To smooth an image we use often use a 3 by 3 matrix of ones divided by the average or mean filter Intuitively I understand that we take the average gray levels value so there will be no "high" or "low" values, but mathematically how can one show it? and how much the noise will be reduced
What you have is a convolution with a box filter. This has a meaningful interpretation in the Fourier domain where the convolution turns into a multiplication and the box filter turns into a sinc that is high near the DC and low for higher frequencies, which means that it attenuates the higher frequencies of your signal. So not only does it diminishes noise but it attenuates all high frequencies regardless whether they are meaningful (textures for example). If you do iterated box filtering it approximates a gaussian and you get a gaussian convolution which can be regarded as a solution to the heat diffusion pde (note that it blurs everything - meaning that it's not edge aware).