International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 37   | 1 | 2 |

Section 1.3.2.4.3.4. Eigenspace decomposition of [L^{2}]

G. Bricognea

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1.3.2.4.3.4. Eigenspace decomposition of [L^{2}]

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Some light can be shed on the geometric structure of these rotations by the following simple considerations. Note that[\eqalign{{\scr F}^{2}[\,f]({\bf x}) &= {\textstyle\int\limits_{{\bb R}^{n}}} {\scr F}[\,f]({\boldxi}) \exp (-2\pi i{\bf x}\cdot {\boldxi}) \,\hbox{d}^{n}{\boldxi}\cr &= \bar{\scr F}[{\scr F}[\,f]](-{\bf x}) = f(-{\bf x})}]so that [{\scr F}^{4}] (and similarly [\bar{\scr F}^{4}]) is the identity map. Any eigenvalue of [{\scr F}] or [\bar{\scr F}] is therefore a fourth root of unity, i.e. ±1 or [\pm i], and [L^{2}({\bb R}^{n})] splits into an orthogonal direct sum[{\bf H}_{0} \otimes {\bf H}_{1} \otimes {\bf H}_{2} \otimes {\bf H}_{3},]where [{\scr F}] (respectively [\bar{\scr F}]) acts in each subspace [{\bf H}_{k}] [(k = 0, 1, 2, 3)] by multiplication by [(-i)^{k}]. Orthonormal bases for these subspaces can be constructed from Hermite functions (cf. Section 1.3.2.4.4.2[link]) This method was used by Wiener (1933[link], pp. 51–71).

References

Wiener, N. (1933). The Fourier Integral and Certain of its Applications. Cambridge University Press. [Reprinted by Dover Publications, New York, 1959.]








































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