Fundamental solution

In a well-posed linear equation, the solution is given in terms of the initial data via a linear operator. The kernel of that operator (which may be only a distribution) is then the fundamental solution of the equation. Another way of viewing this is that the fundamental solution is the solution associated to initial data which is a Dirac delta function. Since every function is (formally, at least) a superposition of delta functions, by the principle of superposition every solution is thus a superposition of fundamental solutions.