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  1. For functions $f,g,h$ that are defined over $\mathbb{R}$, suppose we have a convolution equation: $$ f = g * h. $$

    I would like to convert it into a differential equation. Is it correct that $$ \frac{df}{dt} = \frac{dg}{dt} * h $$ under some conditions (unclear to me yet, differential under integral sign?)?

    Why when the Laplace transform G of g is a rational function, the convolution is generally converted to a higher order differential equation, instead of first order one like above?

    Why when the Laplace transform G of g is not a rational function, the convolution is generally converted to a infinite order differential equation?

  2. Similarly, for functions $f,g,h$ that are defined over $\mathbb{Z}$, suppose we have a convolution equation: $$ f = g * h. $$

    I would like to convert it into a difference equation. Is it correct that $$ df = dg * h $$ where $$ df(n) := f(n+1) - f(n), $$ $$ dg(n) := g(n+1) - g(n) $$ under some conditions (unclear to me yet)?

    Why when the Z-transform G of g is a rational function, the convolution is generally converted to a higher order difference equation, instead of first order one like above?

    Why when the Z-transform G of g is not a rational function, the convolution is generally converted to a infinite order difference equation?

The above questions arose from my difficulty understanding a reply by leonbloy. Thanks and regards!

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