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]] |
$$\mathbf{E}(t)=e_{p}(t,0).$$ | $$\mathbf{E}(t)=e_{p}(t,0).$$ | ||
| − | = | + | =Properties= |
| − | < | + | <center> |
| − | + | {| class="wikitable" | |
| − | + | |+Time Scale Gaussian Bells | |
| + | |- | ||
| + | |$\mathbb{T}$ | ||
| + | |$\mathbf{E}(t)$ | ||
| + | |- | ||
| + | |[[Real_numbers | $\mathbb{R}$]] | ||
| + | |$e^{-\frac{t^2}{2}}$ | ||
| + | |- | ||
| + | |[[Integers | $\mathbb{Z}$]] | ||
| + | |$2^{\frac{-t(t-1)}{2}}$ | ||
| + | |- | ||
| + | |[[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 | $\overline{q^{\mathbb{Z}}}, q < 1$]] | ||
| + | | $\displaystyle\prod_{k=\log_q(t)+1}^{\infty} \dfrac{1}{\left(1+(\frac{1}{q}-1)q^k \right)^{q^k}}$ | ||
| + | |- | ||
| + | |[[Harmonic_numbers | $\mathbb{H}$]] | ||
| + | |$\displaystyle\prod_{k=1}^n \left( \dfrac{k}{k+1} \right)^{H_{k-1}}$ | ||
| + | |} | ||
| + | </center> | ||
| + | |||
| + | =References= | ||
| + | * {{PaperReference|Square Integrability of Gaussian Bells on Time Scales|2005|Lynn Erbe|author2=Allan Peterson|author3=Moritz Simon|prev=findme|next=findme}}: Definition $2.30$ | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 15:03, 21 January 2023
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 $$\mathbf{E}(t)=e_{p}(t,0).$$
Properties
| $\mathbb{T}$ | $\mathbf{E}(t)$ |
| $\mathbb{R}$ | $e^{-\frac{t^2}{2}}$ |
| $\mathbb{Z}$ | $2^{\frac{-t(t-1)}{2}}$ |
| $h\mathbb{Z}$ | $\left[(1+h)^{\frac{1}{h}} \right]^{\frac{-t(t-h)}{2}}$ |
| $\mathbb{Z}^2$ | |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | $\displaystyle\prod_{k=\log_q(t)+1}^{\infty} \dfrac{1}{\left(1+(\frac{1}{q}-1)q^k \right)^{q^k}}$ |
| $\mathbb{H}$ | $\displaystyle\prod_{k=1}^n \left( \dfrac{k}{k+1} \right)^{H_{k-1}}$ |
References
- Lynn Erbe, Allan Peterson and Moritz Simon: Square Integrability of Gaussian Bells on Time Scales (2005)... (previous)... (next): Definition $2.30$
