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Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. The notion of superoscillation is a particular case of that one of supershift. In the recent years, superoscillating functions, that appear for example in weak values in quantum mechanics, have become an interesting and independent field of research in complex analysis and in the theory of infinite order differential operators. The aim of this paper is to study some infinite order differential operators acting on entire functions which naturally arise in the study of superoscillating functions. Such operators are of particular interest because they are associated with the relativistic sum of the velocities and with the Blaschke products. To show that some sequences of functions preserve the superoscillatory behavior it is of crucial importance to prove that their associated infinite order differential operators act continuously on some spaces of entire functions with growth conditions.

We give applications of the theory of superoscillations to various questions, namely extension of positive definite functions, interpolation of polynomials and also of R-functions; we also discuss possible applications to signal theory and prediction theory of stationary stochastic processes. In all cases, we give a constructive procedure, by way of a limiting process, to get the required results.

Superoscillations are band-limited functions that can oscillate faster than their fastest Fourier component. These functions (or sequences) appear in weak values in quantum mechanics and in many fields of science and technology such as optics, signal processing and antenna theory. In this paper, we introduce a new method to generate superoscillatory functions that allows us to construct explicitly a very large class of superoscillatory functions.

In this paper we begin the study of Schur analysis and of de Branges–Rovnyak spaces in the framework of Fueter hyperholomorphic functions. The difference with other approaches is that we consider the class of functions spanned by Appell-like polynomials. This approach is very efficient from various points of view, for example in operator theory, and allows us to make connections with the recently developed theory of slice polyanalytic functions. We tackle a number of problems: we describe a Hardy space, Schur multipliers and related results. We also discuss Blaschke functions, Herglotz multipliers and their associated kernels and Hilbert spaces. Finally, we consider the counterpart of the half-space case, and the corresponding Hardy space, Schur multipliers and Carathéodory multipliers.

In this paper we summarize some known facts on slice topology in the quaternionic case, and we deepen some of them by proving new results and discussing some examples. We then show, following [18], how this setting allows us to generalize slice analysis to the general case of functions with values in a real left alternative algebra, which includes the case of slice monogenic functions with values in Clifford algebra. Moreover, we further extend slice analysis, in one and several variables, to functions with values in a Euclidean space of even dimension. In this framework, we study the domains of slice regularity, we prove some extension properties and the validity of a Taylor expansion for a slice regular function., Comment: This preprint has not undergone peer review or any post-submission improvements or corrections. The Version of Record of this article is published in Adv. Appl. Clifford Algebras, and is available online at https://doi.org/10.1007/s00006-021-01170-3

Infinite-order differential operators appear in different fields of mathematics and physics and in the past decade they turned out to be of fundamental importance in the study of the evolution of superoscillations as initial datum for Schrodinger 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. We will consider two classes of hyperholomorphic functions, both being natural extensions of holomorphic functions of one complex variable. We show that, even though these two notions of hyperholomorphic functions are quite different from each other, in both cases, entire hyperholomorphic functions with exponential bounds play a crucial role in the continuity of infinite-order differential operators acting on these two classes of functions. This is particularly remarkable since the exponential function is not in the kernel of the Dirac operator, but it plays an important role in the theory of entire monogenic functions with growth conditions.

In the literature on slice analysis in the hypercomplex setting, there are two main approaches to define slice regular functions in one variable: one consists in requiring that the restriction to any complex plane is holomorphic (with the same complex structure of the complex plane), the second one makes use of stem and slice functions. So far, in the setting of several hypercomplex variables, only the second approach has been considered, i.e. the one based on stem functions. In this paper, we use instead the first definition on the so-called $n$-dimensional quadratic cone of octonions. These two approaches yield the same class of slice regular functions on axially symmetric slice-domains, however, they are different on other types of domains. We call this new class of functions weak slice regular. We show that there exist weak slice regular functions which are not slice regular in the second approach. Moreover, we study various properties of these functions, including a Taylor expansion., 21 pages

In this paper, we study the evolutions of Aharonov–Berry superoscillations under the radial harmonic oscillator potential. For this model, we know the Green function and, taking advantage of it, we use a method recently developed for the step potential to show how superoscillations evolve in time. Also in this case, the time evolution is studied using the notion of super-shift of functions.

In this paper we continue our study on the density of the set of quaternionic polynomials in function spaces of slice regular functions on the unit ball by considering the case of the Bloch and Besov spaces of the first and of the second kind. Among the results we prove, we show some constructive methods based on the Taylor expansion and on the convolution polynomials. We also provide quantitative estimates in terms of higher order moduli of smoothness and of the best approximation quantity. As a byproduct, we obtain two new results for complex Bloch and Besov spaces.

We prove a new theorem on the continuity of convolution operators with variable coefficients, and we use it to deduce that the limit of a superoscillating sequence maintains the superoscillatory behaviour for all values of time, when evolved according to Schrodinger equations with time-dependent potentials.

In this paper, we describe how the study of longevity of superoscillating functions has developed over the last several years. Specifically, we show how the evolution of superoscillations for the Schrodinger equation for the free particle naturally lead the authors to the construction of a larger class of superoscillating functions. This basic idea, originally presented in Aharonov et al. (J Math Pures Appl 99:165–173, 2013), was subsequently generalized when different differential and convolution operators replace the Laplacian in the Schrodinger equation, and it eventually led to larger classes of superoscillating functions. In this paper, we outline this process, and we show how to extend these ideas to the case of several variables, and we summarize some recent applications of superoscillations to problems of approximation in the Schwartz spaces of functions.

Using a left multiplication defined on a right quaternionic Hilbert space, linear self-adjoint momentum operators on a right quaternionic Hilbert space are defined in complete analogy with their complex counterpart. With the aid of the so-obtained position and momentum operators, we study the Heisenberg uncertainty principle on the whole set of quaternions and on a quaternionic slice, namely on a copy of the complex plane inside the quaternions. For the quaternionic harmonic oscillator, the uncertainty relation is shown to saturate on a neighborhood of the origin in the case we consider the whole set of quaternions, while it is saturated on the whole slice in the case we take the slice-wise approach. In analogy with the complex Weyl-Heisenberg Lie algebra, Lie algebraic structures are developed for the quaternionic case. Finally, we introduce a quaternionic displacement operator which is square integrable, irreducible and unitary, and we study its properties., Comment: to appear in Annals of Physics

In this paper we introduce and study the fractional Hardy spaces of the half space and of the unit ball in the quaternionic setting. In particular, we discuss their properties of invariance and of factorization in terms of functions in the Hardy space of the half space in the first case, and in terms of a suitable reproducing kernel Hilbert space in the case of the unit ball.

We study a characterization of slice Carleson measures and of Carleson measures for both the Hardy spaces Hp(B) and the Bergman spaces Ap(B) of the quaternionic unit ball B. In the case of Bergman spaces, the characterization is done in terms of the axially symmetric completion of a pseudohyperbolic disc in a complex plane. We also show that a characterization in terms of pseudo-hyperbolic balls is not possible.

We show that de Branges–Rovnyak spaces include as special cases a number of spaces, such as the Hardy space, the Fock space, the Hardy–Sobolev space and the Dirichlet space. We present a general framework in which all these spaces can be obtained by specializing a sequence that appears in the construction. We show how to exploit this approach to solve interpolation problems in the Fock space.

The slice Dirac operator over octonions is a slice counterpart of the Dirac operator over quaternions. It involves a new theory of stem functions, which is the extension from the commutative O(1) case to the non-commutative O(3) case. For functions in the kernel of the slice Dirac operator over octonions, we establish the representation formula, the Cauchy integral formula (and, more in general, the Cauchy-Pompeiu formula), and the Taylor as well as the Laurent series expansion formulas.

Superoscillations are band-limited functions with the counterintuitive property that they can vary arbitrarily faster than their fastest Fourier component, over arbitrarily long intervals. Modern studies originated in quantum theory, but there were anticipations in radar and optics. The mathematical understanding - still being explored - recognises that functions are extremely small where they superoscillate; this has implications for information theory. Applications to optical vortices, sub-wavelength microscopy and related areas of nanoscience are now moving from the theoretical and the demonstrative to the practical. This Roadmap surveys all these areas, providing background, current research, and anticipating future developments.

In this paper, we consider holomorphic functions on the m-dimensional Lie ball LB(0, 1) which admit a square integrable extension on the Lie sphere. We then define orthogonal projections of this set onto suitable subsets of functions defined in lower- dimensional spaces to obtain several Radon-type transforms. For all these transforms we provide the kernel and an integral representation, besides other properties. In particular, we introduce and study a generalization to the Lie ball of the Szegő–Radon transform introduced in Colombo et al. (Adv Appl Math 74:1–22, 2016), and various types of Hua–Radon transforms.

Purpose of this note is to give an overview on superoscillating sequences and some of their properties. We discuss their persistence in time under Schrodinger equation, we propose various classes of superoscillating functions and we also briefly mention how they can be used to approximate some generalized functions.Purpose of this note is to give an overview on superoscillating sequences and some of their properties. We discuss their persistence in time under Schrodinger equation, we propose various classes of superoscillating functions and we also briefly mention how they can be used to approximate some generalized functions.

The theory of quaternionic operators has applications in several different fields, such as quantum mechanics, fractional evolution problems, and quaternionic Schur analysis, just to name a few. The main difference between complex and quaternionic operator theory is based on the definition of a spectrum. In fact, in quaternionic operator theory the classical notion of a resolvent operator and the one of a spectrum need to be replaced by the two S S -resolvent operators and the S S -spectrum. This is a consequence of the noncommutativity of the quaternionic setting. Indeed, the S S -spectrum of a quaternionic linear operator T T is given by the noninvertibility of a second order operator. This presents new challenges which make our approach to perturbation theory of quaternionic operators different from the classical case. In this paper we study the problem of perturbation of a quaternionic normal operator in a Hilbert space by making use of the concepts of S S -spectrum and of slice hyperholomorphicity of the S S -resolvent operators. For this new setting we prove results on the perturbation of quaternionic normal operators by operators belonging to a Schatten class and give conditions which guarantee the existence of a nontrivial hyperinvariant subspace of a quaternionic linear operator.

The main purpose of this paper is to prove some density results of polynomials in Fock spaces of slice regular functions. The spaces can be of two different kinds since they are equipped with different inner products and contain different functions. We treat both the cases, providing several results, some of them based on constructive methods which make use of the Taylor expansion and of the convolution polynomials. We also prove quantitative estimates in terms of higher order moduli of smoothness and in terms of the best approximation quantities., To appear in Complex Anal. Oper. Theory

The maximum selection principle allows to give expansions, in an adaptive way, of functions in the Hardy space \(\mathbf H_2\) of the disk in terms of Blaschke products. The expansion is specific to the given function. Blaschke factors and products have counterparts in the unit ball of \(\mathbb C^N\), and this fact allows us to extend in the present paper the maximum selection principle to the case of functions in the Drury–Arveson space of functions analytic in the unit ball of \(\mathbb C^N\). This will give rise to an algorithm which is a variation in this higher dimensional case of the greedy algorithm. We also introduce infinite Blaschke products in this setting and study their convergence.

The well known Fueter theorem allows to construct quaternionic regular functions or monogenic functions with values in a Clifford algebra defined on open sets of Euclidean space R n + 1 , starting from a holomorphic function in one complex variable or, more in general, from a slice hyperholomorphic function. Recently, the inversion of this theorem has been obtained for odd values of the dimension n . The present work extends the result to all dimensions n by using the Fourier multiplier method. More precisely, we show that for any axially monogenic function f defined in a suitable open set in R n + 1 , where n is a positive integer, we can find a slice hyperholomorphic function f → such that f = Δ ( n − 1 ) / 2 f → . Both the even and the odd dimensions are treated with the same, viz., the Fourier multiplier, method. For the odd dimensional cases the result obtained by the Fourier multiplier method coincides with the existing result obtained through the pointwise differential method.

We consider the Cauchy problem for the Schrodinger equation with step potential with jump V 0 at the origin and whose initial datum is a superoscillatory function F n that is traveling from − ∞ in the direction of the barrier V 0 . We assume that the energies E n , k of all the traveling waves e i λ n , k x with | λ n , k | ≤ 1 , are strictly less then the potential V 0 ( n and k are suitable natural numbers), so that all the waves e i λ n , k x are subjected to the tunnel effect for x ≥ 0 . We prove that in the limit, as n tends to infinity, from the infinitely many tunneling waves e i λ n , k x emerges a wave that passes through the potential barrier. In other words, we prove that superoscillations overcome the potential barrier of the step potential even though none of its constituent waves can.

We define and study the counterpart of the Wiener algebra in the quaternionic setting, both for the discrete and continuous case. We prove a Wiener-L\'evy type theorem and a factorization theorem. We give applications to Toeplitz and Wiener-Hopf operators.

Feynman stated that the double-slit experiment "…has in it the heart of quantum mechanics. In reality, it contains the only mystery" and that "nobody can give you a deeper explanation of this phenomenon than I have given; that is, a description of it" [Feynman R, Leighton R, Sands M (1965) The Feynman Lectures on Physics]. We rise to the challenge with an alternative to the wave function-centered interpretations: instead of a quantum wave passing through both slits, we have a localized particle with nonlocal interactions with the other slit. Key to this explanation is dynamical nonlocality, which naturally appears in the Heisenberg picture as nonlocal equations of motion. This insight led us to develop an approach to quantum mechanics which relies on pre- and postselection, weak measurements, deterministic, and modular variables. We consider those properties of a single particle that are deterministic to be primal. The Heisenberg picture allows us to specify the most complete enumeration of such deterministic properties in contrast to the Schrodinger wave function, which remains an ensemble property. We exercise this approach by analyzing a version of the double-slit experiment augmented with postselection, showing that only it and not the wave function approach can be accommodated within a time-symmetric interpretation, where interference appears even when the particle is localized. Although the Heisenberg and Schrodinger pictures are equivalent formulations, nevertheless, the framework presented here has led to insights, intuitions, and experiments that were missed from the old perspective.

In this paper we define the quaternionic Cayley transformation of a densely defined, symmetric, quaternionic right linear operator and formulate a general theory of defect number in a right quaternionic Hilbert space. This study investigates the relation between the defect number and S-spectrum, and the properties of the Cayley transform in the quaternionic setting., Comment: arXiv admin note: text overlap with arXiv:1512.08662"

In this paper we study two-sided (left and right) axially symmetric solutions of a generalized Cauchy-Riemann operator. We present three methods to obtain special solutions: via the Cauchy-Kowalevski extension theorem, via plane wave integrals and Funk-Hecke's formula and via primitivation. Each of these methods is effective enough to generate all the polynomial solutions., 17 pages, accepted for publication in Moscow Mathematical Journal

In this paper, we start the study of stochastic processes over the skew field of quaternions. We discuss the relation between positive definite functions and the covariance of centered Gaussian processes and the construction of stochastic processes and their derivatives. The use of perfect spaces and strong algebras and the notion of Fock space are crucial in this framework.

In this paper we show that for any slice regular function f in a quaternionic ellipsoid containing their real interval of orthogonality, the Chebyshev and Legendre expansions converge uniformly to f in all compact subsets in the interior of the ellipsoid, with the order of a geometric series.

In this paper we consider functions in the Hardy space $\mathbf{H}_2^{p\times q}$ defined in the unit disc of matrix-valued. We show that it is possible, as in the scalar case, to decompose those functions as linear combinations of suitably modified matrix-valued Blaschke product, in an adaptive way. The procedure is based on a generalization to the matrix-valued case of the maximum selection principle which involves not only selections of suitable points in the unit disc but also suitable orthogonal projections. We show that the maximum selection principle gives rise to a convergent algorithm. Finally, we discuss the case of real-valued signals.

In this paper we obtain several extensions to the quaternionic setting of someresultsconcerningtheapproximation bypolynomials of functionscontinuous onacompact set and holomorphic in its interior. The results include approximationon compact starlike sets and compact axially symmetric sets. The cases of someconcrete particular sets are described in details, including quantitative estimatestoo. AMS 2010 Mathematics Subject Classification: Primary 30G35; Secondary 30E10,41A25.Keywords and phrases: Mergelyan’s theorem, quaternions, Riemann mapping, axiallysymmetric sets, approximation by polynomials, convolution operators, Cassini pseudo-metric, Cassini cell, order of approximation, slice regular functions. 1 Introduction It is well-known the fact that the Mergelyan’s approximation theorem is the ultimatedevelopment and generalization of the Weierstrass approximation theorem and Runge’stheorem in the complex plane. It can be stated as follows (see [13]):Theorem 1.1. Let K be a compact subset of the complex plane C such that C\ K isconnected. Then, every continuous function on K, f : K → C, which is holomorphic inthe interior of K, can be approximated uniformly on K by polynomials.Notice that all the known proofs of this result, based on the methods in complex analysisuse the Riemann mapping theorem.1

In this paper, we prove the quaternionic version of the result of Walsh stating that the difference between the partial sums of the Taylor expansion of an analytic function and its interpolation po...

The purpose of this paper is to derive various representations of the Dirac delta distribution, including a Bony-type twisted Radon decomposition, from boundary values of monogenic functions. This leads to a new and simpler approach based on the properties of the analogue of the Cauchy kernel in the context of monogenic functions.

We discuss the Fourier-Borel transform for the dual of spaces of monogenic functions. This transform may be seen as a restriction of the classical Fourier-Borel transform for holomorphic functionals, and it transforms spaces of monogenic functionals into quotients of spaces of entire holomorphic functions of exponential type. We prove that, for the Lie ball, these quotient spaces are isomorphic to spaces of monogenic functions of exponential type.

In this chapter we introduce slice hyperholomorphic functions with values in a quaternionic Banach space. As in the complex case, there are two equivalent notions, namely weak and strong slice hyperholomorphicity. In order to properly define a multiplication between slice hyperholomorphic functions, we give a third characterization in terms of the Cauchy–Riemann system. Operator-valued functions can be obtained by using the so-called S-functional calculus. This calculus is associated with the notions of S-spectrum and S-resolvent, which are introduced and studied. We also present some hyperholomorphic extension results and, finally, we study the Hilbert-space-valued quaternionic Hardy space of the ball and its backward-shift invariant subspaces.

In this paper we extend the H ∞ functional calculus to quaternionic operators and to n-tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated functional calculus, called S-functional calculus. The S-functional calculus has two versions: one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz–Dunford functional calculus based on slice hyperholomorphicity, because it shares with it the most important properties. The S-functional calculus is based on the notion of S-spectrum which, in the case of quaternionic normal operators on a Hilbert space, is also the notion of spectrum that appears in the quaternionic spectral theorem. The main purpose of this paper is to construct the H ∞ functional calculus based on the notion of S-spectrum for both quaternionic operators and for n-tuples of noncommuting operators. We remark that the H ∞ functional calculus for ( n + 1 ) -tuples of operators applies, in particular, to the Dirac operator.

The quaternionic spectral theorem has already been considered in the literature, see e.g. [22], [31], [32], however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see [21], it is not specified which notion of spectrum underlies the theorem. In this paper we prove the quaternionic spectral theorem for unitary operators using the $S$-spectrum. In the case of quaternionic matrices, the $S$-spectrum coincides with the right-spectrum and so our result recovers the well known theorem for matrices. The notion of $S$-spectrum is relatively new, see [17], and has been used for quaternionic linear operators, as well as for $n$-tuples of not necessarily commuting operators, to define and study a noncommutative versions of the Riesz-Dunford functional calculus. The main tools to prove the spectral theorem for unitary operators are the quaternionic version of Herglotz's theorem, which relies on the new notion of $q$-positive measure, and quaternionic spectral measures, which are related to the quaternionic Riesz projectors defined by means of the $S$-resolvent operator and the $S$-spectrum. The results in this paper restore the analogy with the complex case in which the classical notion of spectrum appears in the Riesz-Dunford functional calculus as well as in the spectral theorem.

In classical Schur analysis, operator-valued functions appear naturally in a number of different settings, of which we mention: 1. The characteristic operator function, when the imaginary part of the operator is not of finite rank (but possibly trace class, or of Hilbert–Schmidt class). See for instance [110, 111, 242]. 2. The time-varying case, when the complex numbers are replaced by diagonal operators. See [40, 161, 162, 163]. 3. Interpolation in the Hardy space of the polydisk. One can reduce the problem to an operator-valued problem in one variable. The values are then assumed to be Hilbert-Schmidt operators. See [23, 24].

In the first section of this chapter we give a brief survey of the theory of slice hyperholomorphic functions, and we provide the basic results needed in the sequel. The theory is much more developed and we refer the reader to the Introduction to Part II of the present work for references. Then, the Sections from 6.2 to 6.5 focus on the functions slice hyperholomorphic in the unit ball. We define in particular the Hardy space of the ball, Schur functions, and Blaschke factors. We also study linear fractional transformations and introduce the Wiener algebra of the ball. The last two sections, 6.6 and 6.7, are devoted to the case of functions slice hyperholomorphic in the half-space of quaternions with real positive part. We discuss in particular the Hardy space and Blaschke factors.

The notion of superoscillating functions, or more properly of superoscillatory sequences, is a byproduct of Aharonov’s theory of weak measurements and weak values in quantum mechanics. Recently, many mathematicians and physicists have begun to pay attention to the mathematical significance of such objects, and have been able to begin a theory of superoscillatory behavior. Not surprisingly, this theory is based on some classical results in Fourier analysis, and it displays interesting connections with the theory of convolution equations. In this paper we will put these connections in a larger context, and show how to use this context to generate a large class of superoscillating sequences. As a concrete example we discuss the Cauchy problem with superoscillatory datum for the harmonic oscillator. Finally, we show how this theory can be generalized to the case of several variables.

In this chapter we discuss infinite products of quaternions and of quaternionic functions, also in relation with slice regularity. In particular, we introduce the notion of Blaschke factors, Blaschke products and we provide a version of the Weierstrass theorem.

In this chapter we discuss the finite-dimensional aspects of quaternionic analysis which are needed in the sequel. In the first section we survey the main properties of quaternions. Then we consider quaternionic polynomials and we discuss their zeros. In the third section we discuss quaternionic matrices and basic definitions such as adjoint, transpose, and inverse.We introduce the map χ which to any quaternionic square matrix associates a complex matrix of double size. In particular, this map is also defined for quaternions. We discuss the eigenvalue problem and show that it is associated with the notion of left and right spectrum and with the so-called Sspectrum. Finally, we present the Jordan decomposition of a matrix. In the fourth and last section we consider some matrix equations which appear in the sequel.

The tools developed in the previous chapters allow us to define and study in the operator-valued case the various families of functions appearing in classical Schur analysis. In this section we obtain realization formulas for these functions. These formulas in turn have important consequences, such as the existence of slice hyperholomorphic extensions and results in function theory such as Bohr’s inequality. Recall that all two-sided quaternionic vector spaces are assumed to satisfy condition (5.4). An important tool in this chapter is Shmulyan’s theorem on densely defined contractive relations between Pontryagin spaces with the same index, see Theorem 5.7.8, and this forces us to take two-sided quaternionic Pontryagin spaces with the same index for coefficient spaces, and not Krein spaces. The rational case, studied in the following chapter, corresponds to the setting where both the coefficient spaces and the reproducing kernel Pontryagin spaces associated to the various functions are finite-dimensional.

In this paper we study the convergence of some sequences of operators associated to the Aharonov and Berry’s superoscillating functions. The main tool to define the sequences of operators is the spectral theorem. In particular we discuss the case of sequences of unbounded self-adjoint operators on a Hilbert space. We apply our results to the case where T is the self-adjoint extension of the momentum operator with unbounded spectrum.

In this Note we present an algebraic analysis of the system of differential equations described by the Hermitian Dirac operators, which are two linear first order operators invariant with respect to the action of the unitary group, both in the case of one and several variables. To cite this article: A. Damiano et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).

This chapter is a survey on recent developments in quaternionic Schur analysis. The first part is based on functions which are slice hyperholomorphic in the unit ball of the quaternions, and have modulus bounded by 1. These functions, which by analogy to the complex case are called Schur multipliers, are shown to be (as in the complex case) the source of a wide range of problems of general interest. They also suggest new problems in quaternionic operator theory, especially in the setting of indefinite inner product spaces. This chapter gives an overview on rational functions and their realizations, on the Hardy space of the unit ball, on the half-space of quaternions with positive real part, and on Schur multipliers, also discussing related results. For the purpose of comparison this chapter presents also another approach to Schur analysis in the quaternionic setting, in the framework of Fueter series. To ease the presentation most of the chapter is written for the scalar case, but the reader should be aware that the appropriate setting is often that of vector-valued functions.

Weak measurement and weak values have a very deep meaning in quantum mechanics, and new phenomena associated to them have recently been observed experimentally. These measurements give rise to the notion of superoscillating sequences of functions. In the recent years the authors have started an intensive study of this topic from the mathematical point of view. In this paper we use a generalization of the Schrodinger equation in which the spatial derivative is replaced by a suitable convolution operator to prove the existence of a large class of superoscillating sequences. The method we use also allows us to construct such sequences explicitly.

A boundary Nevanlinna–Pick interpolation problem is posed and solved in the quaternionic setting. Given nonnegative real numbers $${\kappa_1,\ldots,\kappa_N}$$ , quaternions p 1,…,p N all of modulus 1, so that the 2-spheres determined by each point do not intersect and p u ≠ 1 for u = 1,…, N, and quaternions s 1,…, s N , we wish to find a slice hyperholomorphic Schur function s so that $$\lim_{\substack{r\rightarrow 1\\ r\in(0,1)}} s(r p_u) = s_u\quad \hbox{for}\ u=1,\ldots, N,$$ and $$\lim_{\substack{r\rightarrow 1\\ r \in(0,1)}}\frac{1-s(rp_u)\overline{s_u}}{1-r}\le\kappa_u,\quad\hbox{for}\ u=1,\ldots, N.$$ Our arguments rely on the theory of slice hyperholomorphic functions and reproducing kernel Hilbert spaces.