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We present a general setting where wavelet filters and multiresolution decompositions can be defined, beyond the classical $\mathbf L^2(\mathbb R,dx)$ setting. This is done in a framework of {\em iterated function system} (IFS) measures; these include all cases studied so far, and in particular the Julia set/measure cases. Every IFS has a fixed order, say $N$, and we show that the wavelet filters are indexed by the infinite dimensional group $G$ of functions from $X$ into the unitary group $U_N$. We call $G$ the loop group because of the special case of the unit circle.

In this paper we consider the classical $${\overline{\partial }}$$ ∂ ¯ -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 Hörmander type estimates using suitable powers of the Cauchy-Riemann operator. We also compute particular solutions of the $${\overline{\partial }}$$ ∂ ¯ -problem for specific polyanalytic data such as the Itô complex Hermite polynomials and polyanalytic Fock kernels.

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.

In this paper, we consider a family \(\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}\) of rings of hypercomplex numbers, indexed by the real numbers, which contain both the quaternions and the split-quaternions. We consider natural Hilbert-space representations \(\{(\mathbb{C}^{2},\pi_{t})\}_{t\in\mathbb{R}}\) of the hypercomplex system \(\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}\), and study the realizations \(\pi_{t}(h)\) of hypercomplex numbers \(h \in \mathbb{H}_{t}\), as \((2\times 2)\)-matrices acting on \(\mathbb{C}^{2}\), for an arbitrarily fixed scale \(t\in\mathbb{R}\). Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.

Given a weighted ℓ2 space with weights associated with an entire function, we consider pairs of weighted shift operators, whose commutators are diagonal operators, when considered as operators over a general Fock space. We establish a calculus for the algebra of these commutators and apply it to the general case of Gelfond–Leontiev derivatives. This general class of operators includes many known examples, such as classic fractional derivatives and Dunkl operators. This allows us to establish a general framework, which goes beyond the classic Weyl–Heisenberg algebra. Concrete examples for its application are provided.

We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present a general positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysis and metric geometry and provide a number of examples.

This paper is a continuation of the research of our previous work [5] and considers quaternionic generalized Caratheodory functions and the related family of generalized positive functions. It is addressed to a wide audience which includes researchers in complex and hypercomplex analysis, in the theory of linear systems, but also electric engineers. For this reason it includes some results on generalized Caratheodory functions and their factorization in the classic complex case which might be of independent interest. An important new result is a pseudo-spectral factorization and we also discuss some interpolation problems in the class of quaternionic generalized positive functions.

In this paper, we prove that slice polyanalytic functions on quaternions can be considered as solutions of a power of some special global operator with nonconstant coefficients as it happens in the case of slice hyperholomorphic functions. We investigate also an extension version of the Fueter mapping theorem in this polyanalytic setting. In particular, we show that under axially symmetric conditions it is always possible to construct Fueter regular and poly-Fueter regular functions through slice polyanalytic ones using what we call the poly-Fueter mappings. We study also some integral representations of these results on the quaternionic unit ball., Comment: To appear in Anal. Appl. (Singap.)

In this paper we treat realization results for operator-valued functions which are analytic in the complex sense or slice hyperholomorphic over the quaternions. In the complex setting, we prove a realization theorem for an operator-valued function analytic in a neighborhood of the origin with a coisometric state space operator thus generalizing an analogous result in the unitary case. A main difference with previous works is the use of reproducing kernel Krein spaces. We then prove the counterpart of this result in the quaternionic setting. The present work is the first paper which presents a realization theorem with a state space which is a quaternionic Krein space and may open new avenues of research in hypercomplex analysis.

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.

In this paper we introduce a new approach to the study of filtering theory by allowing the system's parameters to have a random character. We use Hida's white noise space theory to give an alternative characterization and a proper generalization to the Wiener filter over a suitable space of stochastic distributions introduced by Kondratiev. The main idea throughout this paper is to use the nuclearity of this space in order to view the random variables as bounded multiplication operators (with respect to the Wick product) between Hilbert spaces of stochastic distributions. This allows us to use operator theory tools and properties of Wiener algebras over Banach spaces to proceed and characterize the Wiener filter equations under the underlying randomness assumptions.

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 machines (SVMs) classification algorithms. Complex analysis techniques allow us to consider several notions linked to the RBF kernels like the feature space and the feature map, using the so-called Segal-Bargmann transform. We show also 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., Comment: to appear in J. Math. Phys

In this paper we develop a framework to extend the theory of generalized stochastic processes in the Hida white noise space to more general probability spaces which include the grey noise space. To obtain a Wiener-Itô expansion we recast it as a moment problem and calculate the moments explicitly. We further show the importance of a family of topological algebras called strong algebras in this context. Furthermore we show the applicability of our approach to the study of non-Gaussian stochastic processes. published

In this paper we first write a proof of the perceptron convergence algorithm for the complex multivalued neural networks (CMVNNs). Our primary goal is to formulate and prove the perceptron convergence algorithm for the bicomplex multivalued neural networks (BMVNNs) and other important results in the theory of neural networks based on a bicomplex algebra.

In this paper we introduce reproducing kernel Hilbert spaces of polyanalytic functions of infinite order. First we study in details the counterpart of the Fock space and related results in this framework. In this case the kernel function is given by $\displaystyle e^{z\overline{w}+\overline{z}w}$ which can be connected to kernels of polyanalytic Fock spaces of finite order. Segal-Bargmann and Berezin type transforms are also considered in this setting. Then, we study the reproducing kernel Hilbert spaces of complex-valued functions with reproducing kernel $\displaystyle\frac{1}{(1-z\overline{w})(1-\overline{z}w)}$ and $\displaystyle\frac{1}{1-2{\rm Re}\, z\overline{w}}$. The corresponding backward shift operators are introduced and investigated.

Given n rational inner functions Φ = ( Φ 1 , . . . , Φ n ) defining a 0-dimensional (hence finite) complete intersection in the polydisk D n and a Weil polyhedron Δ r Φ relatively compact in D n , we combine together the Cauchy-Weil representation formulas respectively in D n (considered as a Weil polyhedron subordinated to the coordinate functions) and Δ r Φ in order to provide integral representations formulas involving a positive kernel provided Φ satisfies some Schur-Agler type conditions. We then study interpolation problems in H 2 ( D n ) along the finite set Φ − 1 ( { 0 _ } ) . One of the main objectives of this paper is to emphasize the important role played here by Cauchy-Weil representation formula (as in computational polynomial geometry), together with its intimate connection with multivariate residue calculus.

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 $[0,2\pi]$ 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 Carath\'eodory 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

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 study the state space realization of a tensor product of a pair of rational functions. At the expense of “inflating” the dimensions, we recover the classical expressions for realization of a regular product of rational functions. Under an additional assumption that the limit at infinity of a given rational function exists and is equal to identity, we introduce an explicit formula for a tensor factorization of this function.

In this paper we provide realizations of Herglotz functions in the quaternionic case where hyperholomorphic functions are meant in the sense of Fueter. We consider the counterparts of the Arveson space and of the Herglotz–Agler functions by introducing suitable positive kernels. Crucial tools are the so-called Fueter variables and the CK-product. Our realization results also show that positivity implies analyticity.

On harmonic function spaces, we define shift operators using zonal harmonics and partial derivatives, and develop their basic properties. These operators turn out to be multiplications by the coordinate variables followed by projections on harmonic subspaces. This duality gives rise to a new identity for zonal harmonics. We introduce large families of reproducing kernel Hilbert spaces of harmonic functions on the unit ball of R n and investigate the action of the shift operators on them. We prove a dilation result for a commuting row contraction which is also what we call harmonic type. As a consequence, we show that the norm of one of our spaces G ˘ is maximal among those spaces with contractive norms on harmonic polynomials. We then obtain a von Neumann inequality for harmonic polynomials of a commuting harmonic-type row contraction. This yields the maximality of the operator norm of a harmonic polynomial of the shift on G ˘ making this space a natural harmonic counterpart of the Drury-Arveson space.

In [2] we initiated the theory of polyanalytic functions of a quaternionic variable in the slice hyperholomorphic setting.

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.

In this paper, we display a family of Gaussian processes, with explicit formulas and transforms. This is presented with the use of duality tools in such a way that the corresponding path-space measures are mutually singular. We make use of a corresponding family of representations of the canonical commutation relations (CCR) in an infinite number of degrees of freedom. A key feature of our construction is explicit formulas for associated transforms; these are infinite-dimensional analogs of Fourier transforms. Our framework is that of Gaussian Hilbert spaces, reproducing kernel Hilbert spaces and Fock spaces. The latter forms the setting for our CCR representations. We further show, with the use of representation theory and infinite-dimensional analysis, that our pairwise inequivalent probability spaces (for the Gaussian processes) correspond in an explicit manner to pairwise disjoint CCR representations.

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.

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.

We introduce the Schur class of functions, discrete analytic on the integer lattice in the complex plane. As a special case, we derive the explicit form of discrete analytic Blaschke factors and solve the related basic interpolation problem.

In this paper, we present the groundwork for an Itô/Malliavin stochastic cal-culus and Hida's white noise analysis in the context of a supersymmetry withZ3-graded algebras. To this end, we establish a ternary Fock space and the cor-responding strong algebra of stochastic distributions and present its applicationin the study of stochastic processes in this context. published

This work is a revised and corrected version of the authors' joint paper with T. Constantinescu (Trans Am Math Soc 355:813-836, 2003). Some of the theorems from the original paper that are withdrawn are recast as new open problems for indefinite interpolation. Partial results are obtained by other methods, including Kronecker's theorem for Hankel operators.

In this section we discuss the case of close-to-anti-self-adjoint operators, which is the case of interest in quaternionic analysis, and we first prove some properties of the characteristic operator function, also providing some examples. We then move to inverse problems under assumptions on the behavior at infinity, but of course other situations might be considered. We believe that this chapter can be developed in various directions and that it opens new avenues of research.

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.

The class of holomorphic self-mappings of the open unit disk (which are contractions with respect to the Poincare metric) admits a natural extension to the class of holomorphic pseudo-contractions. In this paper, we study various inequalities involving the values of derivatives of holomorphic pseudo-contractions at fixed points (particularly, at the Denjoy–Wolff fixed point).

In the complex setting \(\mathcal H(A, B)\) spaces, that is, reproducing kernel spaces with reproducing kernel of the form ( 1.4) (or their analogues with denominator replaced by \(1-z\overline{w}\)) play an important role in the theory of operator models and related topics. In this section we focus on \(\mathcal H(A, B)\) spaces in the quaternionic context.

In the complex setting case (and in the related time-varying case), spaces \(\mathcal L(\Phi )\) and \(\mathcal H(\Theta )\) (and the related \(\mathcal H(A, B)\) spaces) are closely related by linear fractional transformations; this originates with the works of de Branges, see [47] and of de Branges and Rovnyak [50]. In this section we outline the counterpart of such relations in the quaternionic setting.

This chapter contains some basic knowledge on quaternions, Toeplitz and Hankel matrices and we introduce some useful maps which allow to consider, instead of quaternionic matrices, complex matrices of double size. For more information about quaternionic matrices, the interested reader may consult, e.g., Rodman’s book [97]. We also recall the notions of Pontryagin and Krein spaces which will be useful in the sequel.

We define an extension of operator-valued positive definite functions from the real or complex setting to topological algebras, and describe their associated reproducing kernel spaces. The case of entire functions is of special interest, and we give a precise meaning to some power series expansions of analytic functions that appears in many algebras., Comment: Any comments are welcome!

Canonical differential systems and their connections to operators have a long history; see, e.g., [2, 41, 69, 103, 104]. In this section we consider such systems in the quaternionic setting, in particular in the case of rational spectral data. We foresee that these systems have potentially several applications, for example, to non-linear partial differential equations and inverse scattering.