Examples
Many familiar distributions can be written as oscillatory integrals.
- 1) The Fourier inversion theorem implies that the delta function, is equal to
- If we apply the first method of defining this oscillatory integral from above, as well as the Fourier transform of the Gaussian, we obtain a well known sequence of functions which approximate the delta function:
- An operator in this case is given for example by
- where is the Laplacian with respect to the variables, and is any integer greater than . Indeed, with this we have
- and this integral converges absolutely.
- 2) The Schwartz kernel of any differential operator can be written as an oscillatory integral. Indeed if
- where, then the kernel of is given by
Read more about this topic: Oscillatory Integral
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