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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 40   | 1 | 2 |

## Section 1.3.2.5.5. Transposition of basic properties

G. Bricognea

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#### 1.3.2.5.5. Transposition of basic properties

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The duality between differentiation and multiplication by a monomial extends from to by transposition:Analogous formulae hold for , with i replaced by −i.

The formulae expressing the duality between translation and phase shift, e.g.between a linear change of variable and its contragredient, e.g.are obtained similarly by transposition from the corresponding identities in . They give a transposition formula for an affine change of variables with nonsingular matrix A:with a similar result for , replacing −i by +i.

Conjugate symmetry is obtained similarly:with the same identities for .

The tensor product property also transposes to tempered distributions: if ,