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