Poisson kernel

The Poisson kernel is a mathematical tool used in harmonic analysis and complex analysis to solve problems involving how values inside a circular region relate to values on its boundary. It helps reconstruct a smooth, harmonic function inside the circle based on known data around the edge. Think of it as a way to "smoothly" extend information from the boundary inward, ensuring the resulting function behaves nicely and respects certain mathematical properties. This concept is fundamental in fields like physics and engineering, where understanding how boundary conditions influence interior behavior is essential.