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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 41   | 1 | 2 |

## Section 1.3.2.5.8. Multiplication and convolution

G. Bricognea

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#### 1.3.2.5.8. Multiplication and convolution

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Multiplier functions for tempered distributions must be infinitely differentiable, as for ordinary distributions; furthermore, they must grow sufficiently slowly as to ensure that for all and that the map is continuous for the topology of . This leads to choosing for multipliers the subspace consisting of functions of polynomial growth. It can be shown that if f is in , then the associated distribution is in (i.e. is a tempered distribution); and that conversely if T is in is in for all .

Corresponding restrictions must be imposed to define the space of those distributions T whose convolution with a tempered distribution S is still a tempered distribution: T must be such that, for all is in ; and such that the map be continuous for the topology of . This implies that S is `rapidly decreasing'. It can be shown that if f is in , then the associated distribution is in ; and that conversely if T is in is in for all .

The two spaces and are mapped into each other by the Fourier transformation and the convolution theorem takes the form The same identities hold for . Taken together with the reciprocity theorem, these show that and establish mutually inverse isomorphisms between and , and exchange multiplication for convolution in .

It may be noticed that most of the basic properties of and may be deduced from this theorem and from the properties of δ. Differentiation operators and translation operators are convolutions with and ; they are turned, respectively, into multiplication by monomials (the transforms of ) or by phase factors (the transforms of ).

Another consequence of the convolution theorem is the duality established by the Fourier transformation between sections and projections of a function and its transform. For instance, in , the projection of on the x, y plane along the z axis may be written its Fourier transform is then which is the section of by the plane , orthogonal to the z axis used for projection. There are numerous applications of this property in crystallography (Section 1.3.4.2.1.8 ) and in fibre diffraction (Section 1.3.4.5.1.3 ).