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Alí Guzmán Adán

Publications and source records attributed to Alí Guzmán Adán.

10 recordsLinked to original sources
arXiv2022

Pizzetti formulae and the Radon Transform on the Sphere

In this paper, we obtain Pizzetti-type formulae on regions of the the unit sphere $\mathbb{S}^{m-1}$ of $\mathbb{R}^m$, and study their applications to the problem of inverting the spherical Radon transform. In particular, we approach integration over $(m-2)$-dimensional sub-spheres of $\mathbb{S}^{m-1}$, $(m-1)$-dimensional sub-balls, and over $(m-1)$-dimensional spherical caps as the action of suitable concentrated delta distributions. In turn, this leads to Pizzetti formulae that express such integrals in terms of the action of SO$(m-1)$-invariant differential operators. In the last section of the paper, we use some of these expressions to derive the inversion formulae for the Radon transform on $\mathbb{S}^{m-1}$ in a direct way.

math-ph↗
arXiv2022

On the connection between the Fueter-Sce-Qian theorem and the generalized CK-extension

The Fueter-Sce-Qian theorem provides a way of inducing axial monogenic functions in $\mathbb{R}^{m+1}$ from holomorphic intrinsic functions of one complex variable. This result was initially proved by Fueter and Sce for the cases where the dimension $m$ is odd using pointwise differentiation, while the extension to the cases where $m$ is even was proved by Qian using the corresponding Fourier multipliers. In this paper, we provide an alternative description of the Fueter-Sce-Qian theorem in terms of the generalized CK-extension. The latter characterizes axial null solutions of the Cauchy-Riemann operator in $\mathbb{R}^{m+1}$ in terms of their restrictions to the real line. This leads to a one-to-one correspondence between the space of axially monogenic functions in $\mathbb{R}^{m+1}$ and the space of analytic functions of one real variable. We provide explicit expressions for the Fueter-Sce-Qian map in terms of the generalized CK-extension for both cases, $m$ even and $m$ odd. These expressions allow for a plane wave decomposition of the Fueter-Sce-Qian map or, more in particular, a factorization of this mapping in terms of the dual Radon transform. In turn, this decomposition provides a new possibility for extending the Coherent State Transform (CST) to Clifford Analysis. In particular, we construct an axial CST defined through the Fueter-Sce-Qian mapping, and show how it is related to the axial and slice CSTs already studied in the literature.

math.CV↗
arXiv2021

On the Radon transform and the Dirac delta distribution in superspace

In this manuscript, we obtain a plane wave decomposition for the delta distribution in superspace, provided that the superdimension is not odd and negative. This decomposition allows for explicit inversion formulas for the super Radon transform in these cases. Moreover, we prove a more general Radon inversion formula valid for all possible integer values of the superdimension. The proof of this result comes along with the study of fractional powers of the super Laplacian, their fundamental solutions, and the plane wave decompositions of super Riesz kernels.

math-ph↗
arXiv2020

Generalized Cauchy-Kovalevskaya extension and plane wave decompositions in superspace

The aim of this paper is to obtain a generalized CK-extension theorem in superspace for the bi-axial Dirac operator. In the classical commuting case, this result can be written as a power series of Bessel type of certain differential operators acting on a single initial function. In the superspace setting, novel structures appear in the cases of negative even superdimensions. In these cases, the CK-extension depends on two initial functions on which two power series of differential operators act. These series are not only of Bessel type but they give rise to an additional structure in terms of Appell polynomials. This pattern also is present in the structure of the Pizzetti formula, which describes integration over the supersphere in terms of differential operators. We make this relation explicit by studying the decomposition of the generalized CK-extension into plane waves integrated over the supersphere. Moreover, these results are applied to obtain a decomposition of the Cauchy kernel in superspace into monogenic plane waves, which shall be useful for inverting the super Radon transform.

math-ph↗
arXiv2019

Pizzetti and Cauchy formulae for higher dimensional surfaces: a distributional approach

In this paper, we study Pizzetti-type formulas for Stiefel manifolds and Cauchy-type formulas for the tangential Dirac operator from a distributional perspective. First we illustrate a general distributional method for integration over manifolds in $\mathbb R^m$ defined by means of $k$ equations $φ_1(\underline{x})=\ldots=φ_k(\underline{x})=0$. Next, we apply this method to derive an alternative proof of the Pizzetti formulae for the real Stiefel manifolds $SO(m)/SO(m-k)$. Besides, a distributional interpretation to invariant oriented integration is provided. In particular, we obtain a distributional Cauchy theorem for the tangential Dirac operator on an embedded $(m-k)$-dimensional smooth surface.

math-ph↗
arXiv2019

The Spin Group in Superspace

There are two well-known ways of describing elements of the rotation group SO$(m)$. First, according to the Cartan-Dieudonné theorem, every rotation matrix can be written as an even number of reflections. And second, they can also be expressed as the exponential of some anti-symmetric matrix. In this paper, we study similar descriptions of a group of rotations SO${}_0$ in the superspace setting. This group can be seen as the action of the functor of points of the orthosymplectic supergroup OSp$(m|2n)$ on a Grassmann algebra. While still being connected, the group SO${}_0$ is thus no longer compact. As a consequence, it cannot be fully described by just one action of the exponential map on its Lie algebra. Instead, we obtain an Iwasawa-type decomposition for this group in terms of three exponentials acting on three direct summands of the corresponding Lie algebra of supermatrices. At the same time, SO${}_0$ strictly contains the group generated by super-vector reflections. Therefore, its Lie algebra is isomorphic to a certain extension of the algebra of superbivectors. This means that the Spin group in this setting has to be seen as the group generated by the exponentials of the so-called extended superbivectors in order to cover SO${}_0$. We also study the actions of this Spin group on supervectors and provide a proper subset of it that is a double cover of SO${}_0$. Finally, we show that every fractional Fourier transform in n bosonic dimensions can be seen as an element of this spin group.

math.GR↗
arXiv2018

Bochner-Martinelli formula in superspace

In a series of recent papers, a harmonic and hypercomplex function theory in superspace has been established and amply developed. In this paper, we address the problem of establishing Cauchy integral formulae in the framework of Hermitian Clifford analysis in superspace. This allows us to obtain a successful extension of the classical Bochner-Martinelli formula to superspace by means of the corresponding projections on the space of spinor-valued superfunctions.

math.CV↗
arXiv2018

Distributions and Integration in superspace

Distributions in superspace constitute a very useful tool for establishing an integration theory. In particular, distributions have been used to obtain a suitable extension of the Cauchy formula to superspace and to define integration over the superball and the supersphere through the Heaviside and Dirac distributions, respectively. In this paper, we extend the distributional approach to integration over more general domains and surfaces in superspace. The notions of domain and surface in superspace are defined by smooth bosonic phase functions $g$. This allows to define domain integrals and oriented (as well as non-oriented) surface integrals in terms of the Heaviside and Dirac distributions of the superfunction $g$. It will be shown that the presented definition for the integrals does not depend on the choice of the phase function $g$ defining the corresponding domain or surface. In addition, some examples of integration over a super-paraboloid and a super-hyperboloid will be presented. Finally, a new distributional Cauchy-Pompeiu formula will be obtained, which generalizes and unifies the previously known approaches.

math-ph↗
arXiv2017

Hypermonogenic solutions and plane waves of the Dirac operator in Rp x Rq

In this paper we first define hypermonogenic solutions of the Dirac operator in Rp x Rq and study some basic properties, e.g., obtaining a Cauchy integral formula in the unit hemisphere. Hypermonogenic solutions form a natural function class in classical Clifford analysis. After that, we define the corresponding hypermonogenic plane wave solutions and deduce explicit methods to compute these functions.

math.AP↗