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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., Comment: 25 pages, no figures. This is an announcement of some of the results in a sequence of three closely related recent papers arXiv:2006.16213, arXiv:2008.05121, and arXiv:2101.02129. Final version, published in Mathematics Research Reports

In two phase materials, each phase having a non-local response in time, it has been found that for some driving fields the response somehow untangles at specific times, and allows one to directly infer useful information about the geometry of the material, such as the volume fractions of the phases. Motivated by this, and to obtain an algorithm for designing appropriate driving fields, we find approximate, measure independent, linear relations between the values that Markov functions take at a given set of possibly complex points, not belonging to the interval [-1,1] where the measure is supported. The problem is reduced to simply one of polynomial approximation of a given function on the interval [-1,1] and to simplify the analysis Chebyshev approximation is used. This allows one to obtain explicit estimates of the error of the approximation, in terms of the number of points and the minimum distance of the points to the interval [-1,1]. Assuming this minimum distance is bounded below by a number greater than 1/2, the error converges exponentially to zero as the number of points is increased. Approximate linear relations are also obtained that incorporate a set of moments of the measure. In the context of the motivating problem, the analysis also yields bounds on the response at any particular time for any driving field, and allows one to estimate the response at a given frequency using an appropriately designed driving field that effectively is turned on only for a fixed interval of time. The approximation extends directly to Markov-type functions with a positive semidefinite operator valued measure, and this has applications to determining the shape of an inclusion in a body from boundary flux measurements at a specific time, when the time-dependent boundary potentials are suitably tailored., 36 pages, 7 figures

The power moments of a positive measure on the real line or the circle are characterized by the non-negativity of an infinite matrix, Hankel, respectively Toeplitz, attached to the data. Except some fortunate configurations, in higher dimensions there are no non-negativity criteria for the power moments of a measure to be supported by a prescribed closed set. We combine two well studied fortunate situations, specifically a class of curves in two dimensions classified by Scheiderer and Plaumann, and compact, basic semi-algebraic sets, with the aim at enlarging the realm of geometric shapes on which the power moment problem is accessible and solvable by non-negativity certificates.

Multivalued linear operators, also known as linear relations, are studied on a specific class of weighted, composition transforms on Fock space. Basic properties of this class of linear relations, such as closed graph, boundedness, complex symmetry, real symmetry, or isometry are characterized in simple algebraic terms, involving their symbols.

A body $\Theta$ 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 $\Theta$ 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 $\Theta$, 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 $\omega_0$, can be designed to exactly retrieve the response at $\omega_0$., Comment: 35 pages, 8 figures

The Christoffel–Darboux kernel, a central object in approximation theory, is shown to have many potential uses in modern data analysis, including applications in machine learning. This is the first book to offer a rapid introduction to the subject, illustrating the surprising effectiveness of a simple tool. Bridging the gap between classical mathematics and current evolving research, the authors present the topic in detail and follow a heuristic, example-based approach, assuming only a basic background in functional analysis, probability and some elementary notions of algebraic geometry. They cover new results in both pure and applied mathematics and introduce techniques that have a wide range of potential impacts on modern quantitative and qualitative science. Comprehensive notes provide historical background, discuss advanced concepts and give detailed bibliographical references. Researchers and graduate students in mathematics, statistics, engineering or economics will find new perspectives on traditional themes, along with challenging open problems.

A series of physically motivated operations appearing in the study of composite materials are interpreted in terms of elementary continued fraction transforms of matrix valued, rational Stieltjes functions., Comment: 16 pages, 2 figures

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., 8 pages

The composition operators preserving total non-negativity and total positivity for various classes of kernels are classified, following three themes. Letting a function act by post composition on kernels with arbitrary domains, it is shown that such a composition operator maps the set of totally non-negative kernels to itself if and only if the function is constant or linear, or just linear if it preserves total positivity. Symmetric kernels are also discussed, with a similar outcome. These classification results are a byproduct of two matrix-completion results and the second theme: an extension of A.M. Whitney's density theorem from finite domains to subsets of the real line. This extension is derived via a discrete convolution with modulated Gaussian kernels. The third theme consists of analyzing, with tools from harmonic analysis, the preservers of several families of totally non-negative and totally positive kernels with additional structure: continuous Hankel kernels on an interval, P\'olya frequency functions, and P\'olya frequency sequences. The rigid structure of post-composition transforms of totally positive kernels acting on infinite sets is obtained by combining several specialized situations settled in our present and earlier works., Comment: Final version, to appear in Journal d'Analyse Mathematique. 63 pages, 2 tables, no figures

Starting from measured data, we develop a method to compute the fine structure of the spectrum of the Koopman operator with rigorous convergence guarantees. The method is based on the observation that, in the measure-preserving ergodic setting, the moments of the spectral measure associated to a given observable are computable from a single trajectory of this observable. Having finitely many moments available, we use the classical Christoffel–Darboux kernel to separate the atomic and absolutely continuous parts of the spectrum, supported by convergence guarantees as the number of moments tends to infinity. In addition, we propose a technique to detect the singular continuous part of the spectrum as well as two methods to approximate the spectral measure with guaranteed convergence in the weak topology, irrespective of whether the singular continuous part is present or not. The proposed method is simple to implement and readily applicable to large-scale systems since the computational complexity is dominated by inverting an N × N Hermitian positive-definite Toeplitz matrix, where N is the number of moments, for which efficient and numerically stable algorithms exist; in particular, the complexity of the approach is independent of the dimension of the underlying state-space. We also show how to compute, from measured data, the spectral projection on a given segment of the unit circle, allowing us to obtain a finite approximation of the operator that explicitly takes into account the point and continuous parts of the spectrum. Finally, we describe a relationship between the proposed method and the so-called Hankel Dynamic Mode Decomposition, providing new insights into the behavior of the eigenvalues of the Hankel DMD operator. A number of numerical examples illustrate the approach, including a study of the spectrum of the lid-driven two-dimensional cavity flow.

This survey contains a selection of topics unified by the concept of positive semidefiniteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or mass distributions). We put emphasis on entrywise operations which preserve positivity, in a variety of guises. Techniques from harmonic analysis, function theory, operator theory, statistics, combinatorics, and group representations are invoked. Some partially forgotten classical roots in metric geometry and distance transforms are presented with comments and full bibliographical references. Modern applications to high-dimensional covariance estimation and regularization are included.

The Holder continuity of the truncated moment map of a shade function in Euclidean space is established in the vicinity of a principal semi-algebraic set. The proof combines volume bounds of semi-algebraic sets and convex optimization methods. The main estimate is applied to a potential type transform specific to two real variables, for perturbations of quadrature domains.

With a proper function theoretic definition of the cloud of a positive measure with compact support in the real plane, a computational scheme of transforming the moments of the original measure into the moments of the uniformly distributed mass on the cloud is described. The main limiting operation involves exclusively truncated Christoffel-Darboux kernels, while error bounds depend on the spectral asymptotics of a Hankel kernel belonging to the Hilbert-Schmidt class.

The space of entire functions which are integrable with respect to the Gaussian weight, known also as the Fock space, is one of the preferred functional Hilbert spaces for modeling and experimenting harmonic analysis, quantum mechanics or spectral analysis phenomena. This space of entire functions carries a three parameter family of canonical isometric involutions. We characterize the linear differential operators acting on Fock space which are complex symmetric with respect to these conjugations. In parallel, as a basis of comparison, we discuss the structure of self-adjoint linear differential operators. The computation of the point spectrum of some of these operators is carried out in detail.

A generalization of a classical argument of Mark G. Krein leads us to the conclusion that the Neumann–Poincar´e operator associated to the Lame system of linear elastostatics equations in two dimensions has the same spectrum on the Lebesgue space of the boundary as the more natural energy space. A similar result for the Neumann–Poincare operator associated to the Laplace equation was stated by Poincare and was proved rigorously a century ago by means of a symmetrization principle for non-selfadjoint operators. We develop the necessary theoretical framework underlying the spectral analysis of the Neumann–Poincare operator, including also a discussion of spectral asymptotics of a Galerkin type approximation. Several examples from function theory of a complex variable and harmonic analysis are included.

By interpreting a truncated moment problem in two real variables in complex terms one exploits the hyponormality of the coordinate multiplier to obtain characterizations of: the moments of a positive measure, the associated complex orthogonal polynomials and the structure of the Hessenberg matrices. All reflecting the intricate structure of multivariate positive polynomials versus sums of squares or sums of hermitian squares of polynomials. The principal step in the proposed algorithm is stated in terms of a linear matrix inequality, making our approach prone for numerical applications.

Cauchy and exponential transforms are characterized, and constructed, as canonical holomorphic sections of certain line bundles on the Riemann sphere defined in terms of the Schwarz function. A well known natural connection between Schwarz reflection and line bundles defined on the Schottky double of a planar domain is briefly discussed in the same context.

We compare the finite central truncations of a given matrix with respect to two non-equivalent Hilbert space norms. While the limit sets of the finite sections spectra are merely located via numerical range bounds, the weak $$*$$ -limits of the counting measures of these spectra are proven in general to be gravi-equivalent with respect to the logarithmic potential in the complex plane. Classical methods of factorization of Volterra type or Wiener–Hopf type operators lead to a series of effective criteria of asymptotic equivalence, or uniform boundedness of the two sequences of truncations. Examples from function theory, integral equations and potential theory complement the theoretical results.

We study certain dynamical systems which leave invariant an indefinite quadratic form via semigroups or evolution families of complex symmetric Hilbert space operators. In the setting of bounded operators we show that a $$\mathcal {C}$$-selfadjoint operator generates a contraction $$C_0$$-semigroup if and only if it is dissipative. In addition, we examine the abstract Cauchy problem for nonautonomous linear differential equations possessing a complex symmetry. In the unbounded operator framework we isolate the class of complex symmetric, unbounded semigroups and investigate Stone-type theorems adapted to them. On Fock space realization, we characterize all $$\mathcal {C}$$-selfadjoint, unbounded weighted composition semigroups. As a byproduct we prove that the generator of a $$\mathcal {C}$$-selfadjoint, unbounded semigroup is not necessarily $$\mathcal {C}$$-selfadjoint.

In this essay, we present an overview of some important mathematical works of Professor Franciszek Hugon Szafraniec and a survey of his achievements and influence.

International audience; Spectral features of the empirical moment matrix constitute a resourceful tool for unveiling properties of a cloud of points, among which, density, support and latent structures. It is already well known that the empirical moment matrix encodes a great deal of subtle attributes of the underlying measure. Starting from this object as base of observations we combine ideas from statistics, real algebraic geometry, orthogonal polynomials and approximation theory for opening new insights relevant for Machine Learning (ML) problems with data supported on singular sets. Refined concepts and results from real algebraic geometry and approximation theory are empowering a simple tool (the empirical moment matrix) for the task of solving non-trivial questions in data analysis. We provide (1) theoretical support, (2) numerical experiments and, (3) connections to real world data as a validation of the stamina of the empirical moment matrix approach.

International audience; This paper aims at studying how finitely many generalized polarization tensors of an algebraic domain can be used to determine its shape. Precisely, given a planar set with real algebraic boundary, it is shown that the minimal polynomial with real coefficients vanishing on the boundary can be identified as the generator of a one dimensional kernel of a matrix whose entries are obtained from a finite number of generalized polarization tensors. The size of the matrix depends polynomially on the degree of the boundary of the algebraic domain. The density with respect to Hausdorff distance of algebraic domains among all bounded domains invites to extend via approximation our reconstruction procedure beyond its natural context. Based on this, a new algorithm for shape recognition/classification is proposed with some strong hints about its efficiency.

In previous work [Adv. Math. 298, pp. 325-368, 2016], the structure of the simultaneous kernels of Hadamard powers of any positive semidefinite matrix were described. Key ingredients in the proof included a novel stratification of the cone of positive semidefinite matrices and a well-known theorem of Hershkowitz, Neumann, and Schneider, which classifies the Hermitian positive semidefinite matrices whose entries are $0$ or $1$ in modulus. In this paper, we show that each of these results extends to a larger class of matrices which we term $3$-PMP (principal minor positive)., Comment: 12 pages. Final version, to appear in Linear Algebra and its Applications (the journal version is longer)

Thompson's partition of a cyclic subnormal operator into normal and completely non-normal components is combined with a non-commutative calculus for hyponormal operators for separating outliers from the cloud, in rather general point distributions in the plane. The main result provides exact transformation formulas from the power moments of the prescribed point distribution into the moments of the uniform mass carried by the cloud. The proposed algorithm solely depends on the Hessenberg matrix associated to the original data. The robustness of the algorithm is reflected by the insensitivity of the output under trace class, or by a theorem of Voiculescu, under certain Hilbert-Schmidt class, additive perturbations of the Hessenberg matrix.

This paper aims at showing the stability of the recovery of a smooth planar domain with a real algebraic boundary from a finite number of its generalized polarization tensors. It is a follow-up of the work [H. Ammari, M. Putinar, A. Steenkamp, and F. Triki, Math. Ann., 375 (2019), pp. 1337-1354], where it is proved that the minimal polynomial with real coefficients vanishing on the boundary can be identified as the generator of a one-dimensional kernel of a matrix whose entries are obtained from a finite number of generalized polarization tensors. The recovery procedure is implemented without any assumption on the regularity of the domain to be reconstructed, and its performance and limitations are illustrated. © 2019 Society for Industrial and Applied Mathematics. ISSN:1936-4954

In resonance to a recent geometric framework proposed by Douglas and Yang, a functional model for certain linear bounded operators with rank-one self-commutator acting on a Hilbert space is developed. By taking advantage of the refined existing theory of the principal function of a hyponormal operator we transfer the whole action outside the spectrum, on the resolvent of the underlying operator, localized at a distinguished vector. The whole construction turns out to rely on an elementary algebra body involving analytic multipliers and Cauchy transforms. A natural field theory interpretation of the resulting resolvent functional model is proposed.

A case study is added to our recent work on Quillen phe-nomenon. Pointwisepositivityofpolynomialsongeneralizedlemniscatesof the complex plane is related to sums of hermitian squares of rationalfunctions,andviaoperatorquantization, toessential subnormality. 1. Introduction and preliminariesA generalized lemniscate is the level set of the norm of the resolvent of acomplex matrix, localized at a non-zero vector. In analogy with classical lem-niscates (i.e., level sets of the modulus of a polynomial) this class of algebraiccurves exhibits special features well understood in the context of function the-ory, potential theory and fluid mechanics, see the references in [6]. The aim ofthe present note is to add a real algebra touch to these works. In analogy tothe recently outlined example of ellipses [7], a subtle difference between sums ofsquares and hermitian sums of squares decompositions of positive polynomialspersists on generalized lemniscates.1.1. Generalized lemniscatesA generalized lemniscate is areal algebraiccurvein R

Finite rank perturbations $T=N+K$ of a bounded normal operator $N$ on a separable Hilbert space are studied thanks to a natural functional model of $T$; in its turn the functional model solely relies on a perturbation matrix/ characteristic function previously defined by the second author. Function theoretic features of this perturbation matrix encode in a closed-form the spectral behavior of $T$. Under mild geometric conditions on the spectral measure of $N$ and some smoothness constraints on $K$ we show that the operator $T$ admits invariant subspaces, or even it is decomposable., Comment: 33 pages; to appear in Journal of Operator Theory

The non-negativity on the unit disk of the real part of the polar derivative of a polynomial is proved via an integral representation with a positive kernel, or as a consequence of a weighted sum of hermitian squares decomposition.

This survey contains a selection of topics unified by the concept of positive semidefiniteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or mass distributions). We put emphasis on entrywise operations which preserve positivity, in a variety of guises. Techniques from harmonic analysis, function theory, operator theory, statistics, combinatorics, and group representations are invoked. Some partially forgotten classical roots in metric geometry and distance transforms are presented with comments and full bibliographical references. Modern applications to high-dimensional covariance estimation and regularization are included.

Continuing our recent work in [5] we study polynomial masks of multivariate tight wavelet frames from two additional and complementary points of view: convexity and system theory. We consider such polynomial masks that are derived by means of the unitary extension principle from a single polynomial. We show that the set of such polynomials is convex and reveal its extremal points as polynomials that satisfy the quadrature mirror filter condition. Multiplicative structure of this polynomial set allows us to improve the known upper bounds on the number of frame generators derived from box splines. Moreover, in the univariate and bivariate settings, the polynomial masks of a tight wavelet frame can be interpreted as the transfer function of a conservative multivariate linear system. Recent advances in system theory enable us to develop a more effective method for tight frame constructions. Employing an example by S.W. Drury, we show that for dimension greater than 2 such transfer function representations of the corresponding polynomial masks do not always exist. However, for all wavelet masks derived from multivariate polynomials with non-negative coefficients, we determine explicit transfer function representations. We illustrate our results with several examples.