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 [[exponential_function | 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]]
 
$$\mathbf{E}(t)=e_{p}(t,0).$$
 
$$\mathbf{E}(t)=e_{p}(t,0).$$
  
==References==
+
=Properties=
<references>
+
<center>
<ref name=gaussbell>[http://www.math.unl.edu/~apeterson1/pub/epsgauss.pdf 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>
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{| class="wikitable"
</references>
+
|+Time Scale Gaussian Bells
 +
|-
 +
|$\mathbb{T}$
 +
|$\mathbf{E}(t)$
 +
|-
 +
|[[Real_numbers | $\mathbb{R}$]]
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|$e^{-\frac{t^2}{2}}$
 +
|-
 +
|[[Integers | $\mathbb{Z}$]]
 +
|$2^{\frac{-t(t-1)}{2}}$
 +
|-
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|[[Multiples_of_integers | $h\mathbb{Z}$]]
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| $\left[(1+h)^{\frac{1}{h}} \right]^{\frac{-t(t-h)}{2}}$
 +
|-
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| [[Square_integers | $\mathbb{Z}^2$]]
 +
|
 +
|-
 +
|[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q &gt; 1$]]
 +
|
 +
|-
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|[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q &lt; 1$]]
 +
| $\displaystyle\prod_{k=\log_q(t)+1}^{\infty} \dfrac{1}{\left(1+(\frac{1}{q}-1)q^k \right)^{q^k}}$
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|-
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|[[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

Time Scale Gaussian Bells
$\mathbb{T}$ $\mathbf{E}(t)$
$\mathbb{R}$ $e^{-\frac{t^2}{2}}$
$\mathbb{Z}$ $2^{\frac{-t(t-1)}{2}}$
$h\mathbb{Z}$ $\left[(1+h)^{\frac{1}{h}} \right]^{\frac{-t(t-h)}{2}}$
$\mathbb{Z}^2$
$\overline{q^{\mathbb{Z}}}, q > 1$
$\overline{q^{\mathbb{Z}}}, q < 1$ $\displaystyle\prod_{k=\log_q(t)+1}^{\infty} \dfrac{1}{\left(1+(\frac{1}{q}-1)q^k \right)^{q^k}}$
$\mathbb{H}$ $\displaystyle\prod_{k=1}^n \left( \dfrac{k}{k+1} \right)^{H_{k-1}}$

References