Editing Distribution (mathematics) (section)

====Fourier transform====

To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions. The ordinary [[continuous Fourier transform]] <math>F : \mathcal{S}(\R^n) \to \mathcal{S}(\R^n)</math> is a TVS-[[automorphism]] of the Schwartz space, and the '''{{em|Fourier transform}}''' is defined to be its [[transpose]] <math>{}^{t}F : \mathcal{S}'(\R^n) \to \mathcal{S}'(\R^n),</math> which (abusing notation) will again be denoted by <math>F.</math> So the Fourier transform of the tempered distribution <math>T</math> is defined by <math>(FT)(\psi) = T(F \psi)</math> for every Schwartz function <math>\psi.</math> <math>FT</math> is thus again a tempered distribution. The Fourier transform is a TVS isomorphism from the space of tempered distributions onto itself. This operation is compatible with differentiation in the sense that
<math display=block>F \dfrac{dT}{dx} = ixFT</math>
and also with convolution: if <math>T</math> is a tempered distribution and <math>\psi</math> is a {{em|slowly increasing}} smooth function on <math>\R^n,</math> <math>\psi T</math> is again a tempered distribution and
<math display=block>F(\psi T) = F \psi * FT</math>
is the convolution of <math>FT</math> and <math>F \psi.</math> In particular, the Fourier transform of the constant function equal to 1 is the <math>\delta</math> distribution.