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

G. Bricognea

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#### 1.3.2.4.3.4. Eigenspace decomposition of

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Some light can be shed on the geometric structure of these rotations by the following simple considerations. Note thatso that (and similarly ) is the identity map. Any eigenvalue of or is therefore a fourth root of unity, i.e. ±1 or , and splits into an orthogonal direct sumwhere (respectively ) acts in each subspace by multiplication by . Orthonormal bases for these subspaces can be constructed from Hermite functions (cf. Section 1.3.2.4.4.2) This method was used by Wiener (1933, 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.]