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Where $f, g$ are functions, $\delta$ is the Dirac Delta distribution ('aka Delta function'), and L is a linear (LTI) operator,

if $g = L(f)$, then can we say in all cases $g = Li * f$ , where $ Li $ is defined as $L(\delta)$, i.e. the impulse response?

My questions: Is this true?

Any references would be appreciated.

Linear operator References:
http://mathworld.wolfram.com/LinearOperator.html
http://vergil.chemistry.gatech.edu/notes/quantrev/node14.html
https://www.encyclopediaofmath.org/index.php/Linear_operator

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1 Answer 1

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If $L$ is well-defined linear operator $L^1(\mathbb{R}) \to L^\infty(\mathbb{R}) $ then $L[f(y)](x) = h \ast f(x)$ for some $h \in L^\infty(\mathbb{R})$ if and only if $$\forall a, \qquad L[f(y-a)](x) = L[f(y)](x-a)$$ That's why we say "linear and time-invariant (LTI) operators".

The same theoy of LTI operators works in $L^\infty(\mathbb{R}),L^2(\mathbb{R}), C^0(\mathbb{R}), D(\mathbb{R})$ and $D'(\mathbb{R})$. The Dirac delta $\delta$ belongs to this last space : it is a distribution.

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    $\begingroup$ @user45664 Yes $L$ is LTI and if it converges then $h(x) = L[\delta](x) = \lim_{\epsilon \to 0} L[\frac{1_{|y| <\epsilon}}{2 \epsilon}](x)$ $\endgroup$
    – reuns
    Apr 22, 2017 at 18:22
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    $\begingroup$ @user45664 Assume that $f \ast h = 0$ for every $f$ so that $h(x) = 0$ "everywhere". Then $h(0) = 0$ or $h(0) = 1$ doesn't change anything under convolution. So it is unique up to isolated values (sets of measure $0$). $\endgroup$
    – reuns
    Apr 22, 2017 at 18:35
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    $\begingroup$ So you need to state it like this : if $L$ is LTI and $h(x) = \lim_{\epsilon \to 0} L[\frac{1_{|y| <\epsilon}}{2 \epsilon}](x)$ is well-defined, then $L[f] = f \ast h$ @user45664 $\endgroup$
    – reuns
    Apr 22, 2017 at 18:37
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    $\begingroup$ @user45664 Yes, this is a rectangle approaching $\delta$ in the limit, and a rigorous definition of $\delta$. $\endgroup$
    – reuns
    Apr 22, 2017 at 18:51
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    $\begingroup$ is there a 'name' (theorem, or ..?) for what you stated in your answer? :) $\endgroup$
    – user45664
    Jun 7, 2017 at 21:31

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