Fourier transform of unit step
The Fourier transform can be generalized for functions that are not absolutely integrable. We can define a Fourier transform for functions with a constant envelope (e.g., sine, cosine, complex exponential), and even for functions with polynomial growth (but not with exponential growth). In these cases we must be prepared to deal with generalized functions in the expression of the Fourier transform, such as the Dirac delta impulse or its derivatives. This is also true for the Fourier transform of the step function.

The multiplication property of the Fourier transform remains true, albeit with certain restrictions, so we can generally compute the Fourier transform of the convolution of two functions by multiplying their Fourier transforms, provided that the convolution exists.