### Remarks on restriction eigenfunctions in ${\u2102}^{n}$.

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One computes the joint and essential joint spectra of a pair of multiplication operators with bounded analytic functions on the Hardy spaces of the unit ball in ${\u2102}^{n}$.

The paper relates several generalized eigenfunction expansions to classical spectral decomposition properties. From this perspective one explains some recent results concerning the classes of decomposable and generalized scalar operators. In particular a universal dilation theory and two different functional models for related classes of operators are presented.

One proves the density of an ideal of analytic functions into the closure of analytic functions in a ${L}^{p}\left(\mu \right)$-space, under some geometric conditions on the support of the measure $\mu $ and the zero variety of the ideal.

Let H²(bΩ) be the Hardy space of a bounded weakly pseudoconvex domain in ${\u2102}^{n}$. The natural resolution of this space, provided by the tangential Cauchy-Riemann complex, is used to show that H²(bΩ) has the important localization property known as Bishop’s property (β). The paper is accompanied by some applications, previously known only for Bergman spaces.

The most accurate determinateness criteria for the multivariate moment problem require the density of polynomials in a weighted Lebesgue space of a generic representing measure. We propose a relaxation of such a criterion to the approximation of a single function, and based on this condition we analyze the impact of the geometry of the support on the uniqueness of the representing measure. In particular we show that a multivariate moment sequence is determinate if its support has dimension one and...

We adapt the privilege theorem of Douady and Pourcin from polydomains to strictly convex domains in the complex space.

We discuss an exact reconstruction algorithm for time expanding semi-algebraic sets given by a single polynomial inequality. The theoretical motivation comes from the classical $L$-problem of moments, while some possible applications to 2D fluid moving boundaries are sketched. The proofs rely on an adapted co-area theorem and a Hankel form minimization.

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