Page "Dirac delta function" Paragraph 21

:" The greatest drawback of the classical Fourier transformation is a rather narrow class of functions ( originals ) for which it can be effectively computed.

Namely, it is necessary that these functions decrease sufficiently rapidly to zero ( in the neighborhood of infinity ) in order to insure the existence of the Fourier integral.

For example, the Fourier transform of such simple functions as polynomials does not exist in the classical sense.

The extension of the classical Fourier transformation to distributions considerably enlarged the class of functions that could be transformed and this removed many obstacles.