Difference between revisions of "Gaussian bell"
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$$\mathbf{E}(t)=e_{p}(t,0).$$ | $$\mathbf{E}(t)=e_{p}(t,0).$$ | ||
| + | =Properties= | ||
| + | <center> | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Time Scale Gaussian Bells | |+Time Scale Gaussian Bells | ||
| Line 14: | Line 16: | ||
|- | |- | ||
|[[Integers | $\mathbb{Z}$]] | |[[Integers | $\mathbb{Z}$]] | ||
| − | |$ | + | |$2^{\frac{-t(t-1)}{2}}$ |
|- | |- | ||
|[[Multiples_of_integers | $h\mathbb{Z}$]] | |[[Multiples_of_integers | $h\mathbb{Z}$]] | ||
| Line 26: | Line 28: | ||
|- | |- | ||
|[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]] | |[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]] | ||
| − | | $\displaystyle\prod_{\log_q(t)+1}^{\infty} \dfrac{1}{\left(1+(\frac{1}{q}-1)q^k \right)^{q^k}}$ | + | | $\displaystyle\prod_{k=\log_q(t)+1}^{\infty} \dfrac{1}{\left(1+(\frac{1}{q}-1)q^k \right)^{q^k}}$ |
|- | |- | ||
|[[Harmonic_numbers | $\mathbb{H}$]] | |[[Harmonic_numbers | $\mathbb{H}$]] | ||
|$\displaystyle\prod_{k=1}^n \left( \dfrac{k}{k+1} \right)^{H_{k-1}}$ | |$\displaystyle\prod_{k=1}^n \left( \dfrac{k}{k+1} \right)^{H_{k-1}}$ | ||
|} | |} | ||
| + | </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 $2.30$
