Joaquim Ortega-Cerdà

Deformation of Gabor systems

Abstract We introduce a new notion for the deformation of Gabor systems. Such deformations are in general nonlinear and, in particular, include the standard jitter error and linear deformations of phase space. With this new notion we prove a strong deformation result for Gabor frames and Gabor Riesz sequences that covers the known perturbation and deformation results. Our proof of the deformation theorem requires a new characterization of Gabor frames and Gabor Riesz sequences. It is in the style of Beurlings characterization of sets of sampling for bandlimited functions and extends significantly the known characterization of Gabor frames “without inequalities” from lattices to non-uniform sets.

Marcinkiewicz--Zygmund inequalities

We study a generalization of the classical Marcinkiewicz-Zygmund inequalities. We relate this problem to the sampling sequences in the Paley-Wiener space and by using this analogy we give sharp necessary and sufficient computable conditions for a family of points to satisfy the Marcinkiewicz-Zygmund inequalities.

Beurling–Landau densities of weighted Fekete sets and correlation kernel estimates

Let Q be a suitable real function on C. An n-Fekete set corresponding to Q is a subset {z(n vertical bar) , . . . , z(nn)} of C which maximizes the expression Pi(n)(i infinity. In this note we prove that Fekete sets are, in a sense, maximally spread out with respect to the equilibrium measure. In general, our results apply only to a part of the Fekete set, which is at a certain distance away from the boundary of the droplet. However, for the potential Q = vertical bar z vertical bar(2) we obtain results which hold globally, and we conjecture that such global results are true for a wide range of potentials

Beurling–Landauʼs density on compact manifolds

Abstract Given a compact Riemannian manifold M , we consider the subspace of L 2 ( M ) generated by the eigenfunctions of the Laplacian of eigenvalue less than L ⩾ 1 . This space behaves like a space of polynomials and we have an analogy with the Paley–Wiener spaces. We study the interpolating and Marcinkiewicz–Zygmund (M–Z) families and provide necessary conditions for sampling and interpolation in terms of the Beurling–Landau densities. As an application, we prove the equidistribution of the Fekete arrays on some compact manifolds.

Equidistribution of Fekete Points on the Sphere

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 Fekete points on the sphere. The way we proceed is by showing their connection to other arrays of points, the so-called Marcinkiewicz–Zygmund arrays and interpolating arrays, that have been studied recently.

A LOWER BOUND IN NEHARI'S THEOREM ON THE POLYDISC

By theorems of Ferguson and Lacey (d = 2) and Lacey and Terwilleger (d > 2), Nehari’s theorem (i.e., if Hψ is a bounded Hankel form on H2(Dd) with analytic symbol ψ, then there is a function φ in L∞(Td ) such that ψ is the Riesz projection of g4) is known to hold on the polydisc Dd for d > 1. A method proposed in Helson’s last paper is used to show that the constant Cd in the estimate ‖φ‖∞ ≤ Cd ‖Hψ‖ grows at least exponentially with d; it follows that there is no analogue of Nehari’s theorem on the infinite-dimensional polydisc.

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 the 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.

SOME SPECTRAL PROPERTIES OF THE CANONICAL SOLUTION OPERATOR TO ¯ @ ON WEIGHTED FOCK SPACES

Abstract We characterize the Schatten class membership of the canonical solution operator to ∂ ¯ acting on L 2 ( e − 2 ϕ ) , where ϕ is a subharmonic function with Δϕ a doubling measure. The obtained characterization is in terms of Δϕ. As part of our approach, we study Hankel operators with anti-analytic symbols acting on the corresponding Fock space of entire functions in L 2 ( e − 2 ϕ ) .

Energy and discrepancy of rotationally invariant determinantal point processes in high dimensional spheres

We study expected Riesz s-energies and linear statistics of some determinantal processes on the sphere. In particular, we compute the expected Riesz and logarithmic energies of the determinantal processes given by the reproducing kernel of the space of spherical harmonics. This kernel defines the so called harmonic ensemble on the sphere. With these computations we improve previous estimates for the discrete minimal energy of configurations of points in the sphere. We prove a comparison result for Riesz 2-energies of points defined through determinantal point processes associated to isotropic kernels. As a corollary we get that the Riesz 2-energy of the harmonic ensemble is optimal among ensembles defined by isotropic kernels with the same trace. Finally, we study the variance of smooth and rough linear statistics for the harmonic ensemble and compare the results with the variance for the spherical ensemble.

Carleson Measures and Logvinenko-Sereda sets on compact manifolds

Abstract. Given a compact Riemannian manifold M of dimension , we study the space of functions of generated by eigenfunctions of eigenvalues less than associated to the Laplace–Beltrami operator on M. On these spaces we give a characterization of the Carleson measures and the Logvinenko–Sereda sets.