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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 | # 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)
)
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)
)
# 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))
)
)
# 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
)
)
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