Related papers: Fock spaces corresponding to positive definite lin…
In this paper, we study the weighted composition operator on the Fock space $\mf$ of slice regular functions. First, we characterize the boundedness and compactness of the weighted composition operator. Subsequently, we describe all the…
We explicitly construct and study an isometry between the spaces of square integrable functionals of an arbitrary Levy process and a vector-valued Gaussian white noise. In particular, we obtain explicit formulas for this isometry at the…
Gaussian processes are flexible function approximators, with inductive biases controlled by a covariance kernel. Learning the kernel is the key to representation learning and strong predictive performance. In this paper, we develop…
We study representations of positive definite kernels $K$ in a general setting, but with view to applications to harmonic analysis, to metric geometry, and to realizations of certain stochastic processes. Our initial results are stated for…
In this paper we consider the reproducing kernel thesis for boundedness and compactness for various operators on Bergman-type spaces. In particular, the results in this paper apply to the weighted Bergman space on the unit ball, the unit…
We study concentration operators acting on the Fourier symmetric Sobolev space $H$ consisting of functions $f$ such that $\int_{\mathbb{R}} |f(x)|^2(1+x^2) dx + \int_{\mathbb{R}} |\hat{f}(\xi)|^2(1+\xi^2) d\xi < \infty $. We find that the…
We consider Fock representations of the $Q$-deformed commutation relations $$\partial_s\partial^\dag_t=Q(s,t)\partial_t^\dag\partial_s+\delta(s,t), \quad s,t\in T.$$ Here $T:=\mathbb R^d$ (or more generally $T$ is a locally compact Polish…
We consider the bosonic Fock space over the Hilbert space of transversal vector fields in three dimensions. This space carries a canonical representation of the group of rotations. For a certain class of operators in Fock space we show that…
Given a fixed sigma-finite measure space $\left(X,\mathscr{B},\nu\right)$, we shall study an associated family of positive definite kernels $K$. Their factorizations will be studied with view to their role as covariance kernels of a variety…
In this paper we use techniques in Fock spaces theory and compute how the Segal-Bargmann transform acts on special wave functions obtained by multiplying superoscillating sequences with normalized Hermite functions. It turns out that these…
We deal with a class of one-parameter family of integral transforms of Bargmann type arising as dual transforms of fractional Hankel transform. Their ranges are identified to be special subspaces of the weighted hyperholomorphic left…
We characterize a holomorphic positive definite function $f$ defined on a horizontal strip of the complex plane as the Fourier-Laplace transform of a unique exponentially finite measure on $\mathbb{R}$. The classical theorems of Bochner on…
In this paper we study the compactness of operators on the Bergman space of the unit ball and on very generally weighted Bargmann-Fock spaces in terms of the behavior of their Berezin transforms and the norms of the operators acting on…
A recurrent theme in functional analysis is the interplay between the theory of positive definite functions, and their reproducing kernels, on the one hand, and Gaussian stochastic processes, on the other. This central theme is motivated by…
Consider the zero set of the random power series f(z)=sum a_n z^n with i.i.d. complex Gaussian coefficients a_n. We show that these zeros form a determinantal process: more precisely, their joint intensity can be written as a minor of the…
Let $\nu$ 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 $\nu$-square integrable $q$-analytic functions is the closure of…
In this paper we construct an "abstract Fock space" for general Lie types that serves as a generalisation of the infinite wedge $q$-Fock space familiar in type $A$. Specifically, for each positive integer $\ell$, we define a…
We construct the bosonization of the Fock space $\mathit{F^{\otimes \frac{1}{2}}}$ of a single neutral fermion by using a 2-point local Heisenberg field. We decompose the Fock space $\mathit{F^{\otimes \frac{1}{2}}}$ as a direct sum of…
For the full range of index $0<p\leq\infty$, real weight $\alpha$ and real Sobolev order $s$, two types of weighted Fock-Sobolev spaces over $\mathbb C^n$, $F^p_{\alpha, s}$ and $\widetilde F^p_{\alpha,s}$, are introduced through fractional…
Composition operators with analytic symbols on some reproducing kernel Hilbert spaces of entire functions on a complex Hilbert space are studied. The questions of their boundedness, seminormality and positivity are investigated. It is…