A Dirac distribution or Dirac -distribution is the distribution that is given by evaluating a function at a point .
It is closely related to Dirac measures, in the language of measure theory.
The distributional derivative of the Heaviside distribution is the delta distribution :
For any bump function we compute:
(Fourier transform of the delta-distribution)
The Fourier transform (this def.) of the delta distribution, via this example, is the constant function on 1:
This implies by the Fourier inversion theorem (this prop.) that the delta distribution itself has equivalently, in generalized function-notation, the expression
in the sense that for every function with rapidly decreasing partial derivatives we have
which is just the statement of the Fourier inversion theorem for smooth functions (this prop.).
It is clear that:
The delta distribution is a compactly supported distribution, and in fact a point-supported distribution.
Every point-supported distribution with is a finite sum of multiplies of derivatives of the delta distribution at that point:
where , and for the order of .
(e.g. Hörmander 90, theorem 2.3.4)
Last revised on July 13, 2024 at 10:02:48. See the history of this page for a list of all contributions to it.