The Space-time Fourier transform is a special case of the Fourier transform in 3 spatial dimensions plus time. The non-radiation condition and GUTCP rely on the Space-time Fourier transform in a fundamental way. Since the topic is rarely presented even in graduate level courses, this page collects some general information on the Space-time Fourier transform.
The Fourier transform and its inverse establish the equivalence of expressing a function of a real variable as a related function in a frequency domain. The extract from Bracewell (referenced below) provides a short intuitive summary of the Space-time Fourier.
Bracewell, "The Fourier Transform and It's Applications", 1982, Chap. 12