Fourier transform scaling theorem

This establishes the relationship between the density fluctuation expressed as ((A A )2), on the one hand, and the scattering intensity I(q) and the shape cr(r) of the region of volume v being assumed, on the other hand. As the volume v is increased, its Fourier transform Y,(q) becomes more sharply peaked around q = 0, and therefore we see that only the part of the intensity curve I(q) observable at very small q has a bearing on the density fluctuation for large v. We are mainly interested in the density fluctuation on a macroscopic scale, that is, in the limit of v —> oo, in which case H(q) approaches the delta function. Noting, by use of Parseval s theorem, that... 

This theorem shows that if a function/(x) is scaled by shrinking its width by a factor a, its Fourier transform is expanded in width by the same factor (while the height is altered by a factor 1/ a ). It clearly illustrates the reciprocity relationship between fix) and F(s). 

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