arXiv subjects

Irene Sabadini

Publications and source records attributed to Irene Sabadini.

At least 37 records · Page 2Linked to original sources

arXiv2024

Nuclearity and Grothendieck-Lidskii formula for quaternionic operators

We introduce an appropriate notion of trace in the setting of quaternionic linear operators, arising from the well-known companion matrices. We then use this notion to define the quaternionic Fredholm determinant of trace-class operators in Hilbert spaces, and show that an analog of the classical Grothendieck-Lidskii formula, relating the trace of an operator with its eigenvalues, holds. We then extend these results to the so-called $\frac{2}{3}$-nuclear (Fredholm) operators in the context of quaternionic locally convex spaces. While doing so, we develop some results in the theory of topological tensor products of noncommutative modules, and show that the trace defined ad hoc in terms of companion matrices, arises naturally as part of a canonical trace.

math.CA↗

arXiv2024

Super-phenomena in arbitrary quantum observables

Superoscillations occur when a globally band-limited function locally oscillates faster than its highest Fourier coefficient. We generalize this effect to arbitrary quantum mechanical operators as a weak value, where the preselected state is a superposition of eigenstates of the operator with eigenvalues bounded to a range, and the postselection state is a local position. Superbehavior of this operator occurs whenever the operator's weak value exceeds its eigenvalue bound. We give illustrative examples of this effect for total angular momentum and energy. In the later case, we demonstrate a sequence of harmonic oscillator potentials where a finite energy state converges everywhere on the real line, using only bounded superpositions of states whose asymptotic energy vanishes - "energy out of nothing". This limit requires postselecting the particle in a region whose size diverges in the considered limit. We further show that superenergy behavior implies that the state superoscillates in time with a rate given by the superenergy divided by the reduced Planck's constant. This example demonstrates the possibility of mimicking a high-energy state with coherent superpositions of nearly zero-energy states for as wide a spatial region as desired. We provide numerical evidence of these features to further bolster and elucidate our claims.

quant-ph↗

arXiv2024

A Borel-Pompeiu formula in a $(q,q')$-model of quaternionic analysis

The study of $ψ-$hyperholomorphic functions defined on domains in $\mathbb R^4$ with values in $\mathbb H$, namely null-solutions of the $ψ-$Fueter operator, is a topic which captured great interest in quaternionic analysis. This class of functions is more general than that of Fueter regular functions. In the setting of $(q,q')-$calculus, also known as post quantum calculus, we introduce a deformation of the $ψ-$Fueter operator written in terms of suitable difference operators, which reduces to a deformed $q$ calculus when $q'=1$. We also prove the Stokes and Borel-Pompeiu formulas in this context. This work is the first investigation of results in quaternionic analysis in the setting of the $(q,q')-$calculus theory.

math.CV↗

arXiv2023

On the Fueter-Sce theorem for generalized partial-slice monogenic functions

In a recent paper, we introduced the concept of generalized partial-slice monogenic functions. The class of these functions includes both the theory of monogenic functions and of slice monogenic functions with values in a Clifford algebra. In this paper, we prove a version of the Fueter-Sce theorem in this new setting, which allows to construct monogenic functions in higher dimensions starting from generalized partial-slice monogenic functions. We also prove that an alternative construction can be obtained by using the dual Radon transform. To this end, we provide a study of the generalized CK-extension.

math.CV↗

arXiv2023

Slice monogenic theta series

In this paper we introduce a generalization of theta series in the context of the slice monogenic function theory in $\mathbb{R}^{n+1}$ where me make use of the so-called $*$-exponential function in a hypercomplex variable. Together with the Eisenstein and Poincaré series that we introduced in a previous paper, the theta series construction in this paper completes the fundament of the basic theory of modular forms in the slice monogenic setting. We introduce a suitable generalized Poisson summation formula in this framework and we apply a properly adapted Fourier transform. As a direct application we prove a transformation formula for slice monogenic theta series. Then we introduce a family of conjugated theta functions. These are used to construct a slice monogenic generalization of the third power of the Dedekind eta function and of the modular discriminant. We also investigate their transformation behavior. Finally, we show that these theta series are special solutions to a generalization of the heat equation associated with the slice derivative. We round off by discussing the monogenic case.

math.NT↗

arXiv2023

Octonionic monogenic and slice monogenic Hardy and Bergman spaces

In this paper we discuss some basic properties of octonionic Bergman and Hardy spaces. In the first part we review some fundamental concepts of the general theory of octonionic Hardy and Bergman spaces together with related reproducing kernel functions in the monogenic setting. We explain how some of the fundamental problems in well-defining a reproducing kernel can be overcome in the non-associative setting by looking at the real part of an appropriately defined para-linear octonion-valued inner product. The presence of a weight factor of norm $1$ in the definition of the inner product is an intrinsic new ingredient in the octonionic setting. Then we look at the slice monogenic octonionic setting using the classical complex book structure. We present explicit formulas for the slice monogenic reproducing kernels for the unit ball, the right octonionic half-space and strip domains bounded in the real direction. In the setting of the unit ball we present an explicit sequential characterization which can be obtained by applying the special Taylor series representation of the slice monogenic setting together with particular octonionic calculation rules that reflect the property of octonionic para-linearity.

math.CV↗

arXiv2023

Fractional slice regular functions of a quaternionic variable

The theory of slice regular functions of a quaternionic variable on the unit ball of the quaternions was introduced by Gentili and Struppa in 2006 and nowadays it is a well established function theory, especially in view of its applications to operator theory. In this paper, we introduce the notion of fractional slice regular functions of a quaternionic variable defined as null-solutions of a fractional Cauchy-Riemann operators. We present a fractional Cauchy-Riemann operator in the sense of Riemann-Liouville and then in the sense of Caputo, with orders associated to an element of $(0,1)\times \mathbb R \times (0,1)\times \mathbb R$ for some axially symmetric slice domains which are new in the literature. We prove a version of the representation theorem, of the splitting lemma and we discuss a series expansion.

math.FA↗

arXiv2023

Superoscillations and Fock spaces

In this paper we use techniques in Fock spaces theory and compute how the Segal-Bargmann transform acts on special wave functions obtained by multiplying superoscillating sequences with normalized Hermite functions. It turns out that these special wave functions can be constructed also by computing the approximating sequence of the normalized Hermite functions. First, we start by treating the case when a superoscillating sequence is multiplied by the Gaussian function. Then, we extend these calculations to the case of normalized Hermite functions leading to interesting relations with Weyl operators. In particular, we show that the Segal-Bargmann transform maps superoscillating sequences onto a superposition of coherent states. Following this approach, the computations lead to a specific linear combination of the normalized reproducing kernels (coherent states) of the Fock space. As a consequence, we obtain two new integral Bargmann-type representations of superoscillating sequences. We also investigate some results relating superoscillation functions with Weyl operators and Fourier transform.

math-ph↗

arXiv2023

On the generating functions and special functions associated with superoscillations

The aim of this paper is to study generating functions for the coefficients of the classical superoscillatory function associated with weak measurements. We also establish some new relations between the superoscillatory coefficients and many well-known families of special polynomials, numbers, and functions such as Bernstein basis functions, the Hermite polynomials, the Stirling numbers of second kind, and also the confluent hypergeometric functions. Moreover, by using generating functions, we are able to develop a recurrence relation and a derivative formula for the superoscillatory coefficients.

math.CA↗

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↗

arXiv2023

The general theory of superoscillations and supershifts in several variables

In this paper we describe a general method to generate superoscillatory functions of several variables starting from a superoscillating sequence of one variable. Our results are based on the study of suitable infinite order differential operators on holomorphic functions with growth conditions of exponential type, where additional constraints are required when dealing with infinite order differential operators whose symbol is a function that is holomorphic in some open set, but not necessarily entire. The results proved for the superoscillating sequence in several variables are extended to sequences of supershifts in several variables.

math.FA↗

arXiv2022

An approach to the Gaussian RBF kernels via Fock spaces

We use methods from the Fock space and Segal-Bargmann theories to prove several results on the Gaussian RBF kernel in complex analysis. The latter is one of the most used kernels in modern machine learning kernel methods, and in support vector machines (SVMs) classification algorithms. Complex analysis techniques allow us to consider several notions linked to the RBF kernels like the feature space and the feature map, using the so-called Segal-Bargmann transform. We show also how the RBF kernels can be related to some of the most used operators in quantum mechanics and time frequency analysis, specifically, we prove the connections of such kernels with creation, annihilation, Fourier, translation, modulation and Weyl operators. For the Weyl operators, we also study a semigroup property in this case.

math-ph↗

arXiv2022

A Hörmander-Fock space

In a recent paper we used a basic decomposition property of polyanalytic functions of order $2$ in one complex variable to characterize solutions of the classical $\overline{\partial}$-problem for given analytic and polyanalytic data. Our approach suggested the study of a special reproducing kernel Hilbert space that we call the Hörmander-Fock space that will be further investigated in this paper. The main properties of this space are encoded in a specific moment sequence denoted by $η=(η_n)_{n\geq 0}$ leading to a special entire function $\mathsf{E}(z)$ that is used to express the kernel function of the Hörmander-Fock space. We present also an example of a special function belonging to the class ML introduced recently by Alpay et al. and apply a Bochner-Minlos type theorem to this function, thus motivating further connections with the theory of stochastic processes.

math.CV↗

arXiv2022

Universality property of the $S$-functional calculus, noncommuting matrix variables and Clifford operators

The spectral theory on the $S$-spectrum was born out of the need to give quaternionic quantum mechanics (formulated by Birkhoff and von Neumann) a precise mathematical foundation. Then it turned out that this theory has important applications in several fields such as fractional diffusion problems and, moreover, it allows one to define several functional calculi for $n$-tuples of noncommuting operators. With this paper we show that the spectral theory on the $S$-spectrum is much more general and it contains, just as particular cases, the complex, the quaternionic and the Clifford settings. More precisely, we show that the $S$-spectrum is well defined for objects in an algebra that has a complex structure and for operators in general Banach modules. We show that the abstract formulation of the $S$-functional calculus goes beyond quaternionic and Clifford analysis. Indeed we show that the $S$-functional calculus has a certain {\em universality property}. This fact makes the spectral theory on the $S$-spectrum applicable to several fields of operator theory and allows one to define functions of noncommuting matrix variables, and operator variables, as a particular case.

math.FA↗

arXiv2022

Hörmander's $L^2$-method, $\bar{\partial}$-problem and polyanalytic function theory in one complex variable

In this paper we consider the classical $\bar{\partial}$-problem in the case of one complex variable both for analytic and polyanalytic data. We apply the decomposition property of polyanalytic functions in order to construct particular solutions of this problem and obtain new Hörmander type estimates using suitable powers of the Cauchy-Riemann operator. We also compute particular solutions of the $\bar{\partial}$-problem for specific polyanalytic data such as the Itô complex Hermite polynomials and polyanalytic Fock kernels.

math.CV↗

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

Discrete analytic functions, structured matrices and a new family of moment problems

Using Zeilberger generating function formula for the values of a discrete analytic function in a quadrant we make connections with the theory of structured reproducing kernel spaces, structured matrices and a generalized moment problem. An important role is played by a Krein space realization result of Dijksma, Langer and de Snoo for functions analytic in a neighborhood of the origin. A key observation is that, using a simple Moebius transform, one can reduce the study of discrete analytic functions in the upper right quadrant to problems of function theory in the open unit disk. As an example, we associate to each finite positive measure on $[0,2π]$ a discrete analytic function on the right-upper quarter plane with values on the lattice defining a positive definite function. Emphasis is put on the rational case, both when an underlying Carathéodory function is rational and when, in the positive case, the spectral function is rational. The rational case and the general case are linked via the existence of a unitary dilation, possibly in a Krein space

math.CV↗

arXiv2021

Reproducing kernel Hilbert spaces of polyanalytic functions of infinite order

In this paper we introduce reproducing kernel Hilbert spaces of polyanalytic functions of infinite order. First we study in details the counterpart of the Fock space and related results in this framework. In this case the kernel function is given by $\displaystyle e^{z\overline{w}+\overline{z}w}$ which can be connected to kernels of polyanalytic Fock spaces of finite order. Segal-Bargmann and Berezin type transforms are also considered in this setting. Then, we study the reproducing kernel Hilbert spaces of complex-valued functions with reproducing kernel $\displaystyle\frac{1}{(1-z\overline{w})(1-\overline{z}w)}$ and $\displaystyle\frac{1}{1-2{\rm Re}\, z\overline{w}}$. The corresponding backward shift operators are introduced and investigated.

math.CV↗