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Daniel Alpay

Publications and source records attributed to Daniel Alpay.

At least 19 recordsLinked to original sources
arXiv2026

Quantitatively hyper-positive real rational functions IV: The canonical case

In the linear time-invariant framework, passive systems are modeled by positive real functions. Dissipative systems can be modeled by the subset of quantita- tively Hyper-positive real functions, related through nested inclusions. This family was introduced and studied in our three previous works. Here, we further focus our attention on the proper subset of canonical Hyper-Positive functions. Although this family is "small", its exploration is well motivated: First, this set turns to be associated with absolute stability (the Lurie problem). Then, a systematic parametrization of all canonical Hyper-Positive functions is introduced. Moreover each canonical Hyper-Positive function can be viewed as an extreme point of the convex set of Hyper-Positive functions. Specifically a convex combination of canonical Hyper-Positive is Hyper-Positive, but not canonical. These observations hold in both frameworks: of analytic functions and of state-space realization arrays. Technically, some of the analysis is facilitated by employing Quadratic Matrix Inclusions of both, matrices and of matrix-valued rational functions.

math.CV↗
arXiv2026

Hyperpositive functions, sector bounded functions and a trace formula

We study an indexed family of functions closely related to functions ana- lytic and with a real positive part in the right open half plane (the so-called positive functions) and to the functions analytic in the right open half-plane and bounded in modulus by one there (the so-called bounded functions). These two families are related by the Cayley transform. In the present paper we introduce an affine linear relation- ship between a subclass of positive functions and the family of bounded functions, and study the corresponding connections with passivity of linear systems, interpolation and operator models. This is therefore a multidisciplinary paper, with potential readers from engineering, linear system theory and operator theory, and some repetitions of known results are given to allow various audiences to read the work. Reproducing kernel Hilbert spaces of analytic functions are a key tool in the arguments. A special role is played by the de Branges-Rovnyak spaces associated to bounded functions, and we prove a related trace formula connecting an underlying pair of operators.

math.FA↗
arXiv2026

Operator model and a trace formula for pairs of unitary operators

Using the theory of reproducing kernel Hilbert spaces introduced by L. de Branges and J. Rovnyak we prove a trace formula for pairs of operators in Hilbert space in terms of a Carath\'eodory function. We consider the special case, when the latter is rational. An application to the theory of first order discrete systems is given.

math.FA↗
arXiv2026

Schur functions on a rhombic lattice

We extend the study of discrete analytic (DA) Schur functions to rhombic lattices, utilizing suitably defined shift operators. There is a number of important differences with the classical case, including eigenvalues of the backward shift operator. As an application we solve a basic interpolation problem in a weighted Hardy space of DA functions, introducing a discrete counterpart of the Blaschke factor.

math.CV↗
arXiv2026

Brownian motion: the hyperbolic number setting

The purpose of this paper is to define normal Gaussian variables in the setting of hyperbolic probabilities, and introduce an associated Brownian motion, when both the index and the values of the process lie in the real algebra $\mathbb{H}$ of hyperbolic numbers. In Hida's white noise space, we construct two probability measures (say $P_1$ and $P_2$), and associate to them two families of $N(0,1)$ variables $(Z_n)_{n\in\mathbb N_0}$ (independent with respect to $P_1$) and $(W_n)_{n\in\mathbb N_0}$ (independent with respect to $P_2$). An important feature is that the $Z_n$ and $W_m$ need not be mutually independent either with respect to $P_1$ or $P_2$. An hyperbolic normal Gaussian variable is constructed (in non-degenerate cases) from two classical Gaussian variables and the hyperbolic Brownian motion is, in general, composed from two copies of the classical Brownian motion. Using the associated Gelfand triples we also compute the derivative of the hyperbolic Brownian motion as a stochastic distribution. The argument extends to the $\mathbb{H}$-valued fractional Brownian motion, and more generally to a wide family of $\mathbb{H}$-valued stationary-increment second order processes.

math.PR↗
arXiv2026

Magnetic Dirichlet Laplacian on deformed waveguides

It is well known that the spectrum of the Dirichlet Laplacian for a two-dimensional waveguide, which is a local deformation of a straight strip, is unstable with respect to waveguide boundary deformations. This means that, when the waveguide is a straight strip, the spectrum of the Dirichlet Laplacian is purely essential. On the other hand, local boundary perturbations of the straight strip produce eigenvalues below the essential spectrum. This paper considers the Dirichlet-Laplace operator with a compactly supported magnetic field. Furthermore, we omit the condition that the boundary perturbation is local. We prove that, in this case, the spectrum of the magnetic Laplacian is stable under small deformations of the waveguide boundary.

math.SP↗
arXiv2026

Quantitatively hyper-positive real rational functions III

Hyper-Positive Real, matrix-valued, rational functions are associated with absolute stability (the Lurie problem). Here, quantitative subsets of Hyper-positive functions, related through nested inclusions, are introduced. Structurally, this family of functions turns out to be matrix-convex and closed under inversion. A state-space characterization of these functions through a corresponding Kalman-Yakubovich-Popov Lemma, is given. Technically, the classical Linear Matrix Inclusions, associated with passive systems, are here substituted by Quadratic Matrix Inclusions.

math.OC↗
arXiv2026

Hardy decomposition of first order Lipschitz functions by Lam\'e-Navier solutions

The Clifford algebra language allows us to rewrite the Lam\'e-Navier system in terms of the Euclidean Dirac operator. In this paper, the main question we shall be concerned with is whether or not a higher order Lipschitz function on the boundary $\Gamma$ of a Jordan domain $\Omega\subset\mathbb{R}^m$ can be decomposed into a sum of the two boundary values of a solution of the Lam\'e-Navier system with jump across $\Gamma$. Our main tool are the Hardy projections related to a singular integral operator arising in the context of Clifford analysis, which turns out to be an involution operator on the first order Lipschitz classes.

math.AP↗
arXiv2025

Jump problem for generalized Lam\'e-Navier systems in $\mathbb{R}^m$

This paper is devoted to study a fundamental system of equations in Linear Elasticity Theory: the famous Lam\'e-Navier system. The Clifford algebra language allows us to rewrite this system in terms of the Euclidean Dirac operator, which at the same time suggests a very natural generalization involving the so-called structural sets. Our interest lies mainly in the jump problem for these elastic systems. A generalized Teodorescu transform, to be introduced here, provides the means for obtaining the explicit solution of the jump problem for a very wide classes of regions, including those with a fractal boundary.

math.AP↗
arXiv2025

The Kaczmarz Algorithm in Hilbert $C^{*}$-modules

The Kaczmarz algorithm in Hilbert spaces is a classical iterative method for stably recovering vectors from inner product data. In this paper, we extend the algorithm to the setting of Hilbert $C^*$-modules and establish analogues of its effectiveness in both finite-dimensional and stationary cases. Consequently, we demonstrate that continuous families of elements in a Hilbert space can be uniformly recovered using the Kaczmarz algorithm. Additionally, we develop a normalized Cauchy transform for continuous families of measures and use it to provide sufficient conditions under which standard frames in Hilbert $C(X)$-modules can be generated by the Kaczmarz algorithm and realized as orbits of bounded operators.

math.FA↗
arXiv2025

Analytic continuation of time in Brownian motion. Stochastic distributions approach

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 approach to stochastic distributions. This in turn allows us to present an explicit formula for an analytically continued white noise process, realized this way in complex domain. With the use of the Wick product, we then apply our complex white noise analysis in a derivation of a new realization of Hilbert space-valued stochastic integrals

math.PR↗
arXiv2024

Unitary rational functions: The scaled quaternion case

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 quaternionic settings, and in particular, the adjoint is not the classical adjoint and we use properties of real Hilbert spaces. This adjoint allows to define the counterpart of unitarity for matrix-rational functions, and we develop the corresponding theories of realizations and unitary factorizations. We also begin a theory of matrices in the underlying rings.

math.FA↗
arXiv2024

Short-time Fourier transform and superoscillations

In this paper we investigate new results on the theory of superoscillations using time-frequency analysis tools and techniques such as the short-time Fourier transform (STFT) and the Zak transform. We start by studying how the short-time Fourier transform acts on superoscillation sequences. We then apply the supershift property to prove that the short-time Fourier transform preserves the superoscillatory behavior by taking the limit. It turns out that these computations lead to interesting connections with various features of time-frequency analysis such as Gabor spaces, Gabor kernels, Gabor frames, 2D-complex Hermite polynomials, and polyanalytic functions. We treat different cases depending on the choice of the window function moving from the general case to more specific cases involving the Gaussian and the Hermite windows. We consider also an evolution problem with an initial datum given by superoscillation multiplied by the time-frequency shifts of a generic window function. Finally, we compute the action of STFT on the approximating sequences with a given Hermite window.

math.FA↗
arXiv2024

Scaled global operators and Fueter variables on non-zero scaled hypercomplex numbers

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 $\mathbb{H}_t,\, t\in\mathbb{R}^*$, of which the $\mathbb{H}_{-1}=\mathbb{H}$ is the space of quaternions and $\mathbb{H}_{1}$ is the space of split quaternions. We also describe the scaled Fueter-type variables associated to these operators, developing a coherent theory in this field. We use these types of variables to build different types of function spaces on $\mathbb{H}_t$. Counterparts of the Hardy space and of the Arveson space are also introduced and studied in the present setting. The two different adjoints in the scaled hypercomplex numbers lead to two parallel cases in each instance. Finally we introduce and study the notion of rational function.

math.FA↗
arXiv2024

Schur analysis over the unit spectral ball

We define the corresponding Hardy space, Schur multipliers and their realizations, and interpolation. Possible applications of the present work include matrices of quaternions, matrices of split quaternions, and other algebras of hypercomplex numbers.

math.FA↗
arXiv2024

$q$-Rational functions and interpolation with complete Nevanlinna Pick Kernels

In this paper we introduce the concept of matrix-valued $q$-rational functions. In comparison to the classic case we give different characterizations with principal emphasise on realizations and discuss algebraic manipulations. We also study the concept of Schur multipliers and complete Nevanlinna Pick kernels in this context and provide first applications in terms of an interpolation problem using Schur multipliers and complete Nevanlinna Pick kernels.

math.CV↗
arXiv2023

On scaled hyperbolic numbers induced by scaled hyperbolic rings

In this paper, we generalize the well-known hyperbolic numbers to certain numeric structures scaled by the real numbers. Under our scaling of $\mathbb{R}$, the usual hyperbolic numbers are understood to be our 1-scaled hyperbolic numbers. If a scale $t$ is not positive in $\mathbb{R}$, then our $t$-scaled hyperbolic numbers have similar numerical structures with those of the complex numbers, however, if a scale is positive in $\mathbb{R}$, then their numerical properties are similar to those of the classical hyperbolic numbers. We here understand scaled-hyperbolic numbers as elements of the scaled-hypercomplex rings $\{\mathbb{H}_t\}_{t\in \mathbb{R}}$, introduced in [1]. This scaled-hyperbolic analysis is done by algebra, analysis, operator theory, operator-algebra theory and free probability on scaled-hypercomplex numbers

math.RA↗