How exactly do Gaussian Processes (square dist kernel) enforce smoothness? (Aka how are they computed to do so?) From:
http://www.cs.cmu.edu/~16831-f12/notes/F10/16831_lecture22_jlisee/16831_lecture22.jlisee.pdf

"Gaussian Processes artificially introduce correlation between close samples in that vector in order to
enforce some sort of smoothness on the succession of samples."

But how is this computed? Is the function f(x) ~ GP(mu,k(x,x')) performed incrementally?
e.g. 
the n'th calculated value f(xn) uses values f(x-1)...f(x-n) to compute its mean and variance?
It's by definition, when you fit a guassian process you specify the mean function m(x) and the covariance function (or kernel) k(x,x'). Often the mean function is 0 and the covariance is the radial basis function or squared expential kernel which is smooth (in fact infinitly differentiable).