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 1.3.2.4.3.5. The convolution theorem and the isometry property

G. Bricognea

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#### 1.3.2.4.3.5. The convolution theorem and the isometry property

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In , the convolution theorem (when applicable) and the Parseval/Plancherel theorem are not independent. Suppose that f, g, and are all in (without questioning whether these properties are independent). Then may be written in terms of the inner product in as follows:i.e.

Invoking the isometry property, we may rewrite the right-hand side asso that the initial identity yields the convolution theorem.

To obtain the converse implication, note thatwhere conjugate symmetry (Section 1.3.2.4.2.2) has been used.

These relations have an important application in the calculation by Fourier transform methods of the derivatives used in the refinement of macromolecular structures (Section 1.3.4.4.7).