Schur functions and their realizations in the slice hyperholomorphic setting

we start the study of Schur analysis in the quaternionic setting using the theory of slice hyperholomorphic functions. The novelty of our approach is that slice hyperholomorphic functions allows to write realizations in terms of a suitable resolvent, the so called S-resolvent operator and to extend several results that hold in the complex case to the quaternionic case. We discuss reproducing kernels, positive definite functions in this setting and we show how they can be obtained in our setting using the extension operator and the slice regular product. We define Schur multipliers, and find their co-isometric realization in terms of the associated de Branges-Rovnyak space

OMBUDSMAN IN TURKEY: ITS CONTRIBUTIONS AND CRITICISM

Ombudsman institution is an independent, impartial, and a legal body which aims to protect the ruled ones against the trend of lawful authority, eliminate bad management practices, and act upon a complaint or ex officio. Ombudsman has authority in the investigation of incident, analysis, and in the making of announcement to the general public. With the legal arrangement made in Turkey, ombudsman institution has an Ankara-based special budget and legal entity connected to the Grand National Assembly of Turkey. The transition of the ombudsman institution into a constitutional institution has made significant contributions to both public administration and also to the citizens. This paper aims to study the various contributions and criticisms of the ombudsman institution in our country, Turkey

Wiener algebra for the quaternions

We define and study the counterpart of the Wiener algebra in the quaternionic setting, both for the discrete and continuous case. We prove a Wiener-L\'evy type theorem and a factorization theorem. We give applications to Toeplitz and Wiener-Hopf operators

OMBUDSMAN IN TURKEY: ITS CONTRIBUTIONS AND CRITICISM

Ombudsman institution is an independent, impartial, and a legal body which aims to protect the ruled ones against the trend of lawful authority, eliminate bad management practices, and act upon a complaint or ex officio. Ombudsman has authority in the investigation of incident, analysis, and in the making of announcement to the general public. With the legal arrangement made in Turkey, ombudsman institution has an Ankara-based special budget and legal entity connected to the Grand National Assembly of Turkey. The transition of the ombudsman institution into a constitutional institution has made significant contributions to both public administration and also to the citizens. This paper aims to study the various contributions and criticisms of the ombudsman institution in our country, Turkey

On a New Class of Structured Reproducing Kernel Spaces, Journal of Functional Analysis

A class of reproducing kernel spaces with reproducing kernels of the form Kω(λ) = {J − Θ(λ)JΘ(ω)*}/ρω(λ) with pω(λ) = a(λ)a(ω)* is characterized in terms of invariance under a pair of generalized shift operators and a structural identity. This incorporates a characterization of de Branges for the line case and a later analogue due to Ball for the circle case, as well as many other possibilities, by specializing the choice of ρ. These results also permit the extension of some earlier characterizations by the authors of finite dimensional spaces with reproducing kernels of the form given above to the infinite dimensional case. The non-Hermitian case is also considered

On the class SI of J-contractive functions intertwining solutions of linear differential equations

In the PhD thesis of the second author under the supervision of the third author was defined the class SI of J-contractive functions, depending on a parameter and arising as transfer functions of overdetermined conservative 2D systems invariant in one direction. In this paper we extend and solve in the class SI, a number of problems originally set for the class SC of functions contractive in the open right-half plane, and unitary on the imaginary line with respect to some preassigned signature matrix J. The problems we consider include the Schur algorithm, the partial realization problem and the Nevanlinna-Pick interpolation problem. The arguments rely on a correspondence between elements in a given subclass of SI and elements in SC. Another important tool in the arguments is a new result pertaining to the classical tangential Schur algorithm.Comment: 46 page

Factorization of J-unitary matrix polynomials on the line and a Schur algorithm for generalized Nevanlinna functions

AbstractWe prove that a 2×2 matrix polynomial which is J-unitary on the real line can be written as a product of normalized elementary J-unitary factors and a J-unitary constant. In the second part we give an algorithm for this factorization using an analog of the Schur transformation

The Transformation of Issai Schur and Related Topics in an Indefinite Setting

We review our recent work on the Schur transformation for scalar generalized Schur and Nevanlinna functions. The Schur transformation is defined for these classes of functions in several situations, and it is used to solve corresponding basic interpolation problems and problems of factorization of rational J-unitary matrix functions into elementary factors. A key role is played by the theory of reproducing kernel Pontryagin spaces and linear relations in these spaces