8 papers
Brownian motion: the hyperbolic number setting
Daniel Alpay, Ilwoo Cho, Liora Mayats-Alpay
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 th…
The Kaczmarz Algorithm in Hilbert -modules
Daniel Alpay, Chad Berner, Eric S. Weber
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 settin…
Analytic continuation of time in Brownian motion. Stochastic distributions approach
Luis Daniel Abreu, Daniel Alpay, Tryphon Georgiou +1
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 proc…
Unitary rational functions: The scaled quaternion case
Daniel Alpay, Ilwoo Cho, Mihaela Vajiac
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…
Scaled global operators and Fueter variables on non-zero scaled hypercomplex numbers
Daniel Alpay, Ilwoo Cho, Mihaela Vajiac
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 numbe…
Schur analysis over the unit spectral ball
Daniel Alpay, Ilwoo Cho
We define the corresponding Hardy space, Schur multipliers and their realizations, and interpolation. Possible applications of the present work include matrices of quaternions, mat…
On scaled hyperbolic numbers induced by scaled hyperbolic rings
Daniel Alpay, Ilwoo Cho
In this paper, we generalize the well-known hyperbolic numbers to certain numeric structures scaled by the real numbers. Under our scaling of , the usual hyperbolic num…
Regular Functions on the Scaled Hypercomplex Numbers
Daniel Alpay, Ilwoo Cho
In this paper, we study the regularity of -differentiable functions on open connected subsets of the scaled hypercomplex numbers $\left\{ \mathbb{H}_{t}\right\} _{t\in\…