It is proved that quadrature domains are ubiquitous in a very strong sense in the realm of smoothly bounded multiply connected domains in the plane. In fact, they are so dense that one might as well assume that any given smooth domain one is dealing with is a quadrature domain, and this allows access to a host of strong conditions on the classical kernel functions associated to the domain. Following this string of ideas leads to the discovery that the Bergman kernel can be “zipped” down to a strikingly small data set.

The main goal of this paper is to extend the approximation theorem of contiuous functions by Haar polynomials (see Theorem A) to infinite matrices (see Theorem C). The extension to the matricial framework will be based on the one hand on the remark that periodic functions which belong to$L$^{∞}($T$) may be one-to-one identified with Toeplitz matrices from$B$($l$_{2}) (see Theorem 0) and on the other hand on some notions given in the paper. We mention for instance:$ms$—a unital commutative subalgebra of$l$^{∞},$C$($l$_{2}) the matricial analogue of the space of all continuous periodic functions$C$($T$), the matricial Haar polynomials, etc.

We give characterisations of certain positive finite Borel measures with unbounded support on the real axis so that the algebraic polynomials are dense in all spaces$L$_{$p$}($R$,$d$μ),$p$≥1. These conditions apply, in particular, to the measures satisfying the classical Carleman conditions.

Let$Z$be the zero set of a holomorphic section$f$of a Hermitian vector bundle. It is proved that the current of integration over the irreducible components of$Z$of top degree, counted with multiplicities, is a product of a residue factor$R$^{$f$}and a “Jacobian factor”. There is also a relation to the Monge-Ampère expressions ($dd$^{$c$}log|$f$|)^{$k$}, which we define for all positive powers$k$.

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