For integers n, q = 1, 2, 3, . . ., let Pol n,q denote the C-linear space of polynomials in z andz, of degree ≤ n − 1 in z and of degree ≤ q − 1 inz. We supply Pol n,q with the inner product structure ofthe resulting Hilbert space is denoted by Pol m,n,q . Here, it is assumed that m is a positive real. We let K m,n,q denote the reproducing kernel of Pol m,n,q , and study the associated determinantal process, in the limit as m, n → +∞ while n = m + O(1); the number q, the degree of polyanalyticity, is kept fixed. We call these processes polyanalytic Ginibre ensembles, because they generalize the Ginibre ensemble -the eigenvalue process of random (normal) matrices with Gaussian weight. A possible interpretation is that we permit a few higher Landau levels. We consider local blow-ups of the polyanalytic Ginibre ensembles around points in the spectral droplet, which is here the closed unit disk D := {z ∈ C : |z| ≤ 1}. We obtain asymptotics for the blow-up process, using a blow-up to characteristic distance m −1/2 ; the typical distance is the same both for interior and for boundary points ofD. This amounts to obtaining the asymptotical behavior of the generating kernel K m,n,q . Following [2], the asymptotics of the K m,n,q is rather conveniently expressed in terms of the Berezin measure (and density) dB z m,n,q (w) := B z m,n,q (w)dA(w), B z m,n,q (w) = |K m,n,q (z, w)| 2 K m,n,q (z, z) e −m|z| 2 .For interior points |z| < 1, we obtain that dB z m,n,q (w) → dδ z in the weak-star sense, where δ z denotes the unit point mass at z. Moreover, if we blow up to the scale of m −1/2 around z, we get convergence to a measure which is Gaussian for q = 1, but exhibits more complicated Fresnel zone behavior for q > 1. In contrast, for exterior points |z| > 1, we have instead that dB z m,n,q (w) → dω(w, z, D e ), where dω(w, z, D e ) is the harmonic measure at z with respect to the exterior disk D e := {w ∈ C : |w| > 1}. For boundary points, |z| = 1, the Berezin measure dB z m,n,q converges to the unit point mass at z, like for interior points, but the blow-up to the scale m −1/2 exhibits quite different behavior at boundary points compared with interior points. The Fresnel-type pattern appears also for boundary points when q > 1, but then it is not rotationally symmetric.

Abstract. We derive necessary density conditions for sampling and for interpolation in general reproducing kernel Hilbert spaces satisfying some natural conditions on the geometry of the space and the reproducing kernel. If the volume of shells is small compared to the volume of balls (weak annular decay property) and if the kernel possesses some off-diagonal decay or even some weaker form of localization, then there exists a critical density D with the following property: a set of sampling has density ≥ D, whereas a set of interpolation has density ≤ D. The main theorem unifies many known density theorems in signal processing, complex analysis, and harmonic analysis. For the special case of bandlimited function we recover Landau's fundamental density result. In complex analysis we rederive a critical density for generalized Fock spaces. In harmonic analysis we obtain the first general result about the density of coherent frames.

Abstract. We consider the q-analytic functions on a given planar domain Ω, square integrable with respect to a weight. This gives us a q-analytic Bergman kernel, which we use to extend the Bergman metric to this context. We recall that f is q-analytic if∂ q f = 0 for the given positive integer q.We also obtain asymptotic formulae for the q-analytic Bergman kernel in the setting of degenerating power weights e −2mQ , as the positive real parameter m tends to infinity. This is only known for q = 1 in view of the work of Tian, Yau, Zelditch, and Catlin. We remark here that since a q-analytic function may be identified with a vector-valued holomorphic function, the Bergman space of q-analytic functions may be understood as a vector-valued holomorphic Bergman space supplied with a certain singular local metric on the vectors. Finally, we apply the obtained asymptotics for q = 2 to the bianalytic Bergman metrics, and after suitable blow-up, the result is independent of Q for a wide class of potentials Q. We interpret this as an instance of geometric universality. OverviewIn Section 2, we define, in the one-variable context, the weighted Bergman spaces and their polyanalytic extensions, and in Section 3, we consider the various possible ramifications for Bergman metrics. It should be remarked that the polyanalytic Bergman spaces can be understood as vector-valued (analytic) Bergman spaces with singular local inner product matrix.Generally speaking, Bergman kernels are difficult to obtain in explicit form. However, it is sometimes possible to obtain an asymptotic expansion for them, for instance as the weight degenerates in a power fashion. In Sections 5-9, we extend the asymptotic expansion to the polyanalytic context. Our analysis is based on the microlocal PDE approach of Berman, Berndtsson, Sjöstrand [8]. We focus mainly on the biholomorphic (bianalytic) case, and obtain the explicit form of the first few terms of the expansion. In Section 4, we estimate the norm of point evaluations on bianalytic Bergman spaces, which is later needed to estimate the bianalytic Bergman kernel along the diagonal.Finally, in Section 10, we apply the obtained asymptotics for q = 2 to the bianalytic Bergman metrics introduced in Section 3, and after suitable blow-up, the resulting metrics turn out to be independent of the given potential (which defines the power weight). We interpret this as an instance of geometric universality.2. Weighted polyanalytic Bergman spaces and kernels 2.1. Basic notation. We let C denote the complex plane and R the real line. For z 0 ∈ C and positive real r, let D(z 0 , r) be the open disk centered at z 0 with radius r; moreover, we let T(z 0 , r)

We consider sharp inequalities for Bergman spaces of the unit disc, establishing analogues of the inequality in Carleman's proof of the isoperimetric inequality and of Weissler's inequality for dilations. By contractivity and a standard tensorization procedure, the unit disc inequalities yield corresponding inequalities for the Bergman spaces of Dirichlet series. We use these results to study weighted multiplicative Hankel forms associated with the Bergman spaces of Dirichlet series, reproducing most of the known results on multiplicative Hankel forms associated with the Hardy spaces of Dirichlet series. In addition, we find a direct relationship between the two type of forms which does not exist in lower dimensions. Finally, we produce some counter-examples concerning Carleson measures on the infinite polydisc.

We investigate the completeness of Gabor systems with respect to several classes of window functions on rational lattices. Our main results show that the time-frequency shifts of every finite linear combination of Hermite functions with respect to a rational lattice are complete in L 2 (R), thus generalizing a remark of von Neumann (and proved by Bargmann, Perelomov et al.). An analogous result is proven for functions that factor into certain rational functions and the Gaussian. The results are also interesting from a conceptual point of view since they show a vast difference between the completeness and the frame property of a Gabor system. In the terminology of physics we prove new results about the completeness of coherent state subsystems.2010 Mathematics Subject Classification. 42C30,42C15,81R30.