Difference between revisions of "Cylinder transformation"
From timescalewiki
| Line 1: | Line 1: | ||
| − | Let $\mathbb{T}$ be a [[time scale]]. We define the cylinder transformation $\xi_h \colon \mathbb{C}_h \rightarrow \mathbb{Z}_h$, where $\mathbb{C}_h$ denotes the [[Hilger complex plane]] and $\mathbb{Z}_h | + | Let $\mathbb{T}$ be a [[time scale]]. We define the cylinder transformation $\xi_h \colon \mathbb{C}_h \rightarrow \mathbb{Z}_h$, where $\mathbb{C}_h$ denotes the [[Hilger complex plane]] and $\mathbb{Z}_h$ denotes the [[cylinder strip]], and is defined by the formula |
$$\xi_h(z)=\dfrac{1}{h} \mathrm{Log}(1+zh),$$ | $$\xi_h(z)=\dfrac{1}{h} \mathrm{Log}(1+zh),$$ | ||
where $\mathrm{Log}$ denotes the principal logarithm. | where $\mathrm{Log}$ denotes the principal logarithm. | ||
| Line 5: | Line 5: | ||
=See also= | =See also= | ||
[[Delta exponential]]<br /> | [[Delta exponential]]<br /> | ||
| + | |||
| + | =References= | ||
| + | *{{PaperReference|A generalized Fourier transform and convolution on time scales|2008|Robert J. Marks II|author2=Ian A. Gravagne|author3=John M. Davis|prev=Cylinder strip|next=Inverse cylinder transformation}}: Definition $2.3$ | ||
[[Category:Definition]] | [[Category:Definition]] | ||
Latest revision as of 00:52, 30 May 2017
Let $\mathbb{T}$ be a time scale. We define the cylinder transformation $\xi_h \colon \mathbb{C}_h \rightarrow \mathbb{Z}_h$, where $\mathbb{C}_h$ denotes the Hilger complex plane and $\mathbb{Z}_h$ denotes the cylinder strip, and is defined by the formula $$\xi_h(z)=\dfrac{1}{h} \mathrm{Log}(1+zh),$$ where $\mathrm{Log}$ denotes the principal logarithm.
See also
References
- Robert J. Marks II, Ian A. Gravagne and John M. Davis: A generalized Fourier transform and convolution on time scales (2008)... (previous)... (next): Definition $2.3$
