Daniel Alpay

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• Doctor of Philosophy
• Professor (Full) at Chapman University

We are studying the fundamental tools for a quantum calculus based on the Tsallis $q$-exponential In particular we are looking at $q$-Fock spaces, structural identities, as well as rational functions in this context.

Starting from a fixed measure space $$(X, {\mathcal {F}}, \mu )$$ ( X , F , μ ) , with $$\mu $$ μ a positive sigma-finite measure defined on the sigma-algebra $${\mathcal {F}}$$ F , we continue here our study of a generalization $$W^{(\mu )}$$ W ( μ ) of Brownian motion, and introduce a corresponding white-noise process. In detail, the generalized...

With the use of Hida's white noise space theory space theory and spaces of stochastic distributions, we present a detailed analytic continuation theory for classes of Gaussian processes, with focus here on Brownian motion. For the latter, we prove and make use a priori bounds, in the complex plane, for the Hermite functions; as well as a new approa...

In this paper, we study the ranges of the Schwartz space $\mathcal {S}$ and its dual $\mathcal {S}'$ (space of tempered distributions) under the Bargmann transform. The characterization of these two ranges leads to interesting reproducing kernel Hilbert spaces whose reproducing kernels can be expressed, respectively, in terms of the Touchard polyno...

Bergman spaces have been studied in the hypercomplex case, in the context of monogenic functions with values in a Clifford algebra, in particular quaternions. Here we consider Bergman spaces in the slice hyperholomorphic setting. They can be defined in two different ways: one global, one via the complex slices, thus giving the so-called Bergman spa...

This chapter contains a study of the Wiener algebra to the quaternionic setting in the discrete and continuous case, also including an analogue of the Wiener-Lévy theorem and of the Wiener-Hopf factorization.

In this chapter we extend some results in de Branges’ book (Hilbert Spaces of Entire Functions. Prentice-Hall Inc., Englewood Cliffs, 1968) to the slice hyperholomorphic setting over the quaternions.

In the literature there are several possible function theories of several quaternionic or Clifford variables. The first theory that was developed is the theory of several quaternionic variable based on Fueter regularity and in the same way also the theory of several Clifford variables was considered. With the development of the function theory of s...

As a matter of fact, the most widespread impact of the theory of slice hyperholomorphic functions is provided by the applications to operator theory.

In this chapter we study Hardy spaces in the slice hyperholomorphic quaternionic case.

Since the main objects of study of the book are quaternionic Hilbert spaces of functions it is quite natural that the notions of positive definite function and of reproducing kernel Hilbert spaces percolate throughout the work. We have defined the latter spaces in Sect. 2.2 and here we provide a deeper study.

Fock spaces are a very important tool in complex and in quaternionic quantum mechanics, and in this chapter we treat the quaternionic slice hyperholomorphic case.

In this chapter, we review some results on quaternions, quaternionic matrices and slice hyperholomorphic functions.

In this last chapter we summarise the crucial facts regarding hyperholomorphic functions and the associated spectral theories. We summarize how the two main hyperholomorphic function theories induce, via their Cauchy formula, the spectral theory on the S-spectrum and the spectral theory on the monogenic spectrum. This stream of ideas originated the...

The present chapter deals with various aspects of quaternionic functional analysis and in particular supplements our previous works (Alpay et al., Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes. Operator Theory: Advances and Applications, vol. 244, pp. 33–65. Birkhäuser, Basel, 2015, §3; Slice Hyperholomorphic Schur...

The Hilbert spaces studied in this chapter have been considered earlier in the framework of monogenic (or Fueter regular) functions. In this chapter, we shall introduce and study the Bloch, Besov and Dirichlet spaces, in the slice hyperholomorphic case, mainly working on the unit ball \(\mathbb B\)

The theory of polyanalytic functions is an interesting topic in complex analysis studied since over a century. It extends the concept of holomorphic functions to nullsolutions of higher order powers of the Cauchy-Riemann operator. The definition can also be extended to the hypercomplex case, in particular to the quaternionic case, by using the idea...

We develop the theory of minimal realizations and factorizations of rational functions where the coefficient space is a ring of the type introduced in our previous work, the scaled quaternions, which includes as special cases the quaternions and the split quaternions. The methods involved are not a direct generalization of the complex or quaternion...

The purpose of this paper is to develop a new theory of three non-commuting quaternionic variables and its related Schur analysis theory for a modified version of the quaternionic global operator.

In this paper we describe the rise of global operators in the scaled quaternionic case, an important extension from the quaternionic case to the family of scaled hypercomplex numbers Ht,t∈R∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \us...

In this paper we introduce the concept of matrix-valued q-rational functions. In comparison to the classical case, we give different characterizations with principal emphasis on realizations and discuss algebraic manipulations. We also study the concept of Schur multipliers and complete Nevanlinna-Pick kernels in the context of q-deformed reproduc...

In this paper we introduce the concept of matrix-valued q-rational functions. In comparison to the classical case, we give different characterizations with principal emphasis on realizations and discuss algebraic manipulations. We also study the concept of Schur multipliers and complete Nevanlinna–Pick kernels in the context of q-deformed reproduci...

In this paper we extend the concept of tensor product to the bicomplex case and use it to prove the bicomplex counterpart of the classical Choi theorem in the theory of complex matrices and operators. The concept of hyperbolic tensor product is also discussed, and we link these results to the theory of quantum channels in the bicomplex and hyperbol...

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 sh...

The present chapter deals essentially with positive semi-definite matrices. It is the longest chapter of the book, but the reader should be aware that barely the surface of the topic has been touched. Positive semi-definite matrices play a key role in numerous domains, of which we mention in particular machine learning in its various facets. See Re...

Algebraic structures, such as groups, rings, fields, and ideals, play an important role in different parts of the present book and are discussed in this chapter. We begin with a related section on sets and functions between sets.

In probability theory, events are subsets of an underlying set \(\Omega \). Denote by \(\mathcal {P}(\Omega )\) the set of all subsets of \(\Omega \), together with the empty set \(\emptyset \). When \(\Omega \) is not finite, one will not in general take all subsets of \(\Omega \) as possible events but restrict to a family of sets \(\mathcal {C}\...

Functional analysis deals with properties of spaces of functions (or of sequences), rather than of individual objects. To study a specific function is doing real (or complex, or more generally hypercomplex) analysis. To study a vector space of all functions with a given common property pertains to functional analysis and uses the fact that this spa...

The main character of this chapter is the entropy function (1.2.9), introduced via the source partition theorem. Most of the chapter is devoted to the case of random variables defined on a finite probability space (but see Remark 8.1.10 for an interpretation of the result in an infinite probability space).

Among the tools from real analysis to be mastered by a student of machine learning, statistical physics, and thermodynamics, we mention in particular convexity, the main properties of partial derivatives and differentials, and the theory of Lagrange multipliers. Liouville’s theorem on systems of ordinary differential equations plays also an importa...

We first present a number of elementary (and less elementary) exercises on complex numbers. We also discuss briefly the skew field of quaternions and other systems of numbers. We tried to limit analytic methods to a minimum in this chapter. Still, we use the existence of eigenvalues for a matrix, i.e., the fundamental theorem of algebra.

Using q-calculus we study a family of reproducing kernel Hilbert spaces which interpolate between the Hardy space and the Fock space. We give characterizations of these spaces in terms of classical operators such as integration and backward-shift operators, and their q-calculus counterparts. Furthermore, these new spaces allow us to study intertwin...

In this paper we start the study of Schur analysis for Cauchy–Fueter regular quaternionic-valued functions, i.e. null solutions of the Cauchy–Fueter operator in R4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlengt...

In this paper, we study the regularity of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}$$\end{document}-differentiable functions on open connected subset...

In this paper, we consider, and make precise, a certain extension of the Radon-Nikodym derivative operator, to functions which are additive, but not necessarily sigma-additive, on a subset of a given sigma-algebra. We give applications to probability theory; in particular, to the study of \(\mu\)-Brownian motion, to stochastic calculus via generali...

Our focus is the operators of multivariable stochastic calculus, i.e., systems of transfer operators, covariance operators, conditional expectations, stochastic integrals, and the counterpart infinite-dimensional stochastic derivatives. In this paper, we present a new operator algebraic framework which serves to unify the analysis and the interrela...

The main purposes of this paper are (i) to enlarge scaled hypercomplex structures to operator-valued cases, where the operators are taken from a C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin...

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 Hermi...

In this paper, we study the regularity of $\mathbb{R}$-differentiable functions on open connected subsets of the scaled hypercomplex numbers $\left\{ \mathbb{H}_{t}\right\} _{t\in\mathbb{R}}$ by studying the kernels of suitable differential operators $\left\{ \nabla_{t}\right\} _{t\in\mathbb{R}}$, up to scales in the real field $\mathbb{R}$.

Using $q$-calculus we study a family of reproducing kernel Hilbert spaces which interpolate between the Hardy space and the Fock space. We give characterizations of these spaces in terms of classical operators such as integration and backward-shift, and their q-calculus counterparts. We introduce an apparently new family of numbers, close to, but d...

Using results from the theory of operators on a Hilbert space, we prove approximation results for matrix‐valued holomorphic functions on the unit disc and the unit bidisc. The essential tools are the theory of unitary dilation of a contraction and the realization formula for functions in the unit ball of H∞$H^\infty$. We first prove a generalizatio...

Similarity metric which is not positive definite, and present a general theorem which provides a large family of similarity metrics which are positive definite.

In this paper, we extend notions of complex C−R$\mathbb {C} - \mathbb {R}$‐calculus to the bicomplex setting and compare the bicomplex polyanalytic function theory to the classical complex case. Applications of this theory include two bicomplex least mean square algorithms, which extend classical real and complex least mean square algorithms.

In this paper we extend notions of complex C-R-calculus and complex Hermite polynomials to the bicomplex setting and compare the bicomplex polyanalytic function theory to the classical complex case.

We study betweenness of membership functions in the fuzzy setting and for membership functions taking values in the set of hyperbolic numbers.

In this paper we present parallel theories on constructing Wiener algebras in the bicomplex setting. With the appropriate symmetry condition, the bicomplex matrix valued case can be seen as a complex valued case and, in this matrix valued case, we make the necessary connection between bicomplex analysis and complex analysis with symmetry. We also w...

In this paper we extend the concept of tensor product to the bicomplex case and use it to prove the bicomplex counterpart of the classical Choi theorem in the theory of complex matrices and operators. The concept of hyperbolic tensor product is also discussed, and we link these results to the theory of quantum channels in the bicomplex and hyperbol...

In this paper, we consider a family of the hypercomplex rings \({\mathscr {H}}=\left\{ {\mathbb {H}}_{t}\right\} _{t\in {\mathbb {R}}}\) scaled by \({\mathbb {R}}\), and the dynamical system of \({\mathbb {R}}\) acting on \({\mathscr {H}}\) via a certain action \(\theta \) of \({\mathbb {R}}\). i.e., we study an analysis on dynamical system induced...

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}\}...

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 Hermi...

Given a weighted ℓ ² 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. Th...

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 Cau...

In this paper, we consider natural Hilbert-space representations $\left\{ \left(\mathbb{C}^{2},\pi_{t}\right)\right\} _{t\in\mathbb{R}}$ of the hypercomplex system $\left\{ \mathbb{H}_{t}\right\} _{t\in\mathbb{R}}$, and study the realizations $\pi_{t}\left(h\right)$ of hypercomplex numbers $h\in\mathbb{H}_{t}$, as $\left(2\times2\right)$-matrices a...

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 machine classification algorithms. Complex analysis techniques allow us to consider several notions linked...

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 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...

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 notion...

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 sp...

The purpose of this paper is to develop a new theory of three non-commuting quaternionic variables and its related Schur analysis theory for a modified version of the quaternionic global operator.

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...

We study reflection positivity in the context of Hilbert space, and Krein-space theory. Our context is that of triple systems (U,J,M+) assumed to satisfy the axioms for reflection positivity (also known as Osterwalder-Schrader positivity). Applications include quantum field theory and the theory of unitary representations of Lie groups. Our analysi...

We investigate the scale-shift operator for discrete-time signals via the action of the hyperbolic Blaschke group. Practical implementation issues are discussed and given for any arbitrary scale, in the framework of very classical discrete-time linear filtering. Our group theoretical standpoint leads to a purely harmonic analysis definition of the...

We study and introduce new gradient operators in the complex and bicomplex settings, inspired from the well-known Least Mean Square (LMS) algorithm invented in 1960 by Widrow and Hoff for Adaptive Linear Neuron (ADALINE). These gradient operators will be used to formulate new learning rules for the Bicomplex Least Mean Square (BLMS) algorithms and...

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 o...

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 ord...

Given a weighted $\ell^2$ space with weights associated to 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....

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 anal...

In this paper we present parallel theories on constructing Wiener algebras in the bicomplex setting. With the appropriate symmetry condition, the bicomplex matrix valued case can be seen as a complex valued case and, in this matrix valued case, we make the necessary connection between classical bicomplex analysis and complex analysis with symmetry....

We consider, and make precise, a certain extension of the Radon-Nikodym derivative operator, to functions which are additive, but not necessarily sigma-additive, on a subset of a given sigma-algebra. We give applications to probability theory; in particular, to the study of $\mu$-Brownian motion, to stochastic calculus via generalized It\^o-integra...

In this paper, we study a specific system of Clifford–Appell polynomials and, in particular, their product. Moreover, we introduce a new family of quaternionic reproducing kernel Hilbert spaces in the framework of Fueter regular functions. The construction is based on a general idea which allows us to obtain various function spaces by specifying a...

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 o...

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 anal...

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 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 al...

Replacing the Lebesgue measure on an interval by a Stieltjes positive non-atomic measure, we study the corresponding counterpart of the Brownian motion. We introduce a new heat equation associated with the measure and make connections with stationary-increments Gaussian processes. We introduce a new transform analysis, and heat equation, associated...

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 assoc...

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 topol...

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 Fo...

We define and study rational discrete analytic functions and prove the existence of a coisometric realization for discrete analytic Schur multipliers.

In this paper, we present the groundwork for an Itô/Malliavin stochastic calculus and Hida's white noise analysis in the context of a supersymmetry with ℤ3‐graded algebras. To this end, we establish a ternary Fock space and the corresponding strong algebra of stochastic distributions and present its application in the study of stochastic processes...

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...

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 assoc...

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...

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, an...

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.