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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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In the last decade there has been a growing interest in superoscillations in various fields of mathematics, physics and engineering. However, while in applications as optics the local oscillatory behaviour is the important property, some convergence to a plane wave is the standard characterizing feature of a superoscillating function in mathematics and quantum mechanics. Also there exists a certain discrepancy between the representation of superoscillations either as generalized Fourier series, as certain integrals or via special functions. The aim of this work is to close these gaps and give a general definition of superoscillations, covering the well-known examples in the existing literature. Superoscillations will be defined as sequences of holomorphic functions, which admit integral representations with respect to complex Borel measures and converge to a plane wave in the space \mathcal {A}_1(\mathbb {C}) of entire functions of exponential type. [ABSTRACT FROM AUTHOR]

Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. These functions appear in various fields of science and technology, in particular they were discovered in quantum mechanics in the context of weak values introduced by Y. Aharonov and collaborators. The evolution problem of superoscillatory functions as initial conditions for the Schrödinger equation is intensively studied nowadays and the supershift property of the solution of Schrödinger equation encodes the persistence of superoscillatory phenomenon during the evolution. In this paper, we prove that the evolution of a superoscillatory initial datum for spinning particles in a magnetic field has the supershift property. Our techniques are based on the exact propagator of spinning particles, the associated infinite order differential operators and their continuity on suitable spaces of entire functions with growth conditions. [ABSTRACT FROM AUTHOR]

Triangular operators are an essential tool in the study of nonselfadjoint operators that appear in different fields with a wide range of applications. Although the development of a quaternionic counterpart for this theory started at the beginning of this century, the lack of a proper spectral theory combined with problems caused by the underlying noncommutative structure prevented its real development for a long time. In this paper, we give criteria for a quaternionic linear operator to have a triangular representation, namely, under which conditions such operators can be represented as a sum of a diagonal operator with a Volterra operator. To this effect, we investigate quaternionic Volterra operators based on the quaternionic spectral theory arising from the S‐spectrum. This allow us to obtain conditions when a non‐selfadjoint operator admits a triangular representation. [ABSTRACT FROM AUTHOR]

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 machine classification algorithms. Complex analysis techniques allow us to consider several notions linked to the radial basis function (RBF) kernels, such as the feature space and the feature map, using the so-called Segal–Bargmann transform. We also show 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. [ABSTRACT FROM AUTHOR]

We prove various Beurling-Lax type theorems, when the classical backward-shift operator is replaced by a general resolvent operator associated with a rational function. We also study connections to the Cuntz relations. An important tool is a new representation result for analytic functions, in terms of composition and multiplication operators associated with a given rational function. Applications to the theory of de Branges-Rovnyak spaces, also in the indefinite metric setting, are given. [ABSTRACT FROM AUTHOR]

In this paper we define a new function theory of slice monogenic functions of a Clifford variable using the S-functional calculus for Clifford numbers. Previous attempts of such a function theory were obstructed by the fact that Clifford algebras, of sufficiently high order, have zero divisors. The fact that Clifford algebras have zero divisors does not pose any difficulty whatsoever with respect to our approach. The new class of functions introduced in this paper will be called the class of slice monogenic Clifford functions to stress the fact that they are defined on open sets of the Clifford algebra Rn. The methodology can be generalized, for example, to handle the case of noncommuting matrix variables. [ABSTRACT FROM AUTHOR]

Spectral theory on the S -spectrum was born out of the need to give quaternionic quantum mechanics a precise mathematical foundation (Birkhoff and von Neumann [8] showed that a general set theoretic formulation of quantum mechanics can be realized on real, complex or quaternionic Hilbert spaces). Then it turned out that spectral theory on S -spectrum 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. In fact, 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 the S -functional calculus has a certain 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. [ABSTRACT FROM AUTHOR]

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 , 2 π ] 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. [ABSTRACT FROM AUTHOR]

Background and Objectives: Remdesivir is an antiviral agent, which was shown to be safe and effective in treating early COVID-19, but its favourable impact in hospitalised patients with non-critical disease is still under investigation. The present study aimed to assess the effectiveness and safety of remdesivir as a treatment for hospitalised patients with COVID-19 by a propensity score analysis of observational data. Methods: In this monocentric retrospective cohort study, the effectiveness and safety of a 5-day course of remdesivir (200 mg intravenously at Day 1, then 100 mg from Days 2–5) in association with the standard of care were assessed in comparison with the standard of care only. The primary endpoint was the proportion of recovery on Day 14. Results: Of 3662 eligible inpatients who tested positive for the severe acute respiratory syndrome coronavirus 2 genome by nasopharyngeal swab at admission, 861 (24%) non-critical patients were included in a propensity score analysis and 281 (33%) were exposed to remdesivir. In total, 242/281 (86.1%) and 435/580 (75.0%) patients recovered in exposed and non-exposed, respectively, with a relative improvement of 11.1% (95% CI + 5.8 to 16.5%; unadjusted odds ratio: 2.07, 95% CI 1.40–3.05, p = 0.0001; after adjustment by propensity score weighting, odds ratio: 1.92, 95% CI 1.30–2.83, p = 0.001). In treated patients, 1 (0.03%) anaphylactic reaction and 1 (0.03%) acute reaction during drug injection were reported, and 24 (8.5%) patients stopped the treatment due to adverse reactions. No significant differences were found with respect to the secondary efficacy endpoints (in-hospital all-cause death, need for intensive care treatments, clinical improvement score at Day 28) and safety endpoints (any and serious adverse reactions). Conclusion: A 5-day course of remdesivir in association with the standard of care effectively promoted recovery from COVID-19 among non-critical in-hospital patients and had an acceptable safety profile. [ABSTRACT FROM AUTHOR]

Introduction: Atherosclerosis causes a chronic reduction of vascularization with consequent impairment of the performance of organs, like the brain or muscles, which determines the functional and cognitive decline of the elderly and their ability to respond to acute stressful condition. Therefore, our aim was to evaluate if ankle brachial index (ABI) could effectively be a determinant of in-hospital functional status and complications in elderly hospitalized patients. Methods: This is a monocentric cross-sectional study of 189 patients aged 65 years or older. The study was undertaken at the Internal Medicine ward of Niguarda Hospital in Milan. ABI (BOSO ABY-System 100) and in-hospital status (activities of daily living, ADL and instrumental activities of daily living, IADL) were collected on the second day of hospitalization. Complications (falls and delirium episodes) were also recorded during the whole hospitalization period. Results: The average age of patients was 79.3 ± 6.9 years. Among outcomes, only ADL (r = 0.192, p = 0.007) and IADL score (r = 0.200, p = 0.005) showed significant correlation with ABI. Moreover, during the subsequent logistic regression, ABI remained among the statistically significant determinants of both scores (β = 0.231, p = 0.013 and β = 0.314, p = 0.001, respectively). Conclusions: The main result of our study is the finding of ABI as a significant determinant of acute in-hospital functional impairment (evaluated as ADL and IADL scores). The continuous exposure of the brain and muscles to the reduced perfusions induced by vascular atherosclerosis, probably determined the reduced ability to respond to stressful conditions. [ABSTRACT FROM AUTHOR]