Difference between revisions of "Unilateral convolution"
From timescalewiki
| (3 intermediate revisions by the same user not shown) | |||
| Line 9: | Line 9: | ||
[[Shift of unilateral convolution]]<br /> | [[Shift of unilateral convolution]]<br /> | ||
| − | + | =See also= | |
| − | + | [[Shifting problem]] | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | + | =References= | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | + | [[Category:Definition]] | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | [[ | ||
Latest revision as of 15:21, 21 January 2023
For $t \in \mathbb{T}$, the convolution on a time scale is defined by the formula $$(f*g)(t,s)=\displaystyle\int_{s}^t \hat{f}(t,\sigma(\xi))g(\xi)\Delta \xi,$$ where $\hat{f}$ denotes the solution of the shifting problem. The classic definition of the convolution using a shift in the integrand is not appropriate for time scales, since a time scale is not closed under addition and subtraction, but this definition does reduce to the classical definition in the cases of $\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=\mathbb{Z}$.
Properties
Covolution theorem for unilateral Laplace transform
Unilateral convolution is associative
Delta derivative of unilateral convolution
Shift of unilateral convolution
