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. 3839

Gaussian functions are particularly important elements of . In dimension 1, a well known contour integration (Schwartz, 1965, p. 184) yieldswhich shows that the `standard Gaussian' is invariant under and . By a tensor product construction, it follows that the same is true of the standard Gaussianin dimension n:In other words, G is an eigenfunction of and for eigenvalue 1 (Section 1.3.2.4.3.4).
A complete system of eigenfunctions may be constructed as follows. In dimension 1, consider the family of functionswhere D denotes the differentiation operator. The first two members of the familyare such that , as shown above, andhenceWe may thus take as an induction hypothesis thatThe identitymay be writtenand the two differentiation theorems give:Combination of this with the induction hypothesis yieldsthus proving that is an eigenfunction of for eigenvalue for all . The same proof holds for , with eigenvalue . If these eigenfunctions are normalized asthen it can be shown that the collection of Hermite functions constitutes an orthonormal basis of such that is an eigenfunction of (respectively ) for eigenvalue (respectively ).
In dimension n, the same construction can be extended by tensor product to yield the multivariate Hermite functions(where is a multiindex). These constitute an orthonormal basis of , with an eigenfunction of (respectively ) for eigenvalue (respectively ). Thus the subspaces of Section 1.3.2.4.3.4 are spanned by those with .
General multivariate Gaussians are usually encountered in the nonstandard formwhere A is a symmetric positivedefinite matrix. Diagonalizing A as with the identity matrix, and putting , we may writei.e.hence (by Section 1.3.2.4.2.3)i.e.i.e. finally
This result is widely used in crystallography, e.g. to calculate form factors for anisotropic atoms (Section 1.3.4.2.2.6) and to obtain transforms of derivatives of Gaussian atomic densities (Section 1.3.4.4.7.10).
References
Schwartz, L. (1965). Mathematics for the Physical Sciences. Paris: Hermann, and Reading: AddisonWesley.