The Dirichlet kernel for Fourier series is the finite sum
 |  |  |
(1)
|
 |  | ![(sin[(n+1/2)x])/(sin(1/2x)),](/images/equations/DirichletKernel/Inline6.svg) |
(2)
|
where 
is a nonnegative integer and values at 
, with 
an integer, are defined by continuity as 
. The normalized kernel 
has unit integral
over 
and converges to the delta function there in the
sense of generalized functions. On the entire
real line, its limit is the
periodic impulse train 
, a rescaled Shah
function.
For the Fourier transform in 
dimensions, define the normalized Dirichlet kernel
 |
(3)
|
where 
and 
is a positive integer. This is a radial function
of 
.
For 
and 
,
it satisfies
 |
(4)
|
with its value at 
defined by continuity.
