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Antonino De Martino

Publications and source records attributed to Antonino De Martino.

31 records · Page 2Linked to original sources
arXiv2022

A polyanalytic functional calculus of order 2 on the S-spectrum

The Fueter theorem provides a two step procedure to build an axially monogenic function, i.e. a null-solutions of the Cauchy-Riemann operator in $ \mathbb{R}^4$, denoted by $ \mathcal{D}$. In the first step a holomorphic function is extended to a slice hyperholomorphic function, by means of the so-called slice operator. In the second step a monogenic function is built by applying the Laplace operator in four real variables ($Δ$) to the slice hyperholomorphic function. In this paper we use the factorization of the Laplace operator, i.e. $Δ= \mathcal{\overline{D}} \mathcal{D}$ to split the previous procedure. From this splitting we get a class of functions that lies between the set of slice hyperholomorphic functions and the set of axially monogenic functions: the set of axially polyanalytic functions of order 2, i.e. null-solutions of $ \mathcal{D}^2$. We show an integral representation formula for this kind of functions. The formula obtained is fundamental to define the associated functional calculus on the $S$-spectrum. As far as the authors know, this is the first time that a monogenic polyanalytic functional calculus has been taken into consideration.

math.FA↗
arXiv2022

Axially harmonic functions and the harmonic functional calculus on the S-spectrum

The spectral theory on the S-spectrum was introduced to give an appropriate mathematical setting to quaternionic quantum mechanics, but it was soon realized that there were different applications of this theory, for example, to fractional heat diffusion and to the spectral theory for the Dirac operator on manifolds. In this seminal paper we introduce the harmonic functional calculus based on the S-spectrum and on an integral representation of axially harmonic functions. This calculus can be seen as a bridge between harmonic analysis and the spectral theory. The resolvent operator of the harmonic functional calculus is the commutative version of the pseudo S-resolvent operator. This new calculus also appears, in a natural way, in the product rule for the F-functional calculus.

math.SP↗
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

Generalized Appell polynomials and Fueter-Bargmann transforms in the polyanalytic setting

Tnis paper deals with some special integral transforms in the setting of quaternionic valued slice polyanalytic functions. In particular, using the polyanalytic Fueter mappings it is possible to construct a new family of polynomials which are called the generalized Appell polynomials. Furthermore, the range of the polyanalytic Fueter mappings on two different polyanalytic Fock spaces is characterized. Finally, we study the polyanalytic Fueter-Bargmann transforms.

math.CV↗
arXiv2021

The $\mathcal{F}$-resolvent equation and Riesz projectors for the $\mathcal{F}$-functional calculus

The Fueter-Sce-Qian mapping theorem is a two steps procedure to extend holomorphic functions of one complex variable to quaternionic or Clifford algebra-valued functions in the kernel of a suitable generalized Cauchy-Riemann operator. Using the Cauchy formula of slice monogenic functions it is possible to give the Fueter-Sce-Qian extension theorem an integral form and to define the $\mathcal{F}$-functional calculus for $n$-tuples of commuting operators. This functional calculus is defined on the $S$-spectrum but it generates a monogenic functional calculus in the spirit of McIntosh and collaborators. One of the main goals of this paper is to show that the $\mathcal{F}$-functional calculus generates the Riesz projectors. The existence of such projectors is obtained via the $\mathcal{F}$-resolvent equation that we have generalized to the Clifford algebra setting. This equation was known in the quaternionic setting, but the Clifford algebras setting turned out to be much more complicated.

math.CV↗
arXiv2021

The Poisson kernel and the Fourier transform of the slice monogenic Cauchy kernels

The Fueter-Sce-Qian (FSQ for short) mapping theorem is a two-steps procedure to extend holomorphic functions of one complex variable to slice monogenic functions and to monogenic functions. Using the Cauchy formula of slice monogenic functions the FSQ-theorem admits an integral representation for $n$ odd. In this paper we show that the relation $ Δ_{n+1}^{(n-1)/2}S_L^{-1}=\mathcal{F}^L_n $ between the slice monogenic Cauchy kernel $S_L^{-1}$ and the F-kernel $\mathcal{F}^L_n$, that appear in the integral form of the FSQ-theorem for $n$ odd, holds also in the case we consider the fractional powers of the Laplace operator $Δ_{n+1}$ in dimension $n+1$, i.e., for $n$ even. Moreover, this relation is proven computing explicitly Fourier transform of the kernels $S_L^{-1}$ and $\mathcal{F}^L_n$ as functions of the Poisson kernel. Similar results hold for the right kernels $S_R^{-1}$ and of $\mathcal{F}^R_n$.

math.CV↗
arXiv2021

On the Clifford short-time Fourier transform and its properties

In this paper we investigate how the short-time Fourier transform can be extended in a Clifford setting. We prove some of the main properties of the Clifford short-time Fourier transform such as the orthogonality relation, the reconstruction property and the reproducing kernel formula. Moreover, we show the effects of modulating and translating the signal and the window function, respectively. Finally, we demonstrate the Lieb's uncertainty principle for the Clifford short-time Fourier transform.

math.CA↗
arXiv2021

On the polyanalytic short-time Fourier transform in the quaternionic setting

In this paper, we consider a quaternionic short-time Fourier transform (QSTFT) with normalized Hermite functions as windows. It turns out that such a transform is based on the recent theory of slice polyanalytic functions on quaternions. Indeed, we will use the notions of true and full slice polyanalytic Fock spaces and Segal-Bargmann transforms. We prove new properties of this QSTFT including a Moyal formula, a reconstruction formula and a Lieb's uncertainty principle. These results extend a recent paper of the authors which studies a QSTFT having a Gaussian function as a window.

math.CV↗
arXiv2021

On the Quadratic Cone of $\mathbb{R}_3.$

In this paper we study the following type of functions $f: \mathcal{Q}_{\mathbb{R}_{3}} \to \mathbb{R}_{3}$, where $ \mathcal{Q}_{\mathbb{R}_3}$ is the quadratic cone of the algebra $\mathbb{R}_{3}$. From the fact that it is possible to write the algebra $ \mathbb{R}_{3}$ as a direct sum of quaternions, we get the observation that it is possible to find a clever representation for $ \mathcal{Q}_{\mathbb{R}_3}$. By using this result a slice regular theory was introduced and a Cauchy formula is discussed. Moreover, a detailed study of the zeros is performed. Finally, we find a formula for the determinant of a matrix with entries in $ \mathcal{Q}_{\mathbb{R}_3}$.

math.CV↗
arXiv2021

On Brolin's theorem over the quaternions

In this paper we investigate the Brolin's theorem over $\mathbb{H}$, the skew field of quaternions. Moreover, considering a quaternionic polynomial $p$ with real coefficients, we focus on the properties of its equilibrium measure, among the others, the mixing property and the Lyapunov exponents of the measure. We prove a central limit theorem and we compute the topological entropy and measurable entropy with respect to the quaternionic equilibrium measure. We prove that they are equal considering both a quaternionic polynomial with real coefficients and a polynomial with coefficients in a slice but not all real. Brolin's theorems for the one slice preserving polynomials and for generic polynomials are also proved.

math.CV↗
arXiv2020

Notes on Symmetry in Convective Flows

In most fluid dynamics problems, the governing equations are nonlinear because of the presence of convective terms. Nevertheless, existence of solutions can be shown by direct sum provided one identifies, in the relevant Banach space of solutions, particular subspaces which are invariant under time evolution. As an example, we consider classical convection problems and show how symmetry arguments can help in identifying such subspaces, as well as the role it can play in some stability issues.

math-ph↗
arXiv2020

On the quaternionic short-time Fourier and Segal-Bargmann transforms

In this paper, we study a special one dimensional quaternion short-time Fourier transform (QSTFT). Its construction is based on the slice hyperholomorphic Segal-Bargmann transform. We discuss some basic properties and prove different results on the QSTFT such as Moyal formula, reconstruction formula and Lieb's uncertainty principle. We provide also the reproducing kernel associated to the Gabor space considered in this setting.

math.CV↗