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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 $ \bar {\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 $ \eta =(\eta _n)_{n\geq ~0} $ η = (η n) n ≥ 0 leading to a special entire function $ \mathsf {E}(z) $ 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 Mittag-Leffler (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. [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]

In this paper we consider the classical ∂ ¯ -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 ∂ ¯ -problem for specific polyanalytic data such as the Itô complex Hermite polynomials and polyanalytic Fock kernels. [ABSTRACT FROM AUTHOR]

In this article, we describe the invention of double-entry bookkeeping (or partita doppia as it was called in Italian) as a fertile intersection between mathematics and early commerce. We focus our attention on this seemingly simple technique that requires only minimal mathematical expertise, but whose discovery is clearly the result of a mathematical way of thinking, in order to make a conceptual point about the role of mathematics as the humus from which disciplines as different as operations research, computer science, and data science have evolved. [ABSTRACT FROM AUTHOR]

As a reply to the commentary (Lenhard in Found Sci, 2012), we stress here that structural understanding of data analysis techniques is the natural counterpart to the lack of understanding of phenomena in agnostic science. We suggest moreover that the dynamics of computational processes, and their parallels with the dynamics of natural processes, will increasingly be, possibly, the driving force of the development of data analysis. Keywords Philosophy of science * Computational methods * Data analysis, We thank Lenhard for his careful effort in (Lenhard 2012) to put our paper (Napoletani et al. 2011) in its appropriate historical and epistemological context. His paraphrase of our term [...]

As a reply to the commentary (Humphreys in Found Sci, 2012), we explore the methodological implications of seeing artificial neural networks as generic classification tools, we show in which sense the use of descriptions and models in data analysis is not equivalent to the original empirical use of epicycles in describing planetary motion, and we argue that agnostic science is essentially related to the type of problems we ask about a phenomenon and to the processes used to find answers. Keywords Philosophy of science * Neural networks * Computational methods data analysis, The thoughtful and sympathetic commentary (Humphreys 2012) was a welcome opportunity to better define concepts we discussed in our paper (Napoletani et al. 2011), and to introduce some new concepts [...]

In this paper we show that continuous endomorphisms on the space of entire functions with a given order can be expressed as differential operators of infinite order satisfying suitable growth conditions. [ABSTRACT FROM AUTHOR]

Primary productivity (PP) has been recently investigated using remote sensing-based models over quite limited geographical areas of the Red Sea. This work sheds light on how phytoplankton and primary production would react to the effects of global warming in the extreme environment of the Red Sea and, hence, illuminates how similar regions may behave in the context of climate variability. study focuses on using satellite observations to conduct an intercomparison of three net primary production (NPP) models—the vertically generalized production model (VGPM), the Eppley-VGPM, and the carbon-based production model (CbPM)—produced over the Red Sea domain for the 1998–2018 time period. A detailed investigation is conducted using multilinear regression analysis, multivariate visualization, and moving averages correlative analysis to uncover the models’ responses to various climate factors. Here, we use the models’ eight-day composite and monthly averages compared with satellite-based variables, including chlorophyll-a (Chla), mixed layer depth (MLD), and sea-surface temperature (SST). Seasonal anomalies of NPP are analyzed against different climate indices, namely, the North Pacific Gyre Oscillation (NPGO), the multivariate ENSO Index (MEI), the Pacific Decadal Oscillation (PDO), the North Atlantic Oscillation (NAO), and the Dipole Mode Index (DMI). In our study, only the CbPM showed significant correlations with NPGO, MEI, and PDO, with disagreements relative to the other two NPP models. This can be attributed to the models’ connection to oceanographic and atmospheric parameters, as well as the trends in the southern Red Sea, thus calling for further validation efforts. [ABSTRACT FROM AUTHOR]