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Swanhild Bernstein

Publications and source records attributed to Swanhild Bernstein.

At least 19 recordsLinked to original sources
arXiv2025

$q$-Fock Space of $q$-Analytic Functions and its realization in $L^{2}(\mathbb{C}; e^{-z\bar z} \,\mathrm{d}x\,\mathrm{d}y)$

We introduce a $q$-deformation of the Fock space of holomorphic functions on $\mathbb{C}$, based on a geometric definition of $q$-analyticity. This definition is inspired by a standard construction in complex differential geometry. Within this framework, we define $q$-analytic monomials $z_q^n$ and construct the associated $q$-Fock space as a Hilbert space with orthonormal basis $\{z_q^n/\sqrt{[n]_q!]}\}_{n\ge 0}$. The reproducing kernel of this space is computed explicitly, and $q$-position and $q$-momentum operators are introduced, satisfying $q$-deformed commutation relations. We show that the $q$-monomials $z_q^n$ can be expanded in terms of complex Hermite polynomials, thereby providing a realization of the $q$-Fock space as a subspace of $L^2(\mathbb{C}; e^{-|z|^2}\,\mathrm{d}x\,\mathrm{d}y)$. Finally, we define a $q$-Bargmann transform that maps suitable $q$-Hermite functions into our $q$-Fock space and acts as a unitary isomorphism. Our construction offers a geometric and analytic approach to $q$-function theory, complementing recent operator-theoretic models.

math.CV↗
arXiv2025

The $q$-Dirac Operator on Quantum Euclidean Space

This paper provides the foundations of quantum Clifford analysis in $q$-commutative variables with symmetric difference operators. We consider a $q$-Dirac operator on the quantum Euclidean space that factorizes the $U_q(\frak{o})$-invariant Laplacian $\Delta_q.$ Due to the non-commutativity of the multiplication, we need a special Clifford algebra $C\ell_{0,n}^q.$ We define $q$-monogenic functions as null solutions of the $q$-Dirac operator and $q$-spherical monogenic functions. We define an inner Fischer product and decompose the space of homogeneous polynomials of degree $k.$

math.CV↗
arXiv2025

Conjugate $(1/q, q)$-harmonic Polynomials in $q$-Clifford Analysis

We consider the problem of constructing a conjugate $(1/q, q)$-harmonic homogeneous polynomial $V_k$ of degree $k$ to a given $(1/q, q)$-harmonic homogeneous polynomial $U_k$ of degree $k.$ The conjugated harmonic polynomials $V_k$ and $U_k$ are associated to the $(1/q, q)$-mono\-genic polynomial $F = U_k + \overline{e}_0V. $ We investigate conjugate $(1/q, q)$-harmonic homogeneous polynomials in the setting of $q$-Clifford analysis. Starting from a given $(1/q, q)$-harmonic polynomial $U_k$ of degree $k$, we construct its conjugate counterpart $V_k$, such that the Clifford-valued polynomial $F = U_k + e_0 V_k$ is $(1/q, q)$-monogenic, i.e., a null solution of a generalized $q$-Dirac operator. Our construction relies on a combination of Jackson-type integration, Fischer decomposition, and the resolution of a $q$-Poisson equation. We further establish existence and uniqueness results, and provide explicit representations for conjugate pairs, particularly when $U_k$ is real-valued.

math.CV↗
arXiv2025

Magnetic Dirichlet Laplacian in curved waveguides

For a two-dimensional curved waveguide, it is well known that the spectrum of the Dirichlet Laplacian is unstable. Any perturbation of the straight strip produces eigenvalues below the essential spectrum. In this paper, a magnetic field is added. We explicitly prove that the spectrum of the magnetic Laplacian is stable under small but non-local deformations of the waveguide.

math.SP↗
arXiv2024

Paley-Wiener Type Theorems associated to Dirac Operators of Riesz-Feller type

This paper explores Paley-Wiener type theorems within the framework of hypercomplex variables. The investigation focuses on a space-fractional version of the Dirac operator $\mathbf{D}_\theta^{\alpha}$ of order $\alpha$ and skewness $\theta$. The pseudo-differential reformulation of $\mathbf{D}_\theta^{\alpha}$ in terms of the Riesz derivative $(-\Delta)^{\frac{\alpha}{2}}$ and the so-called {\textit Riesz-Hilbert transform} $H$, allows for the description of generalized Hardy spaces on the upper and lower half-spaces of $\mathbf{R}^{n+1}$, $\mathbf{R}^{n+1}_+$ resp. $\mathbb{R}^{n+1}_-$, using L\'evy-Feller type semigroups generated by $-(-\Delta)^{\frac{\alpha}{2}}$, and the boundary values $\mathbf{f}_\pm=\frac{1}{2}\left(\mathbf{f}\pm H\mathbf{f}\right)$. Subsequently, we employ a proof strategy rooted in {\textit real Paley-Wiener methods} to demonstrate that the growth behavior of the sequences of functions $\left(\left(\mathbf{D}_\theta^{\alpha}\right)^k\mathbf{f}_{\pm}\right)_{k\in \mathbb{N}_0}$ effectively captures the relationship between the support of the Fourier transform $\widehat{\mathbf{f}}$ of the $L^p-$function $\mathbf{f}$, in the case where $\mathrm{supp}\widehat{\mathbf{f}}\subseteq \overline{B(0,R)}$, and the solutions of Cauchy problems equipped with the space-time operator $\partial_{x_0} + \mathbf{D}_\theta^{\alpha}$, which are of exponential type $R^\alpha$. Within the developed framework, introducing a hypercomplex analog for the Bernstein spaces $B_R^p$ arises naturally, allowing for the meaningful extension of the results by Kou and Qian as well as Franklin, Hogan, and Larkin. Specifically, leveraging the established Stein-Kolmogorov inequalities for hypercomplex variables enables us to accurately determine the maximum radius $R$ for which $\operatorname{supp}\widehat{\mathbf{f}} \subseteq \overline{B(0, R)}$ holds.

math.CV↗
arXiv2024

Hardy decomposition of higher order Lipschitz classes by polymonogenic functions

In this paper we find a decomposition of higher order Lipschitz functions into the traces of a polymonogenic function and solve a related Riemann-Hilbert problem. Our approach lies in using a cliffordian Cauchy-type operator, which behaves as an involution operator on higher order Lipschitz spaces. The result obtained is a multidimensional sharpened version of the Hardy decomposition of H\"older continuous functions on a simple closed curve in the complex plane.

math.CV↗
arXiv2023

Decomposition of first order Lipschitz functions by Clifford algebra-valued harmonic functions

In this paper we solve the problem on finding a sectionally Clifford algebra-valued harmonic function, zero at infinity and satisfying certain boundary value condition related to higher order Lipschitz functions. Our main tool are the Hardy projections related to a singular integral operator arising in bimonogenic function theory, which turns out to be an involution operator on the first order Lipschitz classes. Our result generalizes the classical Hardy decomposition of Holder continuous functions on a simple closed curve in the complex plane.

math.CV↗
arXiv2023

Magnetic Neumann Laplacian on a domain with hole

This article gives a domain with a small compact set of removed and the magnetic Neumann Laplacian on such set. The main theorem of this article shows the description of the holes which do not change the spectrum drastically. In this article we prove that the spectrum of the magnetic Neumann Laplacian converges in the Hausdorff distance sense to the spectrum of the original operator defined on the unperturbed domain.

math.SP↗
arXiv2022

Brownian motion, martingales and Itô formula in Clifford analysis

Clifford analysis has been the field of active research for several decades resulting in various methods to solve problems in pure and applied mathematics. However, the area of stochastic analysis has not been addressed in its full generality in the Clifford setting, since only a few contributions have been presented so far. Considering that the tools of stochastic analysis play an important role in the study of objects, such as positive definite functions, reproducing kernels and partial differential equations, it is important to develop tools for the study of these objects in the context of Clifford analysis. Therefore, in this work-in-progress paper, we present further steps towards stochastic Clifford analysis by studying random variables, martingales, Brownian motion, and Itô formula in the Clifford setting, as well as their applications in Clifford analysis.

math.PR↗
arXiv2021

The Segal-Bargmann Transform in Clifford Analysis

The Segal-Bargmann transform plays an essential role in signal processing, quantum physics, infinite-dimensional analysis, function theory and further topics. The connection to signal processing is the short-time Fourier transform, which can be used to describe the Segal-Bargmann transform. The classical Segal-Bargmann transform $\mathcal{B}$ maps a square-integrable function to a holomorphic function square-integrable with respect to a Gaussian identity. In signal processing terms, a signal from the position space $L_2(\mathbb{R}^m,\mathbb{R})$ is mapped to the phase space of wave functions, or Fock space, $\mathcal{F}^2(\mathbb{C}^m,\mathbb{C})$. We extend the classical Segal-Bargmann transform to a space of Clifford algebra-valued functions. We show how the Segal-Bargmann transform is related to the short-time Fourier transform and use this connection to demonstrate that $\mathcal{B}$ is unitary up to a constant and maps Sommen's orthonormal Clifford Hermite functions $\left\{ϕ_{l,k,j}\right\}$ to an orthonormal basis of the Segal-Bargmann module $\mathcal{F}^2(\mathbb{C}^m,\mathcal{C}\ell_m^{\mathbb{C}})$. We also lay out that the Segal-Bargmann transform can be expanded to a convergent series with a dictionary of $\mathcal{F}^2(\mathbb{C}^m,\mathcal{C}\ell_m^{\mathbb{C}})$. In other words, we analyse the signal $f$ on one basis and reconstruct it on a basis of the Segal-Bargmann module.

math.CV↗
arXiv2015

Fractional Riesz-Hilbert transforms and fractional monogenic signals

The fractional Hilbert transforms plays an important role in optics and signal processing. In particular the analytic signal proposed by Gabor has as a key component the Hilbert transform. The higher dimensional Hilbert transform is the Riesz-Hilbert transform which was used by Felsberg and Sommer to construct the monogenic signal. We will construct fractional and quaternionic fractional Riesz-Hilbert transforms based on a eigenvalue decomposition. We will prove properties of these transformations such as shift and scale invariance, orthogonality and the semigroup property. Based on the fractional/quaternionic fractional Riesz-Hilbert transform we construct (quaternionic) fractional monogenic signals. These signals are rotated and modulated monogenic signals.

math.FA↗
arXiv2014

Crystallographic and geodesic Radon transforms on SO(3): motivation, generalization, discretization

In this paper we consider the so-called crystallographic Radon transform (or crystallographic $X$-ray transform) and totally geodesic Radon transform on the group of rotations SO(3). As we show both of these transforms naturally appear in texture analysis, i.e. the analysis of preferred crystallographic orientation. Although we discuss only applications to texture analysis both transforms have other applications as well. In section 2 we start with motivations and applications. In sections 3 and 4 we develop a general framework on compact Lie groups. In section 5 we give a detailed analysis of the totally geodesic Radon transform on SO(3). In section \ref{relations} we compare crystallographic Radon transform on SO(3) and Funk transform on $S^{3}$. In section \ref{1} we show non-invertibility of the crystallographic transform. In section 8 we describe an exact reconstruction formula for bandlimited functions, which uses only a finite number of samples of their Radon transform. Some auxiliary results for this section are collected in Appendix.

math.FA↗
arXiv2012

Generalized splines for Radon transform on compact Lie groups with applications to crystallography

The Radon transform Rf of functions f on SO(3) has recently been applied extensively in texture analysis, i.e. the analysis of preferred crystallographic orientation. In practice one has to determine the orientation probability density function f \in L2(SO(3)) from Rf \in L2(S2\times S2) which is known only on a discrete set of points. Since one has only partial information about Rf the inversion of the Radon transform becomes an ill-posed inverse problem. Motivated by this problem we define a new notion of the Radon transform Rf of functions f on general compact Lie groups and introduce two approximate inversion algorithms which utilize our previously developed generalized variational splines on manifolds. Our new algorithms fit very well to the application of Radon transform on SO(3) to texture analysis.

math.FA↗
arXiv2012

Diffusive wavelets on the Spin group

The first part of this article is devoted to a brief review of the results about representation theory of the spin group Spin(m) from the point of view of Clifford analysis. In the second part we are interested in Clifford-valued functions and wavelets on the sphere. The connection of representations of Spin(m) and the concept of diffusive wavelets leads naturally to investigations of a modified diffusion equation on the sphere, that makes use of the Gamma operator. We will achieve to obtain Clifford-valued diffusion wavelets with respect to a modified diffusion operator. Since we are able to characterize all representations of Spin(m) and even to obtain all eigenvectors of the (by representation) regarded Casimir operator in representation spaces, it seems appropriate to look at functions on Spin(m) directly. Concerning this, our aim shall be to formulate eigenfunctions for the Laplace-Beltrami operator on Spin(m) and give the series expansion of the heat kernel on Spin(m) in terms of eigenfunctions.

math.FA↗
arXiv2005

Factorization of the nonlinear Schroedinger equation and applications

We consider factorizations of the stationary and non-stationary Schroedinger equation in R^n which are based on appropriate Dirac operators. These factorizations lead to a Miura transform which is an analogue of the classical one-dimensional Miura transform but also closely related to the Riccati equation. In fact, the Miura transform is a nonlinear Dirac equation. We give an iterative procedure which is based on fix-point principles to solve this nonlinear Dirac equation. The relationship to nonlinear Schroedinger equations like the Gross-Pitaevskii equation are highlighted.

math.CV↗