Fourier transforms of radial functions

The Fourier transform is naturally defined for integrable functions. Otherwise, it should be stipulated in which sense the Fourier transform is understood. We consider some class of radial and, generally speaking, nonintegrable functions. The Fourier transform is calculated as an improper integral and the limit coincides with the Fourier transform in the distributional sense. The inverse Fourier formula is proved as well. Given are some applications of the result obtained.