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Full-Text Articles in Other Mathematics
An Extension Of Herglotz's Theorem To The Quaternions, Daniel Alpay, Fabrizio Colombo, David P. Kimsey, Irene Sabadini, David P. Kimsey
An Extension Of Herglotz's Theorem To The Quaternions, Daniel Alpay, Fabrizio Colombo, David P. Kimsey, Irene Sabadini, David P. Kimsey
Mathematics, Physics, and Computer Science Faculty Articles and Research
A classical theorem of Herglotz states that a function n↦r(n) from Z into Cs×s is positive definite if and only there exists a Cs×s-valued positive measure dμ on [0,2π] such that r(n)=∫2π0eintdμ(t)for n∈Z. We prove a quaternionic analogue of this result when the function is allowed to have a number of negative squares. A key tool in the argument is the theory of slice hyperholomorphic functions, and the representation of such functions which have a positive real part in the unit ball of the quaternions. We study in great detail the case of positive definite functions.
Pontryagin De Branges-Rovnyak Spaces Of Slice Hyperholomorphic Functions, Daniel Alpay, Fabrizio Colombo, Irene Sabadini
Pontryagin De Branges-Rovnyak Spaces Of Slice Hyperholomorphic Functions, Daniel Alpay, Fabrizio Colombo, Irene Sabadini
Mathematics, Physics, and Computer Science Faculty Articles and Research
We study reproducing kernel Hilbert and Pontryagin spaces of slice hyperholomorphic functions which are analogs of the Hilbert spaces of analytic functions introduced by de Branges and Rovnyak. In the first part of the paper we focus on the case of Hilbert spaces, and introduce in particular a version of the Hardy space. Then we define Blaschke factors and Blaschke products and we consider an interpolation problem. In the second part of the paper we turn to the case of Pontryagin spaces. We first prove some results from the theory of Pontryagin spaces in the quaternionic setting and, in particular, …