A Note on Abelian Conversion of Constraints

We show that for a system containing a set of general second class constraints which are linear in the phase space variables, the Abelian conversion can be obtained in a closed form and that the first class constraints generate a generalized shift symmetry. We study in detail the example of a general first order Lagrangian and show how the shift symmetry noted in the context of BV quantization arises.Comment: 10 pgs., UR1369, ER40685-81

Joint similarity to operators in noncommutative varieties

In this paper we solve several problems concerning joint similarity to n-tuples of operators in noncommutative varieties in [B(\cH)^n]_1 associated with positive regular free holomorphic functions in $n$ noncommuting variables and with sets of noncommutative polynomials in $n$ indeterminates, where B(\cH) is the algebra of all bounded linear operators on a Hilbert space \cH. In particular, if $f=X_1+...+X_n$ and \cP=\{0\}, the elements of the corresponding variety can be seen as noncommutative multivariable analogues of Agler's $m$-hypercontractions.Comment: 35 pages, corrected typo

Discrete Hilbert transforms on sparse sequences

Weighted discrete Hilbert transforms $(a_n)_n \mapsto \sum_n a_n v_n/(z-\gamma_n)$ from $\ell^2_v$ to a weighted $L^2$ space are studied, with $\Gamma=(\gamma_n)$ a sequence of distinct points in the complex plane and $v=(v_n)$ a corresponding sequence of positive numbers. In the special case when $|\gamma_n|$ grows at least exponentially, bounded transforms of this kind are described in terms of a simple relative to the Muckenhoupt $(A_2)$ condition. The special case when $z$ is restricted to another sequence $\Lambda$ is studied in detail; it is shown that a bounded transform satisfying a certain admissibility condition can be split into finitely many surjective transforms, and precise geometric conditions are found for invertibility of such two weight transforms. These results can be interpreted as statements about systems of reproducing kernels in certain Hilbert spaces of which de Branges spaces and model subspaces of $H^2$ are prime examples. In particular, a connection to the Feichtinger conjecture is pointed out. Descriptions of Carleson measures and Riesz bases of normalized reproducing kernels for certain "small" de Branges spaces follow from the results of this paper

Surgery of spline-type and molecular frames

We prove a result about producing new frames for general spline-type spaces by piecing together portions of known frames. Using spline-type spaces as models for the range of certain integral transforms, we obtain results for time-frequency decompositions and sampling.Comment: 34 pages. Corrected typo