Suppose we have smooth function (for example temperature) defined on sphere. We know everything about its behaviour on its surface.
Does this determines all the values inside sphere? If so, what the formula looks like? If don't ,how the values generally behaves?
I strongly belive that it determines because it works on complex plane in form of Cauchy integral formula. One can just modify it and the result schould follows.
On a real (unit) sphere $\mathbb{S}^2 \subseteq \mathbb{R}^3$ hardly anything can be said. Consider the constant scalar field $f(x) =1$ and $g(x,y,z) = (x^2,y^2,z^2)$.