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Antonino De Martino

Publications and source records attributed to Antonino De Martino.

At least 19 recordsLinked to original sources
arXiv2026

New Taylor and Laurent series of axially harmonic, Fueter regular and polyanalytic functions

The Fueter-Sce mapping theorem stands as one of the most profound outcomes in complex and hypercomplex analysis, producing hypercomplex generalizations of holomorphic functions. In recent years, delving into the factorization of the second operator appearing in the Fueter-Sce mapping theorem has uncovered its potential to generate novel classes of functions and their respective functional calculi. The sets of functions obtained from this factorization and the associated functional calculi define the so-called {\em fine structures on the $S$-spectrum}. This paper aims to comprehensively investigate the function theories for the fine structures of Dirac type in the quaternionic framework, presenting new series expansions for axially harmonic, Fueter regular, and axially polyanalytic functions. These series expansions are highly nontrivial. In fact, when considering the hypercomplex realm, specifically the quaternionic or the Clifford setting, extending the concept of complex power series expansion is not immediate, and different Taylor and Laurent expansions appear with different sets of convergence. Additionally, our objectives include establishing the representation formulas for these function spaces; such formulas encode the fundamental properties of the functions and have numerous consequences. Finally, in the last section of this paper, we explain the applications of the fine structures in operator theory.

math.CV↗
arXiv2026

A variation of the inverse Fueter theorem and the generalized polyanalytic Cauchy-Kovalevskaya extension of order 2

In this paper, we first establish a generalized Cauchy--Kovalevskaya (GCK) extension for axially polyanalytic functions of order $2$. We prove that the extension can be written as a power series involving differential operators acting on two initial functions. We also study the decomposition of the GCK extension in terms of integrals over the sphere 2-sphere $\mathbb{S}$ involving plane-wave type functions. We further establish a connection between the GCK extension for polyanalytic functions of order $2$ and the Fueter theorem. This is one of the most important results in hypercomplex analysis and can be described in two steps. In the first step, starting from holomorphic functions of one complex variable, the application of a suitable operator yields slice hyperholomorphic functions. In the second step, applying the Laplace operator in four real variables (called Fueter map) one obtains axially monogenic functions, i.e. functions in the kernel of the Fueter operator. A suitable factorization of the Fueter map in terms of the Fueter operator and its conjugate gives rise to two intermediate classes of functions between slice hyperholomorphic functions and axially monogenic functions: axially harmonic functions and axially polyanalytic functions of order $2$. Another goal of this paper is to study the invertibility of the factorizations of the Fueter map for harmonic and polyanalytic functions of order 2 on suitable open sets, and to derive integral representation formulas for the inverse of the factorized Fueter map. These integral representations are based on the Cauchy formula for polyanalytic functions of order $2$ and the Poisson integral formula for harmonic functions.

math.FA↗
arXiv2026

Fractional $\frac{1}{2}$-Laplacian of holomorphic functions via Jacobi polynomials

We give a distributional definition of the $\frac{1}{2}$-fractional Laplacian for smooth functions with power-type singularities at the origin, including negative monomials and certain holomorphic functions with isolated singularities. The construction is based on a suitable space of test functions related to the Lizorkin space of test functions, for which both the test functions and their fractional Laplacians belong to the Schwarz class and vanish to infinite order at the origin, and thus compensate for any singularity of power type at the origin. This allows the $\frac{1}{2}$-fractional Laplacian to be defined by duality without requiring the underlying function to satisfy the usual integrability assumptions. The considered distributional framework is shown to be invariant under the $\frac{1}{2}$-fractional Laplacian, yielding a natural semigroup property. By applying the construction term by term to Laurent series we obtain new series that, remarkably, involve Jacobi polynomials. With this procedure we define the fractional Laplacian for a class of holomorphic functions and derive several applications, with particular emphasis on the recently developed theory of the quaternionic fine structures of the spectral theory on the $S$-spectrum, describing the functional calculi extending the classical holomorphic functional calculus in the various classes of holomorphic-type functions arising from the Fueter-Sce-Qian extension theorem.

math.FA↗
arXiv2026

Representer Theorem in Complex Reproducing Kernel Hilbert Spaces with Applications to Fock and Hardy Spaces and Superoscillations

We introduce a complex-valued counterpart of the representer theorem in machine learning. We study several learning and minimization problems in reproducing kernel Hilbert spaces (RKHSs), with the aim of identifying appropriate input-output data sets that allow specific functions to appear as solutions of regression-type minimization problems. In particular, we recover superoscillations in the Fock space, the Gaussian radial basis function (RBF) kernel in the corresponding RKHS, and finite Blaschke products in the Hardy space setting. We then extend the notion of superoscillations through suitable generalizations of the Fock space and investigate the associated learning problems. This is a seminal work relating superoscillations and machine learning kernel methods via the representer theorem.

math.FA↗
arXiv2026

The resolvent equations for the Harmonic and bi-Harmonic functional calculi in dimension five

The fine structures on the $S$-spectrum constitute a new research area that includes a class of functional calculi based on the $S$-spectrum and on integral transforms determined by the Fueter--Sce mapping theorem and the Cauchy formula for slice hyperholomorphic functions. This strategy, based on integral transforms, allows us to construct functional calculi that include harmonic and polyharmonic functional calculi. The resolvent operators in this setting do not arise directly from a Cauchy kernel, but rather from suitable manipulations of it. For this reason the corresponding resolvent equations differ substantially from those associated with the classical Cauchy kernel. In this paper, we investigate the harmonic and biharmonic resolvent equations in dimension five, as well as the corresponding product rules and Riesz projectors for these functional calculi.

math.FA↗
arXiv2026

On the application of the factorized Fueter-Sce map to the slice hyperholomorphic Cauchy kernel

The Fueter-Sce theorem is one of the most important results in hypercomplex analysis, providing a two-step procedure for constructing axially monogenic functions starting from holomorphic functions of one variable. In the first step, the so-called slice operator is applied to holomorphic functions of one variable, producing the class of slice hyperholomorphic functions. The second step yields the class of axially monogenic functions by applying the pointwise differential operator Delta_{n+1}^{(n-1)/2}, with n odd, known as the Fueter-Sce map. The significance of the Fueter-Sce theorem also lies in the fact that it induces two spectral theories corresponding to the function classes it generates. Over the years, factorizations of the Fueter-Sce map have been studied to identify the intermediate spaces that arise between slice hyperholomorphic functions and axially monogenic functions. Until now, only certain factorizations of the Fueter-Sce map have been considered. In this paper, our goal is to determine the most general factorizations of the Fueter-Sce map and to apply these differential operators to the Cauchy kernel of slice hyperholomorphic functions.

math.FA↗
arXiv2025

Polyanalytic Gaussian Radial Basis Function Kernel and Itô-Hermite Polynomials

We introduce a polyanalytic extension of the Gaussian radial basis function (RBF) kernel by computing the action of the convolution operator on normalized Hermite functions. In particular, using the Zaremba-Bergman formula we derive an explicit closed form for this new reproducing kernel function. We then establish an isomorphism relating the reproducing kernel Hilbert space induced by the polyanalytic Gaussian RBF kernel with the corresponding polyanalytic Fock space. Moreover, we provide a characterization of polyanalytic Gaussian RBF spaces in terms of a Landau-type operator. In addition, we investigate the polyanalytic counterpart of the Weyl operator, which leads to applications involving the Christoffel-Darboux formula for Hermite polynomials and Mehler's kernel. Finally, we discuss the analogue of the Weyl operator in the context of the polyanalytic Gaussian RBF setting.

math.FA↗
arXiv2025

Functions and operators of the polyharmonic and polyanalytic Clifford fine structures on the $S$-spectrum

The spectral theory on the $S$-spectrum originated to give quaternionic quantum mechanics a precise mathematical foundation and as a spectral theory for linear operators in vector analysis. This theory has proven to be significantly more general than initially anticipated, naturally extending to fully Clifford operators and revealing unexpected connections with the spectral theory based on the monogenic spectrum, developed over forty years ago by A. McIntosh and collaborators. In recent years, we have combined slice hyperholomorphic functions with the Fueter-Sce mapping theorem, also called Fueter-Sce extension theorem, to broaden the class of functions and operators to which the theory can be applied. This generalization has led to the definition of what we call the {\em fine structures on the $S$-spectrum}, consisting of classes of functions that admit an integral representation and their associated functional calculi. In this paper, we focus on the fine structures within the Clifford algebra setting, particularly addressing polyharmonic functions, polyanalytic functions, holomorphic Cliffordian functions and their associated functional calculi defined via integral representation formulas. Moreover, we demonstrate that the monogenic functional calculus, defined via the monogenic Cauchy formula, and the $F$-functional calculus of the fine structures, defined via the Fueter-Sce mapping theorem in integral form, yield the same operator.

math.FA↗
arXiv2025

On the harmonic generalized Cauchy-Kovalevskaya extension and its connection with the Fueter-Sce theorem

One of the primary objectives of this paper is to establish a generalized Cauchy-Kovalevskaya extension for axially harmonic functions. We demonstrate that the result can be expressed as a power series involving Bessel-type functions of specific differential operators acting on two initial functions. Additionally, we analyze the decomposition of the harmonic CK extension in terms of integrals over the sphere $ \mathbb{S}^{m-1} $ involving functions of plane wave type. Another key goal of this paper is to explore the relationship between the harmonic Cauchy-Kovalevskaya extension and the Fueter-Sce theorem. The Fueter-Sce theorem outlines a two-step process for constructing axially monogenic functions in $ \mathbb{R}^{m+1}$ starting from holomorphic functions in one complex variable. The first step generates the class of slice monogenic functions, while the second step produces axially monogenic functions by applying the pointwise differential operator $ Δ_{\mathbb{R}{^{m+1}}}^{\frac{m-1}{2}} $ with $m$ being odd, known as the Fueter-Sce map, to a slice monogenic function. By suitably factorizing the Fueter-Sce map, we introduce the set of axially harmonic functions, which serves as an intermediate class between slice monogenic and axially monogenic functions. In this paper, we establish a connection between the harmonic CK extension and the factorization of the Fueter-Sce map. This connection leads to a new notion of harmonic polynomials, which we show to form a basis for the Riesz potential. Finally, we also construct a basis for the space of axially harmonic functions.

math.FA↗
arXiv2024

An introduction to the fine structures on the $S$-spectrum

Holomorphic functions are fundamental in operator theory and their Cauchy formula is a crucial tool for defining functions of operators. The Fueter-Sce extension theorem (often called Fueter-Sce mapping theorem) provides a two-step procedure for extending holomorphic functions to hyperholomorphic functions. In the first step, slice hyperholomorphic functions are obtained, and their associated Cauchy formula establishes the $S$-functional calculus for noncommuting operators on the $S$-spectrum. The second step produces axially monogenic functions, which lead to the development of the monogenic functional calculus. In this review paper we discuss the second operator in the Fueter-Sce mapping theorem that takes slice hyperholomorphic to axially monogenic functions. This operator admits several factorizations which generate various function spaces and their corresponding functional calculi, thereby forming the so-called fine structures of spectral theories on the $S$-spectrum.

math.SP↗
arXiv2024

Short-time Fourier transform and superoscillations

In this paper we investigate new results on the theory of superoscillations using time-frequency analysis tools and techniques such as the short-time Fourier transform (STFT) and the Zak transform. We start by studying how the short-time Fourier transform acts on superoscillation sequences. We then apply the supershift property to prove that the short-time Fourier transform preserves the superoscillatory behavior by taking the limit. It turns out that these computations lead to interesting connections with various features of time-frequency analysis such as Gabor spaces, Gabor kernels, Gabor frames, 2D-complex Hermite polynomials, and polyanalytic functions. We treat different cases depending on the choice of the window function moving from the general case to more specific cases involving the Gaussian and the Hermite windows. We consider also an evolution problem with an initial datum given by superoscillation multiplied by the time-frequency shifts of a generic window function. Finally, we compute the action of STFT on the approximating sequences with a given Hermite window.

math.FA↗
arXiv2023

New aspects of Bargmann transform using Touchard polynomials and hypergeometric functions

In this paper we study the ranges of the Schwartz space $\mathcal S$ and its dual $\mathcal S^\prime$ (space of tempered distributions) under the Segal-Bargmann transform. The characterization of these two ranges lead to interesting reproducing kernel Hilbert spaces whose reproducing kernels can be expressed respectively in terms of the Touchard polynomials and the hypergeometric functions. We investigate the main properties of some associated operators and introduce two generalized Bargmann transforms in this framework. This can be considered as a continuation of an interesting research path that Neretin started earlier in his book on Gaussian integral operators

math.FA↗
arXiv2023

Gaussian RBF-kernels via Fock spaces: quaternionic and several complex variables settings

In this paper we study two extensions of the complex-valued Gaussian radial basis function (RBF) kernel and discuss their connections with Fock spaces in two different settings. First, we introduce the quaternonic Gaussian RBF kernel constructed using the theory of slice hyperholomorphic functions. Then, we consider the case of Gaussian RBF kernels in several complex variables.

math.FA↗
arXiv2023

The Bicomplex Tensor Product, a Bicomplex Choi Theorem and Applications

In this paper we extend the concept of tensor product to the bicomplex case and use it to prove the bicomplex counterpart of the classical Choi theorem in the theory of complex matrices and operators. The concept of hyperbolic tensor product is also discussed, and we link these results to the theory of quantum channels in the bicomplex and hyperbolic case, as well as applications to bicomplex digital signal processing.

cs.IT↗
arXiv2023

The Fueter-Sce mapping and the Clifford-Appell polynomials

The Fueter-Sce theorem provides a procedure to obtain axially monogenic functions, which are in the kernel of generalized Cauchy-Riemann operator in $ \mathbb{R}^{n+1}$. This result is obtained by using two operators. The first one is the slice operator, which extends holomorphic functions of one complex variable to slice monogenic functions in $ \mathbb{R}^{n+1}$. The second one is a suitable power of the Laplace operator in $n+1$ variables. Another way to get axially monogenic functions is the generalized Cauchy-Kovalevskaya (CK) extension. This characterizes axial monogenic functions by their restriction to the real line. In this paper, using the connection between the Fueter-Sce map and the generalized CK-extension, we explicitly compute the actions $Δ_{\mathbb{R}^{n+1}}^{\frac{n-1}{2}} x^k$, where $x \in \mathbb{R}^{n+1}$. The expressions obtained is related to a well-known class of Clifford-Appell polynomials. These are the building blocks to write a Taylor series for axially monogenic functions. Moreover, we focus on some elementary axially monogenic functions, where the action of the Fueter-Sce map and the generalized CK-extension coincide. In order to get algebraic relations between the elementary functions, as in complex analysis, we define a new product between axially monogenic functions. By using the connections between the Fueter-Sce map and the generalized CK extension we characterize the range and the kernel of the Fueter-Sce map. Furthermore, we focus on studying the Clifford-Appell-Fock space and the Clifford-Appell-Hardy space. Finally, using the polyanalytic Fueter-Sce theorems we obtain a new family of polyanalytic monogenic polynomials, which extends to higher dimensions the Clifford-Appell polynomials.

math.FA↗
arXiv2023

Harmonic and polyanalytic functional calculi on the $S$-spectrum for unbounded operators

Harmonic and polyanalytic functional calculi have been recently defined for bounded commuting operators. Their definitions are based on the Cauchy formula of slice hyperholomorphic functions and on the factorization of the Laplace operator in terms of the Cauchy-Fueter operator $\mathcal{D}$ and of its conjugate $\overline{\mathcal{D}}$. Thanks to the Fueter extension theorem when we apply the operator $\mathcal{D}$ to slice hyperholomorphic functions we obtain harmonic functions and via the Cauchy formula of slice hyperholomorphic functions we establish an integral representation for harmonic functions. This integral formula is used to define the harmonic functional calculus on the $S$-spectrum. Another possibility is to apply the conjugate of the Cauchy-Fueter operator to slice hyperholomorphic functions. In this case, with a similar procedure we obtain the class of polyanalytic functions, their integral representation and the associated polyanalytic functional calculus. The aim of this paper is to extend the harmonic and the polyanalytic functional calculi to the case of unbounded operators and to prove some of the most important properties. These two functional calculi belong to so called fine structures on the $S$-spectrum in the quaternionic setting. Fine structures on the $S$-spectrum associated with Clifford algebras constitute a new research area that deeply connects different research fields such as operator theory, harmonic analysis and hypercomplex analysis.

math.SP↗
arXiv2023

The fine structure of the spectral theory on the $S$-spectrum in dimension five

Holomorphic functions play a crucial role in operator theory and the Cauchy formula is a very important tool to define functions of operators. The Fueter-Sce-Qian extension theorem is a two steps procedure to extend holomorphic functions to the hyperholomorphic setting. The first step gives the class of slice hyperholomorphic functions; their Cauchy formula allows to define the so-called $S$-functional calculus for noncommuting operators based on the $S$-spectrum. In the second step this extension procedure generates monogenic functions; the related monogenic functional calculus, based on the monogenic spectrum, contains the Weyl functional calculus as a particular case. In this paper we show that the extension operator from slice hyperholomorphic functions to monogenic functions admits various possible factorizations that induce different function spaces. The integral representations in such spaces allows to define the associated functional calculi based on the $S$-spectrum. The function spaces and the associated functional calculi define the so called {\em fine structure of the spectral theories on the $S$-spectrum}. Among the possible fine structures there are the harmonic and poly-harmonic functions and the associated harmonic and poly-harmonic functional calculi. The study of the fine structures depends on the dimension considered and in this paper we study in detail the case of dimension five, and we describe all of them. The five-dimensional case is of crucial importance because it allows to determine almost all the function spaces will also appear in dimension greater than five, but with different orders.

math.SP↗
arXiv2022

Properties of a polyanalytic functional calculus on the $S$-spectrum

The Fueter mapping theorem gives a constructive way to extend holomorphic functions of one complex variable to monogenic functions, i.e., null solutions of the generalized Cauchy-Riemann operator in $\mathbb{R}^4$, denoted by $\mathcal{D}$. This theorem is divided in two steps. In the first step a holomorphic function is extended to a slice hyperholomorphic function. The Cauchy formula for these type of functions is the starting point of the $S$-functional calculus. In the second step a monogenic function is obtained by applying the Laplace operator in four real variables, namely $ Δ$, to a slice hyperholomorphic function. The polyanalytic functional calculus, that we study in this paper, is based on the factorization of $Δ= \mathcal{D} \mathcal{\overline{D}}$. Instead of applying directly the Laplace operator to a slice hyperholomorphic function we apply first the operator $ \mathcal{\overline{D}}$ and we get a polyanalytic function of order 2, i..e, a function that belongs to the kernel of $ \mathcal{D}^2$. We can represent this type of functions in an integral form and then we can define the polyanalytic functional calculus on $S$-spectrum. The main goal of this paper is to show the principal properties of this functional calculus. In particular, we study a resolvent equation suitable for proving a product rule and generate the Riesz projectors.

math.SP↗