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Results tagged with generalized-functions
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user 39261
4
votes
2
answers
386
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Initial paper of Gel'fand on Generalized Random Processes
The theory of generalized stochastic processes was introduced independently in the 50's by Ito* and Gel'fand in a short paper. The latter then developed his theory more extensively in the fourth tome …
- 2,306
6
votes
1
answer
227
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What are all the stationary and pointwise independent random processes?
In the 60's, I. Gel'fand introduced the concept of generalized stochastic processes (Ch. III, Vol. 4 of his work on Generalized functions). For a generalized stochastic process $\Phi$, he defines the …
- 2,306
5
votes
0
answers
146
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The space of periodizable tempered distribution
The periodization operator $\mathrm{Per}$ is defined for a Schwartz function $\varphi \in \mathcal{S}(\mathbb{R})$ as
\begin{equation}
\mathrm{Per} \{ \varphi \} (x) = \sum_{n \in \mathbb{Z}} \varphi( …
- 2,306
7
votes
0
answers
142
views
Characterization of tempered distributions from tempered sequences
Let $\mathcal{D}(\mathbb{R})$ be the space of compactly supported and infinitely smooth functions for its usual topology. Let
$\mathcal{D}'(\mathbb{R})$ be the topological dual of $\mathcal{D}(\mathb …
- 2,306