Difference between revisions of "Gaussian bell"
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The Gaussian bell $\mathbf{E} \colon \mathbb{T} \rightarrow \mathbb{R}$ is defined<ref name=gaussbell /> to be the [[Exponential_functions | exponential function]] | The Gaussian bell $\mathbf{E} \colon \mathbb{T} \rightarrow \mathbb{R}$ is defined<ref name=gaussbell /> to be the [[Exponential_functions | exponential function]] | ||
$$\mathbf{E}(t)=e_{p}(t,0).$$ | $$\mathbf{E}(t)=e_{p}(t,0).$$ | ||
| + | |||
| + | {| class="wikitable" | ||
| + | |+Time Scale Gaussian Bells | ||
| + | |- | ||
| + | |$\mathbb{T}$ | ||
| + | |$\mathbf{E}(t)$ | ||
| + | |- | ||
| + | |[[Real_numbers | $\mathbb{R}$]] | ||
| + | |$e^{-\frac{t^2}{2}}$ | ||
| + | |- | ||
| + | |[[Integers | $\mathbb{Z}$]] | ||
| + | |$foo(t) = 2^{\frac{-t(t-1)}{2}} $ | ||
| + | |- | ||
| + | |[[Multiples_of_integers | $h\mathbb{Z}$]] | ||
| + | | $\left[(1+h)^{\frac{1}{h}} \right]^{\frac{-t(t-h)}{2}}$ | ||
| + | |- | ||
| + | | [[Square_integers | $\mathbb{Z}^2$]] | ||
| + | | | ||
| + | |- | ||
| + | |[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]] | ||
| + | | | ||
| + | |- | ||
| + | |[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]] | ||
| + | | $\displaystyle\prod_{\log_q(t)+1}^{\infty} \dfrac{1}{\left(1+(\frac{1}{q}-1)q^k \right)^{q^k}}$ | ||
| + | |- | ||
| + | |[[Harmonic_numbers | $\mathbb{H}$]] | ||
| + | |$\displaystyle\prod_{k=1}^n \left( \dfrac{k}{k+1} \right)^{H_{k-1}}$ | ||
| + | |} | ||
==References== | ==References== | ||
Revision as of 00:45, 9 September 2015
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 $$\mathbf{E}(t)=e_{p}(t,0).$$
| $\mathbb{T}$ | $\mathbf{E}(t)$ |
| $\mathbb{R}$ | $e^{-\frac{t^2}{2}}$ |
| $\mathbb{Z}$ | $foo(t) = 2^{\frac{-t(t-1)}{2}} $ |
| $h\mathbb{Z}$ | $\left[(1+h)^{\frac{1}{h}} \right]^{\frac{-t(t-h)}{2}}$ |
| $\mathbb{Z}^2$ | |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | $\displaystyle\prod_{\log_q(t)+1}^{\infty} \dfrac{1}{\left(1+(\frac{1}{q}-1)q^k \right)^{q^k}}$ |
| $\mathbb{H}$ | $\displaystyle\prod_{k=1}^n \left( \dfrac{k}{k+1} \right)^{H_{k-1}}$ |
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>
