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 form where 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 gives for 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 by The support of a tensor product is the Cartesian product of the supports of the two factors.