Given two functions $f(t)=t^2-2 t$ and $g(t)=t$ defined on the positive reals, I computed their convolution as $f * g(t)$ =$\frac{t^4}{12}-\frac{t^3}{3}$.
First of all, is that correct? How can I confirm that?
My main question is now regarding the interpretation of the graphs.
If I understood correctly, one of the functions (preferably the simpler one) is first reversed and then moved along the other function. At each point where the two functions overlap, they are multiplied together. The integral of the resulting products gives the value of the convolution.
So, how can one interpret the graph of the convolution function? What do the minima, maxima and zero points mean? Why is it decreasing in the beginning?
I feel very confused by this topic and would highly appreciate any help to further understand it. Thank you very much.
For checking your calculation I consider the Laplace transforms. $$\int_0^{\infty}e^{-st}\frac{t^n}{n!}\, dt=\frac{1}{s^{n+1}}.$$ Therefore the Laplace transform of your convolution is $$\left(\frac{2}{s^3}-\frac{2}{s^2}\right)\times \frac{1}{s^2}.$$ This is another way to see that your calculation $$\frac{t^4}{12}-\frac{t^3}{3}=\int_0^t(t-u)(u^2-2 u)du$$ is correct. Up to this, I do not understand your question about graphs.