Comment by BigTTYGothGF

Harmonic analysis is "just" working with harmonic functions which you have enough background for: f(x,y) is harmonic if f_xx+f_yy=0 (f_xx=2nd partial derivative w/r/t/ x, and so on). Higher dimensions work the same: f_{x1x1} + f_{x2x2} + f_{x3x3}+.... = 0.