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

2020 ◽  
pp. 209-235
Author(s):  
Bilender P. Allahverdiev ◽  
Hüseyin Tuna
Keyword(s):  
Spectral Function ◽  
Spectral Parameter ◽  
Continuous Functions ◽  
Dirac System ◽  
Parseval Equality ◽  
Expansion Formula

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.

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

2020 ◽  
Vol 28 (1) ◽  
pp. 13-25
Author(s):  
Bilender P. Allahverdiev ◽  
Hüseyin Tuna
Keyword(s):  
Difference Equation ◽  
Spectral Function ◽  
Spectral Theory ◽  
Parseval Equality ◽  
Liouville Type ◽  
Expansion Formula ◽  
Unbounded Interval ◽  
Sturm Liouville

AbstractIn 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.

scholarly journals Asymptotic Behavior of Solutions of the Dirac System with an Integrable Potential

2021 ◽  
Vol 93 (5) ◽  
Author(s):  
Łukasz Rzepnicki
Keyword(s):  
Asymptotic Behavior ◽  
Spectral Parameter ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Dirac System ◽  
Eigenvalues And Eigenfunctions ◽  
Integrable Potential ◽  
Sturm Liouville ◽  
Complex Valued ◽  
Liouville Operators

AbstractWe consider the Dirac system on the interval [0, 1] with a spectral parameter $$\mu \in {\mathbb {C}}$$ μ ∈ C and a complex-valued potential with entries from $$L_p[0,1]$$ L p [ 0 , 1 ] , where $$1\le p$$ 1 ≤ p . We study the asymptotic behavior of its solutions in a strip $$|\mathrm{Im}\,\mu |\le d$$ | Im μ | ≤ d for $$\mu \rightarrow \infty $$ μ → ∞ . These results allow us to obtain sharp asymptotic formulas for eigenvalues and eigenfunctions of Sturm–Liouville operators associated with the aforementioned Dirac system.

scholarly journals Spectral properties and a Parseval’s equality in the singular case for q-Dirac problem

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Seyfollah Mosazadeh
Keyword(s):  
Hilbert Space ◽  
Spectral Function ◽  
Spectral Properties ◽  
Dirac System ◽  
Singular Case ◽  
Parseval’S Equality

AbstractThis paper is devoted to studying a q-analog of the singular Dirac problem. First, we investigate some spectral properties of the problem. Then we prove the existence of a spectral function and establish a Parseval’s equality, for the singular q-Dirac system in a Hilbert space. Although there were given some results for this type of problem, we think that Parseval’s equality has not been studied yet.

scholarly journals Computation of Spectral Parameter of Discontinuous Dirac Systems with a Gaussian Multiplier

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Mohammed M. Tharwat ◽  
Mohammed A. Alghamdi
Keyword(s):  
Boundary Conditions ◽  
Spectral Parameter ◽  
Worked Examples ◽  
New Technique ◽  
Dirac System ◽  
Sinc Method ◽  
Dirac Systems

We aim in this paper to apply a sinc-Gaussian technique to compute the eigenvalues of a Dirac system which has a discontinuity at one point and contains a spectral parameter in all boundary conditions. We establish the needed properties of eigenvalues of our problem. The error of this method decays exponentially in terms of the number of involved samples. Therefore the accuracy of the new technique is higher than the classical sinc-method. Numerical worked examples with tables and illustrative figures are given at the end of the paper.

scholarly journals ON THE EXPANSION FORMULA FOR A SINGULAR STURM-LIOUVILLE OPERATOR

2021 ◽  
Vol 21 (1) ◽  
pp. 67-76
Author(s):  
ULVIYE DEMIRBILEK ◽  
KHANLAR R. MAMEDOV
Keyword(s):  
Boundary Condition ◽  
Green Function ◽  
Spectral Parameter ◽  
Liouville Operator ◽  
Resolvent Operator ◽  
Liouville Problem ◽  
Scattering Data ◽  
Expansion Formula ◽  
Sturm Liouville

In this study, on the semi-axis, Sturm - Liouville problem under boundary condition depending on spectral parameter is considered. In what follows scattering data is defined and its properties are given for the problem. The kernel of resolvent operator which is Green function is constructed. Using Titchmarsh method, expansion is obtained according to eigenfunctions and expansion formula is expressed with the scattering data.

Approximation in statistical sense to B -continuous functions by positive linear operators

2010 ◽  
Vol 47 (3) ◽  
pp. 289-298 ◽  
Author(s):  
Fadime Dirik ◽  
Oktay Duman ◽  
Kamil Demirci
Keyword(s):  
Compact Subset ◽  
Real Line ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Statistical Approximation ◽  
Classical Version ◽  
Statistical Sense

In the present work, using the concept of A -statistical convergence for double real sequences, we obtain a statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued B -continuous functions on a compact subset of the real line. Furthermore, we display an application which shows that our new result is stronger than its classical version.

Almost Somewhat Near Continuity and Near Regularity

2021 ◽  
Vol 7 (1) ◽  
pp. 88-99
Author(s):  
Zanyar A. Ameen
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
One To One ◽  
Nearly Continuous

AbstractThe notions of almost somewhat near continuity of functions and near regularity of spaces are introduced. Some properties of almost somewhat nearly continuous functions and their connections are studied. At the end, it is shown that a one-to-one almost somewhat nearly continuous function f from a space X onto a space Y is somewhat nearly continuous if and only if the range of f is nearly regular.

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