Sampling of real multivariate polynomials and pluripotential theory

We consider the problem of stable sampling of multivariate real polynomials of large degree in a general framework where the polynomials are defined on an affine real algebraic variety $M$, equipped with a weighted measure. In particular, this framework contains the well-known setting of trigonometric polynomials (when $M$ is a torus equipped with its invariant measure), where the limit of large degree corresponds to a high frequency limit, as well as the classical setting of one-variable orthogonal algebraic polynomials (when $M$ is the real line equipped with a suitable measure), where the sampling nodes can be seen as generalizations of the zeros of the corresponding orthogonal polynomials. It is shown that a necessary condition for sampling, in the general setting, is that the asymptotic density of the sampling points is greater than the density of the corresponding weighted equilibrium measure of $M$, as defined in pluripotential theory. This result thus generalizes the well-known Landau type results for sampling on the torus, where the corresponding critical density corresponds to the Nyqvist rate, as well as the classical result saying that the zeros of orthogonal polynomials become equidistributed with respect to the logarithmic equilibrium measure, as the degree tends to infinity

Equidistribution of the Fekete points on the sphere

The Fekete points are the points that maximize a Vandermonde-type determinant that appears in the polynomial Lagrange interpolation formula. They are well suited points for interpolation formulas and numerical integration. We prove the asymptotic equidistribution of the Fekete points in the sphere. The way we proceed is by showing their connection with other array of points, the Marcinkiewicz-Zygmund arrays and the interpolating arrays, that have been studied recently

Interpolation in non-positively curved K\"ahler manifolds

We extend to any simply connected K\"ahler manifold with non-positive sectional curvature some conditions for interpolation in $\mathbb{C}$ and in the unit disk given by Berndtsson, Ortega-Cerd\`a and Seip. The main tool is a comparison theorem for the Hessian in K\"ahler geometry due to Greene, Wu and Siu, Yau.Comment: 9 pages, Late

Equidistribution of the Fekete points on the sphere

The Fekete points are the points that maximize a Vandermonde-type determinant that appears in the polynomial Lagrange interpolation formula. They are well suited points for interpolation formulas and numerical integration. We prove the asymptotic equidistribution of the Fekete points in the sphere. The way we proceed is by showing their connection with other array of points, the Marcinkiewicz-Zygmund arrays and the interpolating arrays, that have been studied recently

Pointwise estimates for the Bergman kernel of the weighted Fock space

We prove upper pointwise estimates for the Bergman kernel of the weighted Fock space of entire functions in $L^2(e^{-2\phi})$ where $\phi$ is a subharmonic function with $\Delta \phi$ a doubling measure. We derive estimates for the canonical solution operator to the inhomogeneous Cauchy-Riemann equation and we characterize the compactness of this operator in terms of $\Delta \phi$

Impact of IPDE-SQ personality disorders on the healthcare and societal costs of fibromyalgia patients: A cross-sectional study

Background: Data is lacking on comorbid personality disorders (PD) and fibromyalgia syndrome (FMS) in terms of prevalence, and associated healthcare and societal costs. The main aim of this study was to assess the prevalence of PD in FMS patients and to analyse whether the presence of comorbid PD is related to worse functional impairment and greater healthcare (medical visits, drug consumption, and medical tests) and societal costs. Methods: A cross-sectional study was performed using the baseline data of 216 FMS patients participating in a randomized, controlled trial carried out in three primary health care centres situated in the region of Barcelona, Spain. Measurement instruments included the International Personality Disorder Examination - Screening Questionnaire (IPDE-SQ), the Fibromyalgia Impact Questionnaire (FIQ), the Client Service Receipt Inventory (CSRI), and a socio-demographic questionnaire. Results: Most patients (65 %) had a potential PD according to the IPDE-SQ. The most prevalent PD were the avoidant (41.4 %), obsessive-compulsive (33.1 %), and borderline (27 %). We found statistically significant differences in functional impairment (FIQ scores) between FMS patients with potential PD vs non-PD (59.2 vs 51.1; p < 0.001). Multivariate regression analyses revealed that higher FIQ total scores and the presence of potential PD were related to more healthcare costs (primary and specialised care visits). Conclusions: As expected, PD are frequent comorbid conditions in patients with FMS. Our results suggest that the screening of comorbid PD in patients with FMS might be recommendable in order to detect potential frequent attenders to primary and specialised care

Sampling measures

We give a description of all measures such that for any function ia weighted Fock spaces the Lp norm with respect to the measure is equivalent to the usual norm in the space. We do so by a process of discretization that reduces the problem to the description of sampling sequences. The same kind of result holds for weighted Bergman spaces and the Paley-Wiener space

Sampling of real multivariate polynomials and pluripotential theory

We consider the problem of stable sampling of multivariate real polynomials of large degree in a general framework where the polynomials are defined on an affine real algebraic variety $M$, equipped with a weighted measure. In particular, this framework contains the well-known setting of trigonometric polynomials (when $M$ is a torus equipped with its invariant measure), where the limit of large degree corresponds to a high frequency limit, as well as the classical setting of one-variable orthogonal algebraic polynomials (when $M$ is the real line equipped with a suitable measure), where the sampling nodes can be seen as generalizations of the zeros of the corresponding orthogonal polynomials. It is shown that a necessary condition for sampling, in the general setting, is that the asymptotic density of the sampling points is greater than the density of the corresponding weighted equilibrium measure of $M$, as defined in pluripotential theory. This result thus generalizes the well-known Landau type results for sampling on the torus, where the corresponding critical density corresponds to the Nyqvist rate, as well as the classical result saying that the zeros of orthogonal polynomials become equidistributed with respect to the logarithmic equilibrium measure, as the degree tends to infinity