Difference between revisions of "Delta derivative of unilateral convolution"
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==Theorem== | ==Theorem== | ||
If $f$ is [[delta derivative|$\Delta$-differentiable]], then | If $f$ is [[delta derivative|$\Delta$-differentiable]], then | ||
| − | $$(f*g)^{\Delta}=f^{\Delta}*g+f(t_0)g | + | $$(f*g)^{\Delta}=f^{\Delta}*g+f(t_0)g,$$ |
| − | + | and if $g$ is [[delta derivative|$\Delta$-differentiable]], then | |
| − | $$(f*g)^{\Delta}=f*g^{\Delta}+fg(t_0) | + | $$(f*g)^{\Delta}=f*g^{\Delta}+fg(t_0),$$ |
| + | where $(f*g)$ denotes [[unilateral convolution]]. | ||
==Proof== | ==Proof== | ||
Revision as of 13:41, 20 January 2023
Theorem
If $f$ is $\Delta$-differentiable, then $$(f*g)^{\Delta}=f^{\Delta}*g+f(t_0)g,$$ and if $g$ is $\Delta$-differentiable, then $$(f*g)^{\Delta}=f*g^{\Delta}+fg(t_0),$$ where $(f*g)$ denotes unilateral convolution.
