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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 introduce fractional powers of quaternionic operators. Their definition is based on the theory of slice hyperholomorphic functions and on the S-resolvent operators of the quaternionic functional calculus. The integral representation formulas of the fractional powers and the quaternionic version of Kato’s formula are based on the notion of S-spectrum of a quaternionic operator. The proofs of several properties of the fractional powers of quaternionic operators rely on the S-resolvent equation. This equation, which is very important and of independent interest, has already been introduced in the case of bounded quaternionic operators, but for the case of unbounded operators some additional considerations have to be taken into account. Moreover, we introduce a new series expansion for the pseudo-resolvent, which is of independent interest and allows to investigate the behavior of the S-resolvents close to the S-spectrum. The paper is addressed to researchers working in operator theory and in complex analysis. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the Schatten class and the Berezin transform of quaternionic operators. The first topic is of great importance in operator theory, but it is also necessary to study the second one, which requires the notion of trace class operators, a particular case of the Schatten class. Regarding the Berezin transform, we give the general definition and properties. Then we concentrate on the setting of weighted Bergman spaces of slice hyperholomorphic functions. Our results are based on the S-spectrum of quaternionic operators, which is the notion of spectrum that appears in the quaternionic version of the spectral theorem and in the quaternionic S-functional calculus. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The S -functional calculus is based on the theory of slice hyperholomorphic functions and it defines functions of n -tuples of not necessarily commuting operators or of quaternionic operators. This calculus relays on the notion of S -spectrum and of S -resolvent operator. Since most of the properties that hold for the Riesz–Dunford functional calculus extend to the S -functional calculus, it can be considered its non commutative version. In this paper we show that the Taylor formula of the Riesz–Dunford functional calculus can be generalized to the S -functional calculus. The proof is not a trivial extension of the classical case because there are several obstructions due to the non commutativity of the setting in which we work that have to be overcome. To prove the Taylor formula we need to introduce a new series expansion of the S -resolvent operators associated to the sum of two n -tuples of operators. This result is a crucial step in the proof of our main results, but it is also of independent interest because it gives a new series expansion for the S -resolvent operators. This paper is addressed to researchers working in operator theory and in hypercomplex analysis. [ABSTRACT FROM AUTHOR]

The theory of slice hyperholomorphic functions, introduced in recent years, has important applications in operator theory. The quaternionic version of this function theory and its Cauchy formula yield to a definition of the quaternionic version of the Riesz-Dunford functional calculus which is based on the notion of S-spectrum. This quaternionic functional calculus allows to define the quaternionic evolution operator which appears in the quaternionic version of quantum mechanics proposed by J. von Neumann and later developed by S. L. Adler. Generation results such as the Hille-Phillips-Yosida theorem have been recently proved. In this paper, we study the perturbation of the generator. The motivation of this study is that, as it happens in the classical case of closed complex linear operators, to verify the generation conditions of the Hille-Phillips-Yosida theorem, in the concrete cases, is often difficult. Thus in this paper we study the generation problem from the perturbation point of view. Precisely, given a quaternionic closed operator T that generates the evolution operator we study under which condition a closed operator P is such that T + P generates the evolution operator . This paper is addressed to people working in different research areas such as hypercomplex analysis and operator theory. [ABSTRACT FROM AUTHOR]

In the recent years the theory of slice hyperholomorphic functions has become an important tool to study two functional calculi for n -tuples of operators and also for its applications to Schur analysis. In particular, using the Cauchy formula for slice hyperholomorphic functions, it is possible to give the Fueter–Sce mapping theorem an integral representation. With this integral representation it has been defined a monogenic functional calculus for n -tuples of bounded commuting operators, the so called F -functional calculus. In this paper we show that it is possible to define this calculus also for n -tuples containing unbounded operators and we obtain an integral representation formula analogous to the one of the Riesz–Dunford functional calculus for unbounded operators acting on a complex Banach space. As we will see, it is not an easy task to provide the correct definition of the F -functional calculus in the unbounded case. This paper is addressed to a double audience, precisely to people with interests in hypercomplex analysis and also to people working in operator theory. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study a functional calculus for bicomplex linear bounded operators. The study is based on the decomposition of bicomplex numbers and of linear operators using the two nonreal idempotents. We show that, due to the presence of zero divisors in the bicomplex numbers, the spectrum of a bounded operator is unbounded. We therefore introduce a different spectrum (called reduced spectrum) which is bounded and turns out to be the right tool to construct the bicomplex holomorphic functional calculus. Finally we provide some properties of the calculus. [ABSTRACT FROM AUTHOR]

The aim of this paper is to provide a characterization of the dual of the ℝ n-module of slice monogenic functions on a class of compact sets in the Euclidean space . Despite the fact that the Cauchy formulas which are essential to such a characterization are based on different kernels, depending on whether one considers right or left slice monogenic functions, we are still able to establish a duality theorem which, since holomorphic functions are a very special case of slice monogenic functions, is the generalization of Köthe's theorem. The duality results are also obtained in the setting of quaternionic valued slice regular functions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we prove a new Representation Formula for slice regular functions, which shows that the value of a slice regular function f at a point can be recovered by the values of f at the points and for any choice of imaginary units . This result allows us to extend the known properties of slice regular functions defined on balls centered on the real axis to a much larger class of domains, called axially symmetric domains. We show, in particular, that axially symmetric domains play, for slice regular functions, the role played by domains of holomorphy for holomorphic functions. [Copyright &y& Elsevier]

Abstract: A proper generalization of complex function theory to higher dimension is Clifford analysis and an analogue of holomorphic functions of several complex variables were recently described as the space of solutions of several Dirac equations. The four-dimensional case has special features and is closely connected to functions of quaternionic variables. In this paper we present an approach to the Dolbeault sequence for several quaternionic variables based on symmetries and representation theory. In particular we prove that the resolution of the Cauchy–Fueter system obtained algebraically, via Gröbner bases techniques, is equivalent to the one obtained by R.J. Baston (J. Geom. Phys. 1992). [Copyright &y& Elsevier]

In this paper we study the time dependent Schrödinger equation with all possible self-adjoint singular interactions located at the origin, which include the δ and δ ′ -potentials as well as boundary conditions of Dirichlet, Neumann, and Robin type as particular cases. We derive an explicit representation of the time dependent Green's function and give a mathematical rigorous meaning to the corresponding integral for holomorphic initial conditions, using Fresnel integrals. Superoscillatory functions appear in the context of weak measurements in quantum mechanics and are naturally treated as holomorphic entire functions. As an application of the Green's function we study the stability and oscillatory properties of the solution of the Schrödinger equation subject to a generalized point interaction when the initial datum is a superoscillatory function. [ABSTRACT FROM AUTHOR]