International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 47-48
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Let be such that has compact support ( is said to be band-limited). Then is -periodic, and is such that only a finite number of points of have a non-zero Fourier coefficient attached to them. We may therefore find a decimation of such that the distinct translates of Supp by vectors of do not intersect.
The distribution Φ can be uniquely recovered from by the procedure of Section 1.3.2.7.1, and we may write: these rearrangements being legitimate because and have compact supports which are intersection-free under the action of . By virtue of its -periodicity, this distribution is entirely characterized by its `motif' with respect to :
Similarly, φ may be uniquely recovered by Shannon interpolation from the distribution sampling its values at the nodes of is a subdivision of ). By virtue of its -periodicity, this distribution is completely characterized by its motif:
Let and , and define the two sets of coefficients Define the two distributions and The relation between ω and Ω has two equivalent forms:
By (i), . Both sides are weighted lattice distributions concentrated at the nodes of , and equating the weights at gives Since , is an integer, hence
By (ii), we have Both sides are weighted lattice distributions concentrated at the nodes of , and equating the weights at gives Since , is an integer, hence
Now the decimation/subdivision relations between and may be written: so that with , hence finally
Denoting by and by , the relation between ω and Ω may be written in the equivalent form where the summations are now over finite residual lattices in standard form.
Equations (i) and (ii) describe two mutually inverse linear transformations and between two vector spaces and of dimension . [respectively ] is the discrete Fourier (respectively inverse Fourier) transform associated to matrix N.
The vector spaces and may be viewed from two different standpoints:
These two spaces are said to be `isomorphic' (a relation denoted ≅), the isomorphism being given by the one-to-one correspondence:
The second viewpoint will be adopted, as it involves only linear algebra. However, it is most helpful to keep the first one in mind and to think of the data or results of a discrete Fourier transform as representing (through their sets of unique weights) two periodic lattice distributions related by the full, distribution-theoretic Fourier transform.
We therefore view (respectively ) as the vector space of complex-valued functions over the finite residual lattice (respectively ) and write: since a vector such as ψ is in fact the function .
The two spaces and may be equipped with the following Hermitian inner products: which makes each of them into a Hilbert space. The canonical bases and and and are orthonormal for their respective product.