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We utilize a method using frequency combs to construct waves that feature superoscillations - local regions of the wave that exhibit a change in phase that the bandlimits of the wave should not otherwise allow. This method has been shown to create superoscillating regions that mimic any analytic function - even ones well outside the bandlimits - to an arbitrary degree of accuracy. We experimentally demonstrate that these waves are extremely robust against noise, allowing for accurate reconstruction of a superoscillating target function thoroughly buried in noise. We additionally show that such a construction can be easily used to range-resolve a signal well below the commonly accepted fundamental limit.

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., Comment: to appear in Applied and Computational Harmonic Analysis

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.

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 exponentially bounded entire functions.

We show how the birth of perspective painting in the Italian Renaissance led to a new way of interpreting space that resulted in the creation of projective geometry. Unlike other works on this subject, we explicitly show how the craft of the painters implied the introduction of new points and lines (points and lines at infinity) and their projective coordinates to complete the Euclidean space to what is now called projective space. We demonstrate this idea by looking at original paintings from the Renaissance, and by carrying out the explicit analytic calculations that underpin those masterpieces.

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 H\"ormander-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 $\eta=(\eta_n)_{n\geq 0}$ leading to a special entire function $\mathsf{E}(z)$ that is used to express the kernel function of the H\"ormander-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.

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 H\"ormander 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 It\^o complex Hermite polynomials and polyanalytic Fock kernels.

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., Comment: 9 pages, 5 figures, expanded treatment of time

We introduce a notion of isolated units, elementary particles or more general physical phenomena that do not significantly affect their surrounding environment, and we build a primitive ontology to describe their evolution and interaction. We further introduce a notion of indistinguishability of distinct spacetime paths of a unit, for which the evolution of the state variables of the unit is the same, and a generalization of the equivalence principle based on indistinguishability. Under a time invertibility condition on the whole set of indistinguishable paths of a unit, we show that the quantization of motion of spinless elementary particles in a general potential field can be derived in this framework, in the limiting case of weak fields and low velocities. Extrapolating this approach to include weak relativistic effects, we explore possible experimental consequences., Comment: 34 pages., minor corrections, added new references and a conclusion

Infinite order differential operators appear in different fields of Mathematics and Physics and in the last decades they turned out to be of fundamental importance in the study of the evolution of superoscillations as initial datum for Schr\"odinger equation. Inspired by the operators arising in quantum mechanics, in this paper we investigate the continuity of a class of infinite order differential operators acting on spaces of entire hyperholomorphic functions. The two classes of hyperholomorphic functions, that constitute a natural extension of functions ofone complex variable to functions of paravector variables are illustrated by the Fueter-Sce-Qian mapping theorem. We show that, even though the two notions of hyperholomorphic functions are quite different from each other, entire hyperholomorphic functions with exponential bounds play a crucial role in the continuity of infinite order differential operators acting on these two classes of entire hyperholomorphic functions. We point out the remarkable fact that the exponential function of a paravector variable is not in the kernel of the Dirac operator but entire monogenic functions with exponential bounds play an important role in the theory., Comment: 28 pages

We construct in this paper a large class of superoscillating sequences, more generally of $\mathscr F$-supershifts, where $\mathscr F$ is a family of smooth functions (resp. distributions, hyperfunctions) indexed by a real parameter $\lambda\in \R$. The key model we introduce in order to generate such families is the evolution through a Schr\"odinger equation $(i\partial/\partial t - \mathscr H(x))(\psi)=0$ with a suitable hamiltonian $\mathscr H$, in particular a suitable potential $V$ when $\mathscr H(x) = -(\partial^2/\partial x^2)/2 + V(x)$. The family $\mathscr F$ is in this case $\mathscr F= \{(t,x) \mapsto \varphi_\lambda(t,x)\,;\, \lambda \in \R\}$, where $\varphi_\lambda$ is evolved from the initial datum $x\mapsto e^{i\lambda x}$. Then $\mathscr F$-supershifts will be of the form $\{\sum_{j=0}^N C_j(N,a) \varphi_{1-2j/N}\}_{N\geq 1}$ for $a\in \R\setminus [-1,1]$, taking $C_j(N,a) =\binom{N}{j}(1+a)^{N-j}(1-a)^j/2^N$. We prove the locally uniform convergence of derivatives of the supershift towards corresponding derivatives of its limit. We analyse in particular the case of the quantum harmonic oscillator, which forces us, in order to take into account singularities of the evolved datum, to enlarge the notion of supershifts for families of functions to a similar notion for families of hyperfunctions, thus beyond the frame of distributions.

We develop some aspects of the theory of hyperholomorphic functions whose values are taken in a Banach algebra over a field -- assumed to be the real or the complex numbers -- and which contains the field. Notably, we consider Fueter expansions, Gleason's problem, the theory of hyperholomorphic rational functions, modules of Fueter series, and related problems. Such a framework includes many familiar algebras as particular cases. The quaternions, the split quaternions, the Clifford algebras, the ternary algebra, and the Grassmann algebra are a few examples of them.

We begin a study of Schur analysis in the setting of the Grassmann algebra, when the latter is completed with respect to the $1$-norm. We focus on the rational case. We start with a theorem on invertibility in the completed algebra, and define a notion of positivity in this setting. We present a series of applications pertaining to Schur analysis, including a counterpart of the Schur algorithm, extension of matrices and rational functions. Other topics considered include Wiener algebra, reproducing kernels Banach modules, and Blaschke factors., Comment: 35 pages

We associate with the Grassmann algebra a topological algebra of distributions, which allows the study of processes analogous to the corresponding free stochastic processes with stationary increments, as well as their derivatives., Comment: We added an outline of the proof of Corollary 4.9, and corrected some misprints

The aim of this paper is to characterize continuous endomorphisms in the space of entire functions of exponential type of order $p>0$. Let $A_p$ denote the space of entire functions of $n$ complex variables $z\in{\mathbb C}^n$ of order $p$ of normal type. We consider an endomorphism $F$ in the space, which is considered to be a DFS-space. We show that there is a unique linear differential operator $P$ of infinite order with coefficients in the space which realizes $F$, that is, $Ff=Pf$ holds for any $f\in A_p$. The coefficients satisfy certain growth conditions and conversely, if a formal differential operator of infinite order with coefficients in $A_p$ satisfy these conditions, then it induces a continuous endomorphism., Comment: 13 pages

In this paper we prove that, for any $n\in \mathbb N$, the ideal generated by $n$ slice regular functions $f_1,\ldots,f_n$ having no common zeros concides with the entire ring of slice regular functions. The proof required the study of the non-commutative syzygies of a vector of regular functions, that manifest a different character when compared with their complex counterparts., Comment: 13 pages

Entire functions in one complex variable are extremely relevant in several areas ranging from the study of convolution equations to special functions. An analog of entire functions in the quaternionic setting can be defined in the slice regular setting, a framework which includes polynomials and power series of the quaternionic variable. In the first chapters of this work we introduce and discuss the algebra and the analysis of slice regular functions. In addition to offering a self-contained introduction to the theory of slice-regular functions, these chapters also contain a few new results (for example we complete the discussion on lower bounds for slice regular functions initiated with the Ehrenpreis-Malgrange, by adding a brand new Cartan-type theorem). The core of the work is Chapter 5, where we study the growth of entire slice regular functions, and we show how such growth is related to the coefficients of the power series expansions that these functions have. It should be noted that the proofs we offer are not simple reconstructions of the holomorphic case. Indeed, the non-commutative setting creates a series of non-trivial problems. Also the counting of the zeros is not trivial because of the presence of spherical zeros which have infinite cardinality. We prove the analog of Jensen and Carath\'eodory theorems in this setting.