 Eigenfunction expansion

An eigenfunction expansion represents a function as a sum or integral of eigenfunctions of a specified linear operator. In a Hilbert space with a complete orthonormal basis of eigenfunctions ${\varphi }_n$, one writes $f={\sum }_n{c}_n{\varphi }_n$, where $c_n=⟨{\varphi }_n,f⟩$ under the convention that the inner product is linear in its second argument. Convergence is initially in the Hilbert space norm; pointwise convergence needs additional hypotheses.

For the operator $-d^2/d{x}^2$ on $(0,\pi )$ with zero endpoint values, the normalized eigenfunctions are $\sqrt{2/\pi }\sin (nx)$ with eigenvalues $n^2$. The heat equation then evolves each coefficient independently as $c_n(t)=e^{-n^{2}t}{c}_n(0)$. This is why eigenfunction expansions turn a spatial differential equation into scalar evolution equations. Not every linear operator has such an eigenbasis.