Gaussian bell
From timescalewiki
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 $2.30$
