Difference between revisions of "Unilateral convolution"

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Let $f,g \colon \mathbb{R} \rightarrow \mathbb{R}$ be [[Lebesgue integrable]] on $\mathbb{R}$. The classical (i.e. [[time scale]] $\mathbb{T}=\mathbb{R}$) convolution of $f$ and $g$ is the function $f*g \colon \mathbb{R} \rightarrow \mathbb{R}$ given by
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For $t \in \mathbb{T}$, the convolution on a [[time scale]] is defined by the formula
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$$(f*g)_{\mathbb{R}}(t)=\displaystyle\int_{\mathbb{R}} f(\tau)g(t-\tau) \mathrm{d}\tau.$$
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$$(f*g)(t,s)=\displaystyle\int_{s}^t \hat{f}(t,\sigma(\xi))g(\xi)\Delta \xi,$$
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The reason the convolution is of interest is because of the so-called convolution theorem for the classical Laplace transform:
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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 $\mathbb{T}=\mathbb{Z}$.  
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$$\mathscr{L}_{\mathbb{R}}\{f*g\}(z)=\mathscr{L}_{\mathbb{R}}\{f\}(z)\mathscr{L}_{\mathbb{R}}\{g\}(z).$$
 
  
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Let $\mathbb{T}$ be any time scale and $f,g \colon \mathbb{T} \rightarrow \mathbb{C}$ be [[delta integrable]] on $\mathbb{T}$. We cannot simply use the definition of the convolution for time scales because an arbitrary time scale is not closed under addition and subtraction. The integrand $f(s)g(t-s) \mathrm{d}s$ will be written in the time scale convolution as $\hat{f}(t,\sigma(s))g(s) \Delta s$. Thus we define for $t \in \mathbb{T}$,
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=Properties=
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$$(f*g)_{\mathbb{T}}(t;t_0)=\displaystyle\int_{t_0}^t \hat{f}(t,\sigma(s))g(s)\Delta s.$$
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[[Covolution theorem for unilateral Laplace transform]]<br />
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[[Unilateral convolution is associative]]<br />
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[[Delta derivative of unilateral convolution]]<br />
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[[Shift of unilateral convolution]]<br />
  
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=Properties=
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=See also=
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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[[Shifting problem]]
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<strong>Theorem:</strong> (Convolution theorem) The following formula holds:
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=References=
  
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where $\mathscr{L}_{\mathbb{T}}$ denotes the [[Laplace transform|Laplace transform]].
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[[Category:Definition]]
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> █
 
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</div>
 
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</div>
 

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

See also

Shifting problem

References

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