Equidistribution estimates for Fekete points on complex manifolds

We study the equidistribution of Fekete points in a compact complex manifold. These are extremal point configurations defined through sections of powers of a positive line bundle. Their equidistribution is a known result. The novelty of our approach is that we relate them to the problem of sampling and interpolation on line bundles, which allows us to estimate the equidistribution of the Fekete points quantitatively. In particular we estimate the Kantorovich-Wasserstein distance of the Fekete points to its limiting measure. The sampling and interpolation arrays on line bundles are a subject of independent interest, and we provide necessary density conditions through the classical approach of Landau, that in this context measures the local dimension of the space of sections of the line bundle. We obtain a complete geometric characterization of sampling and interpolation arrays in the case of compact manifolds of dimension one, and we prove that there are no arrays of both sampling and interpolation in the more general setting of semipositive line bundles.Comment: Improved version with a sharp decay rate in the estimate of the Kantorovich-Wasserstein distance of the Fekete points to its limiting measure (Theorem 2

Carleson Measures and Logvinenko-Sereda sets on compact manifolds

Given a compact Riemannian manifold $M$ of dimension $m\geq 2$, we study the space of functions of $L^2(M)$ generated by eigenfunctions of eigenvalues less than $L\geq 1$ associated to the Laplace-Beltrami operator on $M$. On these spaces we give a characterization of the Carleson measures and the Logvinenko-Sereda sets

Configurations of balls in Euclidean space that Brownian motion cannot avoid

We consider a collection of balls in Euclidean space and the problem of determining if Brownian motion has a positive probability of avoiding all the ball

Sèries de potències (aleatòries)

La construcció de funcions aleatòries amb propietats interessants és un tema de llarga tradició en les matemàtiques. Darrerament, i resseguint l'estudi dels valors propis de matrius aleatòries, s'està duent a terme l'estudi dels zeros de polinomis i funcions aleatòries holomorfes.The construction of random functions with interesting properties is a classical subject in Mathematics. Lately, there has been a renewed interest in the random zeros of polynomials and of entire functions spurred in part by the growing literature on the spectrum of random matrices

Interpolating and sampling sequences in finite Riemann surfaces

We provide a description of the interpolating and sampling sequences on a space of holomorphic functions on a finite Riemann surface, where a uniform growth restriction is imposed on the holomorphic functions

Sampling measures

We give a description of all measures such that for any function in a weighted Fock spaces the $L^p$ 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