Quantum Measurement of Gravity for Geodesists and Geophysicists by Fernando Sansò & Federica Migliaccio

Author:Fernando Sansò & Federica Migliaccio
Language: eng
Format: epub
ISBN: 9783030428389
Publisher: Springer International Publishing

(4.47)

and

(4.48)

moreover

(4.49)

and

(4.50)

(see [5, XI, 14]).

However, to continue our formalism, we need one more restrictive hypothesis.

We assume that there is a distribution valued kernel u(x, a) (we consider it as the kernel of realizing the identity (4.49)) such that

(4.51)

or dually

(4.52)

and, moreover,

(4.53)

Remark 4.3

Notice that, as some elementary examples prove, we need to assume that u(x, a) is in fact a distribution; this is clear if, for instance, we assume that . Yet if the relation (4.47) holds for a set of smooth functions (like, for instance, the space of fast-decreasing functions) which is also dense in , the relations (4.51), (4.52) can be extended by continuity to the whole ; so, for instance we can correctly write, after a limit process,

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