Difference between revisions of "Unilateral convolution"
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For $t \in \mathbb{T}$, the convolution on a [[time scale]] is defined by the formula | For $t \in \mathbb{T}$, the convolution on a [[time scale]] is defined by the formula | ||
$$(f*g)(t,s)=\displaystyle\int_{s}^t \hat{f}(t,\sigma(\xi))g(\xi)\Delta \xi,$$ | $$(f*g)(t,s)=\displaystyle\int_{s}^t \hat{f}(t,\sigma(\xi))g(\xi)\Delta \xi,$$ | ||
| − | 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. | + | 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= | =Properties= | ||
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[[Unilateral convolution is associative]]<br /> | [[Unilateral convolution is associative]]<br /> | ||
[[Delta derivative of unilateral convolution]]<br /> | [[Delta derivative of unilateral convolution]]<br /> | ||
| + | [[Shift of unilateral convolution]]<br /> | ||
| − | + | =See also= | |
| − | + | [[Shifting problem]] | |
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| − | + | =References= | |
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| − | + | [[Category:Definition]] | |
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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
