MHB What Else Can the Poisson Kernel Achieve Beyond the Dirichlet Problem?

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The Poisson kernel is crucial in solving the Dirichlet problem by providing a means to construct harmonic functions on a domain with given boundary values. Beyond this, it plays a significant role in operator theory and harmonic analysis, serving as a tool for understanding various properties of functions and their behavior. Its applications extend to potential theory and the study of boundary value problems, highlighting its versatility in mathematical analysis. The discussion emphasizes the importance of the Poisson kernel in both theoretical and practical contexts. Overall, the Poisson kernel's significance transcends its initial application to the Dirichlet problem.
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What is the significance of the Poisson kernel (besides solving the Dirichlet problem)?

What is the Poisson's role in solving the Dirichlet problem? I know it is the solution but what is meant by its role?
 
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dwsmith said:
What is the significance of the Poisson kernel (besides solving the Dirichlet problem)?

Perhaps >>this<< might interest you. It describes the applications of Poisson's Kernel in operator theory and harmonic analysis.
 

Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
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