International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 4849

Finally, let us consider the relations between the Fourier transforms of these lattice distributions. Recalling the basic relation of Section 1.3.2.6.5,i.e.and similarly:
Thus (respectively ), a decimated version of (respectively ), is transformed by into a subdivided version of (respectively ).
The converse is also true:i.e.and similarly
Thus (respectively ), a subdivided version of (respectively ) is transformed by into a decimated version of (respectively ). Therefore, the Fourier transform exchanges subdivision and decimation of period lattices for lattice distributions.
Further insight into this phenomenon is provided by applying to both sides of (iv) and (v) and invoking the convolution theorem:These identities show that multiplication by the transform of the periodsubdividing distribution (respectively ) has the effect of decimating to (respectively to ). They clearly imply that, if and , thenTherefore, the duality between subdivision and decimation may be viewed as another aspect of that between convolution and multiplication.
There is clearly a strong analogy between the sampling/periodization duality of Section 1.3.2.6.6 and the decimation/subdivision duality, which is viewed most naturally in terms of subgroup relationships: both sampling and decimation involve restricting a function to a discrete additive subgroup of the domain over which it is initially given.