I have a discrete signal (an image actually), which I am convolving/deconvolving with a zero-mean Gaussian kernel. I would like some proof that these operations do not alter the signal mean. Well, it would suffice in continuous case. Does this have to do something with the result's DC component being the product/fraction of the two signals' DC components? I have searched extensively, but found nothing of relevance. Please help.
EDIT: So, to avoid confusion zero-mean Gaussian is one with zero expected value, meaning it's symmetric to the y axis, and it's not about its DC component.
I've found out that a normalized Gaussian kernel (with its full-domain integral equal to 1) has its DC component as 1, leading to an unchanged signal mean.