arXiv subjects

Francesco Mantovani

Publications and source records attributed to Francesco Mantovani.

7 recordsLinked to original sources

arXiv2026

Approximation of Fractals via Lagrange-type Superoscillations

We study the approximation of the Weierstrass function by means of superoscillating sequences. Superoscillatory functions are band-limited functions whose local oscillation rate can exceed the highest frequency contained in their Fourier spectrum. Starting from Lagrange-type interpolation at nodes in $[-1,1]$, we construct a double-indexed family $\mathcal{W}_{N,n}(x)$ that approximates the truncated Weierstrass function $W_N(x)$ for each fixed truncation order~$N$. We prove that if the number of interpolation nodes $n_N$ grows sufficiently fast relative to the highest frequency $b^Nπ$, namely $b^Nπ/n_N\to 0$, then $\mathcal{W}_{N,n_N}$ converges uniformly to the full Weierstrass function on every compact set. We also show that the two limits in $N$ and $n$ do \emph{not} commute: for any fixed~$n$ the series $\lim_{N\to\infty}\mathcal{W}_{N,n}(x)$ diverges for every $x\neq 0$, a phenomenon called the Divergence Wall.

math.CV↗

arXiv2026

Nonlocal Fourier Laws for Heat Propagation via Fractional powers of Vector Operators

The present work is devoted to the study of fractional powers of vector operators, with particular emphasis on the gradient operator with non-constant coefficients. Within the setting of Clifford algebra $\mathbb{R}_n$, this operator turns out to have bisectorial properties. By applying the spectral theory on the $S$-spectrum, we address a fundamental mathematical challenge: unlike sectorial operators, bisectorial operators involve fractional powers that are not analytic on the negative real line. To circumvent this, we introduce a novel definition of the fractional power function in this setting. Building upon previous works on bisectorial vector operators and weak solutions, we extend the definition of fractional powers to abstract vector operators. The core contribution of this work is the application of the functional calculus for vector operators to the gradient operator, showing that these fractional powers provide a rigorous mathematical foundation for nonlocal Fourier laws in heat propagation.

math.FA↗

arXiv2026

Slice hyperholomorphicity of the $S$-resolvent operators and boundary conditions

The foundation of spectral theory on the $S$-spectrum can be traced back to the quaternionic framework of quantum mechanics. The concept of $S$-spectrum for quaternionic operators emerged as the natural spectrum in slice hyperholomorphic functional calculi, known as the $S$-functional calculus and also utilized in the quaternionic spectral theorem. This spectral theory extends to Clifford operators. A key distinction from classical complex spectral theory lies in the definition of the $S$-spectrum, which is second order in the operator $T$, and in the $S$-resolvent operators that turns out to be the product of two different operators. This study delves into the analyticity of the $S$-resolvent operators under specified boundary conditions for the $S$-spectral problem. The spectral theory on the $S$-spectrum also provides deeper insights into classical spectral theory.

math.SP↗

arXiv2025

Quadratic estimates for the $H^\infty$-functional calculus of bisectorial Clifford operators

The $H^\infty$-functional calculus is a two-step procedure, introduced by A. McIntosh, that allows the definition of functions of sectorial operators in Banach spaces. It plays a crucial role in the spectral theory of differential operators, as well as in their applications to evolution equations and various other fields of science. An extension of the $H^\infty$-functional calculus also exists in the hypercomplex setting, where it is based on the notion of $S$-spectrum. Originally this was done for sectorial quaternionic operators, but then also generalized all the way to bisectorial fully Clifford operators. In the latter setting and in Hilbert spaces, this paper now characterizes the boundedness of the $H^\infty$-functional calculus through certain quadratic estimates. Due to substantial differences in the definitions of the $S$-spectrum and the $S$-resolvent operators, the proofs of quadratic estimates in this setting face additional challenges compared to the classical theory of complex operators.

math.SP↗

arXiv2025

The $H^\infty$-functional calculus for right slice hyperholomorphic functions and right linear Clifford operators

In 2016, the spectral theory on the $S$-spectrum was used to establish the $H^\infty$-functional calculus for quaternionic or Clifford operators. This calculus applies for example to sectorial or bisectorial right linear operators $T$ and left slice hyperholomorphic functions $f$ that can grow as polynomials. It relies on the product of the two operators $e(T)^{-1}$ and $(ef)(T)$, both defined via some underlying $S$-functional calculus (also called $ω$-functional calculus). For left slice holomorphic functions $f$ this definition does not depend on the choice of the regularizer function $e$. However, due to the non-commutative multiplication of Clifford numbers, it was unclear how to extend this definition to right slice hyperholomorphic functions. This paper addresses this significant unresolved issue and shows how right linear operators can possess the $H^\infty$-functional calculus also for right slice hyperholomorphic functions.

math.SP↗

arXiv2025

The $H^\infty$-functional calculus for bisectorial Clifford operators

The aim of this article is to introduce the H-infinity functional calculus for unbounded bisectorial operators in a Clifford module over the algebra R_n. While recent studies have focused on bounded operators or unbounded paravector operators, we now investigate unbounded fully Clifford operators and define polynomially growing functions of them. We first generate the omega-functional calculus for functions that exhibit an appropriate decay at zero and at infinity. We then extend to functions with a finite value at zero and at infinity. Finally, using a subsequent regularization procedure, we can define the H-infinity functional calculus for the class of regularizable functions, which in particular include functions with polynomial growth at infinity and, if T is injective, also functions with polynomial growth at zero.

math.FA↗

arXiv2024

Spectral properties of the gradient operator with nonconstant coefficients

In mathematical physics, the gradient operator with nonconstant coefficients encompasses various models, including Fourier's law for heat propagation and Fick's first law, that relates the diffusive flux to the gradient of the concentration. Specifically, consider $n\geq 3$ orthogonal unit vectors $e_1,\dots,e_n\in\mathbb{R}^n$, and let $Ω\subseteq\mathbb{R}^n$ be some (in general unbounded) Lipschitz domain. This paper investigates the spectral properties of the gradient operator $T=\sum_{i=1}^ne_ia_i(x)\frac{\partial}{\partial x_i}$ with nonconstant positive coefficients $a_i:\overlineΩ\to(0,\infty)$. Under certain regularity and growth conditions on the $a_i$, we identify bisectorial or strip-type regions that belong to the $S$-resolvent set of $T$. Moreover, we obtain suitable estimates of the associated resolvent operator. Our focus lies in the spectral theory on the $S$-spectrum, designed to study the operators acting in Clifford modules $V$ over the Clifford algebra $\mathbb{R}_n$, with vector operators being a specific crucial subclass. The spectral properties related to the $S$-spectrum of $T$ are linked to the inversion of the operator $Q_s(T):=T^2-2s_0T+|s|^2$, where $s\in\mathbb{R}^{n+1}$ is a paravector, i.e., it is of the form $s=s_0+s_1e_1+\dots+s_ne_n$. This spectral problem is substantially different from the complex one, since it allows to associate general boundary conditions to $Q_s(T)$, i.e., to the squared operator $T^2$.

math.FA↗