THE POLYANALYTIC REPRODUCING KERNELS

Let ν be a rotation invariant Borel probability measure on the complex plane having moments of all orders. Given a positive integer q, it is proved that the space of ν-square integrable q-analytic functions is the closure of q-analytic polynomials, and in particular it is a Hilbert space. We establish a general formula for the corresponding polyanalytic reproducing kernel. New examples are given and all known examples, including those of the analytic case are covered. In particular, weighted Bergman and Fock type spaces of polyanalytic functions are introduced. Our results have a higher dimensional generalization for measure on C p which are in rotation invariant with respect to each coordinate

Dixmier trace and the Fock space

We give criteria for products of Toeplitz and Hankel operators on the Fock (Segal-Bargmann) space to belong to the Dixmier class, and compute their Dixmier trace. At the same time, analogous results for the Weyl pseudodifferential operators are also obtained.Comment: 23 pages, no figure

A COVARIANCE EQUATION

Let S be a commutative semigroup with identity e and let Γ be a compact subset in the pointwise convergence topology of the space S of all non-zero multiplicative functions on S. Given a continuous function F : Γ → C and a complex regular Borel measure µ on Γ such that µ(Γ) = 0. It is shown that µ(Γ) Γ (s)(t)|F | 2 ()dµ() = Γ (s)F ()dµ() Γ (t)F ()dµ() for all (s, t) ∈ S × S if and only if for some γ ∈ Γ, the support of µ is contained is contained in {F = 0} ∪ {γ}. Several applications of this characterization are derived. In particular, the reduction of our theorem to the semigroup of non-negative integers (N 0 , +) solves a problem posed by El Fallah, Klaja, Kellay, Mashregui and Ransford in a more general context. More consequences of results are given, some of them illustrate the probabilistic flavor behind the problem studied herein and others establish extremal properties of analytic kernels