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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 33-34   | 1 | 2 |

## Section 1.3.2.3.9.6. Tensor product of distributions

G. Bricognea

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#### 1.3.2.3.9.6. Tensor product of distributions

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The purpose of this construction is to extend Fubini's theorem to distributions. Following Section 1.3.2.2.5, we may define the tensor product as the vector space of finite linear combinations of functions of the formwhere and .

Let and denote the distributions associated to f and g, respectively, the subscripts x and y acting as mnemonics for and . It follows from Fubini's theorem (Section 1.3.2.2.5) that , and hence defines a distribution over ; the rearrangement of integral signs givesfor all . In particular, if with , then

This construction can be extended to general distributions and . Given any test function , let denote the map ; let denote the map ; and define the two functions and . Then, by the lemma on differentiation under the sign of Section 1.3.2.3.9.1, , and there exists a unique distribution such that is called the tensor product of S and T.

With the mnemonic introduced above, this definition reads identically to that given above for distributions associated to locally integrable functions:

The tensor product of distributions is associative:Derivatives may be calculated byThe support of a tensor product is the Cartesian product of the supports of the two factors.