[kurlberg]

Convolution alone does not smooth. Eg consider a random variable supported on the pts 0 and 1 (delta masses at 2 pts.) No matter how many convolutions you do, you still have support on integers - not smooth at all. You need appropriate rescaling for a gaussian.

Also, convolving a distribution with itself is NOT a linear operation, hence cannot be described by a matrix multiplication with a fixed matrix.

[srean]

You are absolutely right. Even edge detection can be written as a convolution. That's why I mention averaging.

I address scaling, very peripherally, towards the end. Of course, depending on how you scale you end up with distinctly different limit laws.