Reproducing kernel function-based Filon and Levin methods for solving highly oscillatory integral

2021 ◽  
Vol 397 ◽  
pp. 125980
Author(s):  
F.Z. Geng ◽  
X.Y. Wu
Keyword(s):  
Kernel Function ◽  
Reproducing Kernel ◽  
Oscillatory Integral ◽  
Highly Oscillatory Integral ◽  
Highly Oscillatory ◽  
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.

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