definiteness and strict monotonicity of Schur function ratios

We continue the study of real polynomials acting entrywise on matrices of fixed dimension to preserve positive semidefiniteness, together with the related analysis of order properties of Schur polynomials. Previous work has shown that, given a real polynomial with positive coefficients that is perturbed by adding a higher-degree monomial, there exists a negative lower bound for the coefficient of the perturbation which characterizes when the perturbed polynomial remains positivity preserving. We show here that, if the perturbation coefficient is strictly greater than this bound then the transformed matrix becomes positive definite given a simple genericity condition that can be readily verified. We identity a slightly stronger genericity condition that ensures positive definiteness occurs at the boundary. The analysis is complemented by computing the rank of the transformed matrix in terms of the location of the original matrix in a Schubert cell-type stratification that we have introduced and explored previously. The proofs require enhancing to strictness a Schur monotonicity result of Khare and Tao, to show that the ratio of Schur polynomials is strictly increasing along each coordinate on the positive orthant and non-decreasing on its closure whenever the defining tuples satisfy a coordinate-wise domination condition.

math.CA↗
arXiv2023

Negativity-preserving transforms of tuples of symmetric matrices

Compared to the entrywise transforms which preserve positive semidefiniteness, those leaving invariant the inertia of symmetric matrices reveal a surprising rigidity. We first obtain the classification of negativity preservers by combining recent advances in matrix analysis with some novel arguments relying on well chosen test matrices, Sidon sets from number theory, and analytic properties of absolutely monotone functions. We continue with the analogous classification in the multi-variable setting, revealing for the first time a striking separation of variables, with absolute monotonicity on one side and only homotheties on the other. We conclude with the complex analogue of this result.

math.CA↗
arXiv2023

Moment indeterminateness: the Marcel Riesz variational principle

The discrete data encoded in the power moments of a positive measure, fast decaying at infinity on euclidean space, is incomplete for recovery, leading to the concept of moment indeterminateness. On the other hand, classical integral transforms (Fourier-Laplace, Fantappiè, Poisson) of such measures are complete, often invertible via an effective inverse operation. The gap between the two non-uniqueness/ uniqueness phenomena is manifest in the dual picture, when trying to extend the measure, regarded as a positive linear functional, from the polynomial algebra to the full space of continuous functions. This point of view was advocated by Marcel Riesz a century ago, in the single real variable setting. Notable advances in functional analysis have root in Riesz' celebrated four notes devoted to the moment problem. A key technical ingredient being there the monotone approximation by polynomials of kernels of integral transforms. With inherent new obstacles we reappraise in the context of several real variables M. Riesz' variational principle. The result is an array of necessary and sufficient moment indeterminateness criteria, some raising real algebra questions, others involving intriguing analytic problems, all gravitating around the concept of moment separating function.

math.FA↗
arXiv2023

Determining the volume fraction in 2-phase composites and bodies using time varying applied fields

A body $Θ$ containing two phases, which may form a periodic composite with microstructure much smaller that the body, or which may have structure on a length scale comparable to the body, is subjected to slowly time varying boundary conditions that would produce an approximate uniform field in $Θ$ were it filled with homogeneous material. Here slowly time varying means that the wavelengths and attenuation lengths of waves at the frequencies associated with the time variation are much larger than the size of $Θ$, so that we can make a quasistatic approximation. At least one of the two phase does not have an instantaneous response but rather depends on fields at prior times. The fields may be those associated with electricity, magnetism, fluid flow in porous media, or antiplane elasticity. We find, subject to these approximations, that the time variation of the boundary conditions can be designed so boundary measurements at a specific time $t=t_0$ exactly yield the volume fractions of the phases, independent of the detailed geometric configuration of the phases. Moreover, for specially tailored time variations, the volume fraction can be exactly determined frommeasurements at any time $t$, not just at the specific time $t=t_0$. We also show how time varying boundary conditions, not oscillating at the single frequency $ω_0$, can be designed to exactly retrieve the response at $ω_0$.

math-ph↗
arXiv2022

Preservers of totally positive kernels and Polya frequency functions

Fractional powers and polynomial maps preserving structured totally positive matrices, one-sided Polya frequency functions, or totally positive kernels are treated from a unifying perspective. Besides the stark rigidity of the polynomial transforms, we unveil an ubiquitous separation between discrete and continuous spectra of such inner fractional powers. Classical works of Schoenberg, Karlin, Hirschman, and Widder are completed by our classification. Concepts of probability theory, multivariate statistics, and group representation theory naturally enter into the picture.

math.FA↗
arXiv2022

Jörg Eschmeier's mathematical work

An outline of Jörg Eschmeier's main mathematical contributions is organized both on a historical perspective, as well as on a few distinct topics. The reader can grasp from our essay the dynamics of spectral theory of commutative tuples of linear operators during the last half century. Some clear directions of future research are also underlined.

math.HO↗
arXiv2022

Hirschman-Widder densities

Hirschman and Widder introduced a class of Pólya frequency functions given by linear combinations of one-sided exponential functions. The members of this class are probability densities, and the class is closed under convolution but not under pointwise multiplication. We show that, generically, a polynomial function of such a density is a Pólya frequency function only if the polynomial is a homothety, and also identify a subclass for which each positive-integer power is a Pólya frequency function. We further demonstrate connections between the Maclaurin coefficients, the moments of these densities, and the recovery of the density from finitely many moments, via Schur polynomials.

math.CA↗
arXiv2022

Matrix compression along isogenic blocks

A matrix-compression algorithm is derived from a novel isogenic block decomposition for square matrices. The resulting compression and inflation operations possess strong functorial and spectral-permanence properties. The basic observation that Hadamard entrywise functional calculus preserves isogenic blocks has already proved to be of paramount importance for thresholding large correlation matrices. The proposed isogenic stratification of the set of complex matrices bears similarities to the Schubert cell stratification of a homogeneous algebraic manifold. An array of potential applications to current investigations in computational matrix analysis is briefly mentioned, touching concepts such as symmetric statistical models, hierarchical matrices and coherent matrix organization induced by partition trees.

math.RA↗
arXiv2021

Approximation in the mean on rational curves

In the presence of a positive, compactly supported measure on an affine algebraic curve, we relate the density of polynomials in Lebesgue $L^2$-space to the existence of analytic bounded point evaluations. Analogues to the complex plane results of Thomson and Brennan are obtained on rational curves.

math.CV↗