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We characterize sampling and interpolating sets with derivatives in weighted Fock spaces on the complex plane in terms of their weighted Beurling densities., Comment: 27 pages. Some minor typographical errors were corrected

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., Comment: This paper has been accepted for publication in Transactions of the AMS

Answering a question of Lindholm, we prove strict density inequalities for sampling and interpolation in Fock spaces of entire functions in several complex variables defined by a plurisubharmonic weight. In particular, these spaces do not admit a set that is simultaneously sampling and interpolating. To prove optimality of the density conditions, we construct sampling sets with a density arbitrarily close to the critical density. The techniques combine methods from several complex variables (estimates for $\bar \partial$) and the theory of localized frames in general reproducing kernel Hilbert spaces (with no analyticity assumed). The abstract results on Fekete points and deformation of frames may be of independent interest., 33 pages

We propose a methodology that detects signal components of a signal in white noise based on a hyperbolic triangulation of its wavelet transform (WT). The theoretical background is a connection between analyticity inducing wavelets and Gaussian analytic functions. This relation allows us to obtain some useful details on the random distribution of the zeros of the wavelet transformed signal. We apply our method to some acoustic signals and observe that many signal components are found but as predicted by the theory there is no guarantee to find all signal components.

We consider mainly the Hilbert space of bianalytic functions on a given domain in the plane, square integrable with respect to a weight. We show how to obtain the asymptotic expansion of the corresponding bianalytic Bergman kernel for power weights, under the standard condition on those weights. This is known only in the analytic setting, from the work of e.g. Tian, Yau, Zelditch, Catlin, et al. We remark that a bianalytic function may be identified with a vector-valued analytic function, supplied with a locally singular metric on the vectors. We also apply our findings to two bianalytic Bergman metrics introduced here., 40 pages

For a weight function Q:C→R and a positive scaling parameter m, we study reproducing kernels Kq,mQ,n of the polynomial spaces Aq,mQ,n2:=spanC{z¯rzj|0⩽r⩽q−1,0⩽j⩽n−1} equipped with the inner product from the space L2(e−mQ(z)dA(z)). Here dA denotes a suitably normalized area measure on C. For a point z0 belonging to the interior of certain compact set S and satisfying ΔQ(z0)>0, we define the rescaled coordinates z=z0+ξmΔQ(z0),w=z0+λmΔQ(z0). The following universality result is proved in the case q=2: 1mΔQ(z0)|Kq,mQ,n(z,w)|e−12mQ(z)−12mQ(w)→|Lq−11(|ξ−λ|2)|e−12|ξ−λ|2 as m,n→∞ while n⩾m−M for any fixed M>0, uniformly for (ξ,λ) in compact subsets of C2. The notation Lq−11 stands for the associated Laguerre polynomial with parameter 1 and degree q−1. This generalizes a result of Ameur, Hedenmalm and Makarov concerning analytic polynomials to bianalytic polynomials. We also discuss how to generalize the result to q>2. Our methods include a simplification of a Bergman kernel expansion algorithm of Berman, Berndtsson and Sjostrand in the one compex variable setting, and extension to the context of polyanalytic functions. We also study off-diagonal behaviour of the kernels Kq,mQ,n.

We study fluctuations of linear statistics in Polyanalytic Ginibre ensembles, a family of point processes describing planar free fermions in a uniform magnetic field at higher Landau levels. Our main result is asymptotic normality of fluctuations, extending a result of Rider and Vir\'ag. As in the analytic case, the variance is composed of independent terms from the bulk and the boundary. Our methods rely on a structural formula for polyanalytic polynomial Bergman kernels which separates out the different pure $q$-analytic kernels corresponding to different Landau levels. The fluctuations with respect to these pure $q$-analytic Ginibre ensembles are also studied, and a central limit theorem is proved. The results suggest a stabilizing effect on the variance when the different Landau levels are combined together.

We consider a polyanalytic generalization of the Ginibre ensemble. This models allowing higher Landau levels (the Ginibre ensemble corresponds to the lowest Landau level). We study the local behavior of this point process under blow-ups., Comment: 31 pages