scholarly journals The construction of a two-dimensional reproducing kernel function and its application in a biomedical model

2016 ◽  
Vol 24 (s2) ◽  
pp. S477-S486
Author(s):  
Qi Guo ◽  
Shu-Ting Shen
Keyword(s):  
Kernel Function ◽  
Reproducing Kernel ◽  
Biomedical Model ◽  
Two Dimensional ◽  
Reproducing Kernel Function

DIRECT AND REVERSE LOG-SOBOLEV INEQUALITIES IN μ-DEFORMED SEGAL–BARGMANN ANALYSIS

2007 ◽  
Vol 10 (04) ◽  
pp. 539-571 ◽  
Author(s):  
CARLOS ERNESTO ANGULO AGUILA ◽  
STEPHEN BRUCE SONTZ
Keyword(s):  
Kernel Function ◽  
Shannon Entropy ◽  
Dirichlet Form ◽  
Reproducing Kernel ◽  
Integral Kernel ◽  
Sobolev Inequalities ◽  
Dirichlet Energy ◽  
Reverse Inequality ◽  
Main Method ◽  
Reproducing Kernel Function

Both direct and reverse log-Sobolev inequalities, relating the Shannon entropy with a μ-deformed energy, are shown to hold in a family of μ-deformed Segal–Bargmann spaces. This shows that the μ-deformed energy of a state is finite if and only if its Shannon entropy is finite. The direct inequality is a new result, while the reverse inequality has already been shown by the authors but using different methods. Next the μ-deformed energy of a state is shown to be finite if and only if its Dirichlet form energy is finite. This leads to both direct and reverse log-Sobolev inequalities that relate the Shannon entropy with the Dirichlet energy. We obtain that the Dirichlet energy of a state is finite if and only if its Shannon entropy is finite. The main method used here is based on a study of the reproducing kernel function of these spaces and the associated integral kernel transform.

scholarly journals Using Reproducing Kernel for Solving a Class of Fractional Order Integral Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Zhiyuan Li ◽  
Meichun Wang ◽  
Yulan Wang ◽  
Jing Pang
Keyword(s):  
Fractional Order ◽  
Kernel Function ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Jacobi Polynomials ◽  
Kernel Method ◽  
Schmidt Orthogonalization ◽  
Integral Differential Equations ◽  
First Time ◽  
Reproducing Kernel Function

This paper is devoted to the numerical scheme for a class of fractional order integrodifferential equations by reproducing kernel interpolation collocation method with reproducing kernel function in the form of Jacobi polynomials. Reproducing kernel function in the form of Jacobi polynomials is established for the first time. It is implemented as a reproducing kernel method. The numerical solutions obtained by taking the different values of parameter are compared; Schmidt orthogonalization process is avoided. It is proved that this method is feasible and accurate through some numerical examples.

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