Bargmann Fock Space Research Articles

On a Sampling Problem for a Bargmann-Fock Space

The purpose of the present article is to provide geometric sufficient conditions for discrete points to be a sampling sequence for a generalized Hilbert Bargmann-Fock space in several complex variables.

Beurling-Type Density Criteria for System Identification

This paper addresses the problem of identifying a linear time-varying (LTV) system characterized by a (possibly infinite) discrete set of delay-Doppler shifts without a lattice (or other “geometry-discretizing”) constraint on the support set. Concretely, we show that a class of such LTV systems is identifiable whenever the upper uniform Beurling density of the delay-Doppler support sets, measured “uniformly over the class”, is strictly less than 1/2. The proof of this result reveals an interesting relation between LTV system identification and interpolation in the Bargmann-Fock space. Moreover, we show that the density condition we obtain is also necessary for classes of systems invariant under time-frequency shifts and closed under a natural topology on the support sets. We furthermore find that identifiability guarantees robust recovery of the delay-Doppler support set, as well as the weights of the individual delay-Doppler shifts, both in the sense of asymptotically vanishing reconstruction error for vanishing measurement error.

Gaussian Gabor frames, Seshadri constants and generalized Buser–Sarnak invariants

We investigate the frame set of regular multivariate Gaussian Gabor frames using methods from Kähler geometry such as Hörmander’s {overline{partial }}-L^2 estimate with singular weight, Demailly’s Calabi–Yau method for Kähler currents and a Kähler-variant generalization of the symplectic embedding theorem of McDuff–Polterovich for ellipsoids. Our approach is based on the well-known link between sets of interpolation for the Bargmann-Fock space and the frame set of multivariate Gaussian Gabor frames. We state sufficient conditions in terms of a certain extremal type Seshadri constant of the complex torus associated to a lattice to be a set of interpolation for the Bargmann-Fock space, and give also a condition in terms of the generalized Buser-Sarnak invariant of the lattice. In particular, we obtain an effective Gaussian Gabor frame criterion in terms of the covolume for almost all lattices, which is the first general covolume criterion in multivariate Gaussian Gabor frame theory. The recent Berndtsson–Lempert method and the Ohsawa–Takegoshi extension theorem also allow us to give explicit estimates for the frame bounds in terms of certain Robin constant. In the one-dimensional case we obtain a sharp estimate of the Robin constant using Faltings’ theta metric formula for the Arakelov Green functions.

Riesz–Kolmogorov Type Compactness Criteria in Function Spaces with Applications

We present forms of the classical Riesz–Kolmogorov theorem for compactness that are applicable in a wide variety of settings. In particular, our theorems apply to classify the precompact subsets of the Lebesgue space $$L^2$$ , Paley–Wiener spaces, weighted Bargmann–Fock spaces, and a scale of weighted Besov–Sobolev spaces of holomorphic functions that includes weighted Bergman spaces of general domains as well as the Hardy space and the Dirichlet space. We apply the compactness criteria to characterize the compact Toeplitz operators on the Bergman space, deduce the compactness of Hankel operators on the Hardy space, and obtain general umbrella theorems.

On the Bergman kernel in weighted monogenic Bargmann-Fock spaces

In this paper, we study the Bergman kernel Bφ(x,y) of generalized Bargmann-Fock spaces in the setting of Clifford algebra. The versions of L2-estimate method and weighted subharmonic inequality for Clifford algebra are established. Consequently we show the existence of Bφ(x,y) and then give some estimates on- and off- the diagonal. As a by-product, we also obtain an upper estimate of the weighted harmonic Bergman kernel.

A fractal uncertainty principle for the short-time Fourier transform and Gabor multipliers

We study the fractal uncertainty principle in the joint time-frequency representation, and we prove a version for the Short-Time Fourier transform with Gaussian window on the modulation spaces. This can equivalently be formulated in terms of projection operators on the Bargmann-Fock spaces of entire functions. Specifically for signals in L 2 ( R d ) , we obtain norm estimates of Daubechies' time-frequency localization operator localizing on porous sets. The proof is based on the maximal Nyquist density of such sets, which we also use to derive explicit upper bound asymptotes for the multidimensional Cantor iterates, in particular. Finally, we translate the fractal uncertainty principle to discrete Gaussian Gabor multipliers.

Random Walk on Quantum Blobs

We describe the action of the symplectic group on the homogeneous space of squeezed states (quantum blobs) and extend this action to the semigroup. We then extend the metaplectic representation to the metaplectic (or oscillator) semigroup and study the properties of such an extension using Bargmann-Fock space. The shape geometry of squeezing is analyzed and noncommuting elements from the symplectic semigroup are proposed to be used in simultaneous monitoring of noncommuting quantum variables — which should lead to fractal patterns on the manifold of squeezed states.

Heisenberg’s uncertainty principle associated with the Caputo fractional derivative

In this paper, we establish the Heisenberg’s uncertainty inequality associated with the Caputo derivative of order q ∈ (1, ∞) in the generalized Bargmann–Fock space. We also determine exactly when the equality occurs in the uncertainty inequality. It is done by estimating the growth of the eigenvalues of the commutator [Dq,zq]. We prove that the sequence of eigenvalues of [Dq,zq] tends to ∞ when the fractional order q belongs to (1, ∞). However, the sequence converges to zero when q belongs to (0, 1), which shows different behavior. Hence, only a weak uncertainty inequality is obtained for the latter case.

A Wiener Tauberian theorem for operators and functions

We prove variants of Wiener's Tauberian theorem in the framework of quantum harmonic analysis, i.e. for convolutions between an absolutely integrable function and a trace class operator, or of two trace class operators. Our results include Wiener's Tauberian theorem as a special case. Applications of our Tauberian theorems are related to localization operators, Toeplitz operators, isomorphism theorems between Bargmann-Fock spaces and quantization schemes with consequences for Shubin's pseudodifferential operator calculus and Born-Jordan quantization. Based on the links between localization operators and Tauberian theorems we note that the analogue of Pitt's Tauberian theorem in our setting implies compactness results for Toeplitz operators in terms of the Berezin transform. In addition, we extend the results on Toeplitz operators to other reproducing kernel Hilbert spaces induced by the short-time Fourier transform, known as Gabor spaces. Finally, we establish the equivalence of Wiener's Tauberian theorem and the condition in the characterization of compactness of localization operators due to Fernández and Galbis.

Toeplitz Operators with Positive Operator-Valued Symbols on Vector-Valued Generalized Fock Spaces

In this article, we study some characterizations of Toeplitz operators with positive operator-valued function as symbols on the vector-valued generalized Bargmann-Fock spaces $$F_\varphi^2$$ . Main results including Fock-Carleson condition, bounded Toeplitz operators, compact Toeplitz operators, and Toeplitz operators in the Schatten-p class are all considered.

Sampling of entire functions of several complex variables on a lattice and multivariate Gabor frames

ABSTRACT We give a general construction of entire functions in d complex variables that vanish on a lattice of the form for an invertible complex-valued matrix. As an application, we exhibit a class of lattices of density >1 that fail to be a sampling set for the Bargmann–Fock space in . By using an equivalent real-variable formulation, we show that these lattices fail to generate a Gabor frame.

NONUNIFORMLY FLAT AFFINE ALGEBRAIC HYPERSURFACES

The relationship between interpolation and separation properties of hypersurfaces in Bargmann–Fock spaces over $\mathbb{C}^{n}$ is not well understood except for $n=1$. We present four examples of smooth affine algebraic hypersurfaces that are not uniformly flat, and show that exactly two of them are interpolating.

Non-trivial 1d and 2d Segal–Bargmann transforms

ABSTRACTSpecial generating functions and Mehler's formula for the univariate complex Hermite polynomials are obtained and next employed to introduce and study some one- and two-dimensional integral transforms of Segal–Bargmann type in the framework of some specific functional Hilbert spaces; including the so-called generalized Bargmann–Fock spaces that are realized as -eigenspaces of a special magnetic Schrödinger operator.

Correlations between zeros and critical points of random analytic functions

We study the two-point correlation K n m ( z , w ) K^m_n(z,w) between zeros and critical points of Gaussian random holomorphic sections s n s_n over Kähler manifolds. The critical points are points ∇ h n s n = 0 \nabla _{h^n} s_n=0 , where ∇ h n \nabla _{h^n} is the smooth Chern connection with respect to the Hermitian metric h n h^n on line bundle L n L^n . The main result is that the rescaling limit of K n m ( z 0 + u n , z 0 + v n ) K^m_n(z_0+\frac u{\sqrt n}, z_0+\frac v{\sqrt n}) for any z 0 ∈ M z_0\in M is universal as n n tends to infinity. In fact, the universal rescaling limit is the two-point correlation between zeros and critical points of Gaussian analytic functions for the Bargmann–Fock space of level 1 1 . Furthermore, in the length scale of order n − 1 2 n^{-\frac 12} , there is a “repulsion" between zeros and critical points for the short range, and a “neutrality" for the long range.

Analytic and arithmetic properties of the $$(\Gamma ,\chi )$$ ( Γ , χ ) -automorphic reproducing kernel function and associated Hermite–Gauss series

We consider the reproducing kernel function of the theta Bargmann–Fock Hilbert space associated with given full-rank lattice and pseudo-character, and we deal with some of its analytical and arithmetical properties. Specially, the distribution and the discreteness of its zeros are examined. The analytic sets of zeros of the theta Bargmann–Fock space inside a given fundamental cell is characterized and shown to be finite and of cardinal less or equal to its dimension. Moreover, we obtain some remarkable lattice sums by evaluating the so-called complex Hermite–Gauss coefficients. Some of them generalize some of the arithmetic identities given by Perelomov in the framework of coherent states for the specific case of von Neumann lattice. Such complex Hermite–Gauss coefficients are nontrivial examples of the so-called lattice’s functions according the Serre terminology. The perfect use of the basic properties of the complex Hermite polynomials is crucial in this framework.

Composition of Segal–Bargmann transforms

ABSTRACTWe introduce and discuss some basic properties of some integral transforms in the framework of specific functional Hilbert spaces, the holomorphic Bargmann–Fock spaces on and and the slice hyperholomorphic Bargmann–Fock space on . The first one is a natural integral transform mapping isometrically the standard Hilbert space on the real line into the two-dimensional Bargmann–Fock space. It is obtained as a composition of the one- and two-dimensional Segal–Bargmann transforms and reduces further to an extreme integral operator that looks like a composition operator of the one-dimensional Segal–Bargmann transform with a specific symbol. We study its basic properties, including the identification of its image and the determination of a like-left inverse defined on the whole two-dimensional Bargmann–Fock space. We examine their combination with the Fourier transform which lead to special integral transforms connecting the two-dimensional Bargmann–Fock space and its analogue on the complex plane. We also investigate the relationship between special subspaces of the two-dimensional Bargmann–Fock space and the slice-hyperholomorphic one on the quaternions by introducing appropriate integral transforms. We identify their image and their action on the reproducing kernel.

The Mittag Leffler reproducing kernel Hilbert spaces of entire and analytic functions

This paper investigates the function theoretic properties of two reproducing kernel functions based on the Mittag-Leffler function that are related through a composition. Both spaces provide one parameter generalizations of the traditional Bargmann–Fock space. In particular, the Mittag-Leffler space of entire functions yields many similar properties to the Bargmann–Fock space, and several results are demonstrated involving zero sets and growth rates. The second generalization, the Mittag-Leffler space of the slitted plane, is a reproducing kernel Hilbert space (RKHS) of functions for which Caputo fractional differentiation and multiplication by zq (for q>0) are densely defined adjoints of one another.

Zero Sets for Spaces of Analytic Functions

We show that under mild conditions, a Gaussian analytic function F that a.s. does not belong to a given weighted Bergman space or Bargmann–Fock space has the property that a.s. no non-zero function in that space vanishes where F does. This establishes a conjecture of Shapiro [21] on Bergman spaces and allows us to resolve a question of Zhu [24] on Bargmann–Fock spaces. We also give a similar result on the union of two (or more) such zero sets, thereby establishing another conjecture of Shapiro [21] on Bergman spaces and allowing us to strengthen a result of Zhu [24] on Bargmann–Fock spaces.

Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions

We study nonuniform sampling in shift-invariant spaces and the construction of Gabor frames with respect to the class of totally positive functions whose Fourier transform factors as hat{g}(xi )= prod _{j=1}^n (1+2pi idelta _jxi )^{-1} , e^{-c xi ^2} for delta _1,ldots ,delta _nin mathbb {R}, c >0 (in which case g is called totally positive of Gaussian type). In analogy to Beurling’s sampling theorem for the Paley–Wiener space of entire functions, we prove that every separated set with lower Beurling density >1 is a sampling set for the shift-invariant space generated by such a g. In view of the known necessary density conditions, this result is optimal and validates the heuristic reasonings in the engineering literature. Using a subtle connection between sampling in shift-invariant spaces and the theory of Gabor frames, we show that the set of phase-space shifts of g with respect to a rectangular lattice alpha mathbb {Z}times beta mathbb {Z} forms a frame, if and only if alpha beta <1. This solves an open problem going back to Daubechies in 1990 for the class of totally positive functions of Gaussian type. The proof strategy involves the connection between sampling in shift-invariant spaces and Gabor frames, a new characterization of sampling sets “without inequalities” in the style of Beurling, new properties of totally positive functions, and the interplay between zero sets of functions in a shift-invariant space and functions in the Bargmann–Fock space.

Percolation of random nodal lines

We prove a Russo-Seymour-Welsh percolation theorem for nodal domains and nodal lines associated to a natural infinite dimensional space of real analytic functions on the real plane. More precisely, let $U$ be a smooth connected bounded open set in $\mathbf{R}^{2}$ and $\gamma, \gamma '$ two disjoint arcs of positive length in the boundary of $U$ . We prove that there exists a positive constant $c$ , such that for any positive scale $s$ , with probability at least $c$ there exists a connected component of the set $\{x\in \smash{\bar{U}}, f(sx) > 0\} $ intersecting both $\gamma $ and $\gamma '$ , where $f$ is a random analytic function in the Wiener space associated to the real Bargmann-Fock space. For $s$ large enough, the same conclusion holds for the zero set $\{x\in \smash{\bar{U}}, f(sx) = 0\} $ . As an important intermediate result, we prove that sign percolation for a general stationary Gaussian field can be made equivalent to a correlated percolation model on a lattice.