This version of the theorem is used in the proof of the Fourier inversion theorem for tempered distributions ( see below ).
12.
The space of "'tempered distributions "'is defined as the ( continuous ) dual of the Schwartz space.
13.
A precise version of this result, given below, holds for distributions of compact support, tempered distributions, and general distributions.
14.
There, it is explained that the sum of exponentials only agrees with the sum of deltas in the sense of tempered distributions.
15.
Therefore, this map defines, as it is obviously linear, a continuous functional on the Schwartz space and therefore a tempered distribution.
16.
For the definition of the Fourier transform of a tempered distribution, let and be integrable functions, and let and be their Fourier transforms respectively.
17.
Tempered distributions generalize the bounded ( or slow-growing ) locally integrable functions; all distributions with compact support and all square-integrable functions are tempered distributions.
18.
Tempered distributions generalize the bounded ( or slow-growing ) locally integrable functions; all distributions with compact support and all square-integrable functions are tempered distributions.
19.
Generally, the Fourier transform can be defined for any tempered distribution; moreover, any distribution of compact support " v " is a tempered distribution.
20.
Generally, the Fourier transform can be defined for any tempered distribution; moreover, any distribution of compact support " v " is a tempered distribution.
