Difference between revisions of "Gaussian bell"
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Let $\mathbb{T}$ be a [[time_scale | time scale]] with $0 \in \mathbb{T}$. Let $p \colon \mathbb{T} \rightarrow \mathbb{R}$ be [[regressive_function | regressive]] and defined by | Let $\mathbb{T}$ be a [[time_scale | time scale]] with $0 \in \mathbb{T}$. Let $p \colon \mathbb{T} \rightarrow \mathbb{R}$ be [[regressive_function | regressive]] and defined by | ||
$$p(t)=\ominus(t \odot 1).$$ | $$p(t)=\ominus(t \odot 1).$$ | ||
| − | The Gaussian bell $\mathbf{E} \colon \mathbb{T} \rightarrow \mathbb{R}$ is defined<ref name=gaussbell /> to be the [[ | + | The Gaussian bell $\mathbf{E} \colon \mathbb{T} \rightarrow \mathbb{R}$ is defined<ref name=gaussbell /> to be the [[Exponential_functions | exponential function]] |
$$\mathbf{E}(t)=e_{p}(t,0).$$ | $$\mathbf{E}(t)=e_{p}(t,0).$$ | ||
Revision as of 08:55, 26 May 2014
Let $\mathbb{T}$ be a time scale with $0 \in \mathbb{T}$. Let $p \colon \mathbb{T} \rightarrow \mathbb{R}$ be regressive and defined by $$p(t)=\ominus(t \odot 1).$$ The Gaussian bell $\mathbf{E} \colon \mathbb{T} \rightarrow \mathbb{R}$ is defined<ref name=gaussbell /> to be the exponential function $$\mathbf{E}(t)=e_{p}(t,0).$$
References
<references> <ref name=gaussbell>Erbe, L.; Peterson, A.;Simon, M. Square integrability of Gaussian bells on time scales. Comput. Math. Appl. 49 (2005), no. 5-6, 871--883. </ref> </references>
