Extreme problems in the space of meromorphic functions of finite order in the half plane. II

The extremal problems in the space of meromorphic functions of order $\rho>0$ in upper half-plane are studed.The method for studying is based on the theory of Fourier coefficients of meromorphic functions. The concept of just meromorphic function of order $\rho>0$ in upper half-plane is introduced. Using Lemma on the P\'olya peaks and the Parseval equality, sharp estimate from below of the upper limits of relations Nevanlinna characteristics of meromorphic functions in the upper half plane are obtained.

Spectral Theory of Singular Hahn Difference Equation of the Sturm-Liouville Type

In this work, we consider the singular Hahn difference equation of the Sturm-Liouville type. We prove the existence of the spectral function for this equation. We establish Parseval equality and an expansion formula for this equation on a semi-unbounded interval.

The Parseval Equality and Expansion Formula for Singular Hahn-Dirac System

This work studies the singular Hahn-Dirac system given by Here 𝜇 is a complex spectral parameter, p(.) and r(.) are real-valued continuous functions at 𝜔0, defined on [𝜔0,∞) and q∈(0,1), , 𝜔>0, x∈[𝜔0,∞). The existence of a spectral function for this system is proved. Further, a Parseval equality and an expansion formula in eigenfunctions are proved in terms of the spectral function.

On Asymptotically Orthonormal Sequences

An asymptotically orthonormal sequence is a sequence that is nearly orthonormal in the sense that it satisfies the Parseval equality up to two constants close to one. In this paper, we explore such sequences formed by normalized reproducing kernels for model spaces and de Branges– Rovnyak spaces.

The Modified Parseval Equality of Sturm-Liouville Problems with Transmission Conditions

We consider the Sturm-Liouville (S-L) problems with very general transmission conditions on a finite interval. Firstly, we obtain the sufficient and necessary condition forλbeing an eigenvalue of the S-L problems by constructing the fundamental solutions of the problems and prove that the eigenvalues of the S-L problems are bounded below and are countably infinite. Furthermore, the asymptotic formulas of the eigenvalues and eigenfunctions of the S-L problems are obtained. Finally, we derive the eigenfunction expansion for Green's function of the S-L problems with transmission conditions and establish the modified Parseval equality in the associated Hilbert space.