Difference between revisions of "Unilateral convolution"
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[[Delta derivative of unilateral convolution]]<br /> | [[Delta derivative of unilateral convolution]]<br /> | ||
[[Shift of unilateral convolution]]<br /> | [[Shift of unilateral convolution]]<br /> | ||
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=See also= | =See also= | ||
[[Shifting problem]] | [[Shifting problem]] | ||
Revision as of 13:48, 20 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
