# Poisson, Dirichlet and Fejer kernels

# See
# Eric W. Weisstein. "Poisson Kernel." From MathWorld--A Wolfram Web Resource.
# http://mathworld.wolfram.com/PoissonKernel.html

SetHelp("PoissonKernel","functions","Poisson kernel on D(0,1) (not normalized to 1, that is integral of this is 2pi)");
function PoissonKernel(r,sigma) = (
	(1 - r^2) / (1 - 2*r*cos(sigma) + r^2)
)
protect("PoissonKernel")

SetHelp("PoissonKernelRadius","functions","Poisson kernel on D(0,R) (not normalized to 1)");
function PoissonKernelRadius(r,sigma) = (
	(R^2 - r^2) / (R^2 - 2*R*r*cos(sigma) + r^2)
)
protect("PoissonKernelRadius")

# See Planetmath http://planetmath.org/encyclopedia/DirchletKernel.html
SetHelp("DirichletKernel","functions","Dirichlet kernel of order n");
function DirichletKernel(n,t) = (
	if t == 0 then (
		2*n+1
	) else (
		( sin ((n+0.5)*t) ) / ( sin(t/2))
	)
)
protect("DirichletKernel")

# See Planetmath http://planetmath.org/encyclopedia/FejerKernel.html
SetHelp("FejerKernel","functions","Fejer kernel of order n");
function FejerKernel(n,t) = (
	if t == 0 then (
		n
	) else (
		(1/n) *
		( ( sin ((n*t)/2) ) / ( sin(t/2)) )^2
	)
)
protect("FejerKernel")