pr.probability - What is convolution intuitively?

Here is my take on this. $newcommand{bZ}{mathbb{Z}}$ $newcommand{bR}{mathbb{R}}$ Discretize the real axis and thing of it as the collection of point $Lambda_hbar:=hbar bZ$, where $hbar>0$ is a small number. Then a function $f:Lambda_hbarto bR$ is determined by its generating function, i.e., the formal power series $newcommand{ii}{boldsymbol{i}}$

$$G^hbar_f(t)=sum_{ninbZ}f(nhbar)t^n. $$

Then

$$G^hbar_{f_0ast f_1}(t)= G^hbar_{f_0}(t)cdot G^hbar_{f_1}(t).tag{1} $$

Observe that if we set $t=e^{-iixi hbar}$, then

$$G^hbar_f(t)=sum_{xinLambda_hbar} f(x) e^{-ii xi x}. $$

Moreover

$$ hbar G^hbar_f(e^{-iixi hbar})=sum _{nin bZ} hbar f(nhbar) e^{-iixi(nhbar)}, tag{2}$$

and the expression in the right hand sum is a "Riemann sum" approximating

$$int_{bR} f(x)^{-iixi x} dx. $$

Above we recognize the Fourier transform of $f$. If we let $hbarto 0$ in (ref{2}) and we use (ref{1}) we obtain the wellknown fact that the Fourier transform maps the convolution to the usual pointwise product of functions. (The fact that this rather careless passing to the limit can be rigorous is what the Poisson formula is all about.)

The above argument shows that we can regard $hbar G_f^hbar(1)$ as an approximation for $int_{bR} f(x) dx$.

Denote by $delta(x)$ the Delta function 9concentrated at $0$. The Delta function concentrated at $x_0$ is then $delta(x-x_0)$. What could be the generating function of $delta(x)$, $Gdelta^hbar$? First, we know that $delta(x)=0$, $forall xneq 0$ so that

$$G_delta^hbar(t) =ct^0=c. $$

The constant $c$ can be determined from the equality

$$ 1= int_{bR} delta(x) dx=hbar G_delta^hbar(1)=hbar c$$

Hence $hbar G_delta^hbar(1)=1$. Similarly

$$ G^hbar_{delta(cdot-nhbar)} =frac{1}{hbar} t^n. $$

Putting together all of the above we obtain an equivalemt description for the generating functon af a function $f:Lambda_hbartobR$. More precisely

$$ G_f(t)=hbarsum_{lambdainLambda_hbar}f(lambda) G_{delta(cdot-lambda)}(t). $$

The last equality suggests an interpretation for the generating function as an algebraic encoding of the fact that $f:Lambda_hbartobR$ is a superposition of $delta$ functions concentrated along the points of the lattice $Lambda_hbar$.