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In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson.
A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that e −x 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.
The conjugate Poisson kernel has two important properties for ε small [,] | |. (,) | () | Exactly the same reasoning as before shows that the two integrals tend to 0 as ε → 0. Combining these two limit formulas it follows that H ε f tends pointwise to Hf on the common Lebesgue points of f and Hf and therefore almost everywhere.
The next steps in the study of the Dirichlet's problem were taken by Karl Friedrich Gauss, William Thomson (Lord Kelvin) and Peter Gustav Lejeune Dirichlet, after whom the problem was named, and the solution to the problem (at least for the ball) using the Poisson kernel was known to Dirichlet (judging by his 1850 paper submitted to the ...
For example, every integral transform is a linear operator, since the integral is a linear operator, and in fact if the kernel is allowed to be a generalized function then all linear operators are integral transforms (a properly formulated version of this statement is the Schwartz kernel theorem).
The g function is a non-linear operator on L p (R n) that can be used to control the L p norm of a function f in terms of its Poisson integral. The Poisson integral u(x,y) of f is defined for y > 0 by (,) = () where the Poisson kernel P on the upper half space {(;) + >} is given by
[1] [2] In another context, the term refers to a certain wavelet which involves a form of the Poisson integral kernel. [3] In still another context, the terminology is used to describe a family of complex wavelets indexed by positive integers which are connected with the derivatives of the Poisson integral kernel. [4]
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