 Solution

A smooth $2\pi$-periodic function has coefficients $g_k=(2\pi {)}^{-1}{\int }_0^{2\pi }g(x)e^{-ikx}dx$ decaying faster than any power, by repeated integration by parts. Thus its Fourier series and every differentiated series converge absolutely and uniformly to the corresponding function and derivatives. Solving each spatial Fourier mode givesThe zero-frequency term is linear in time, not obtained by division by zero. Uniform convergence on compact time intervals permits termwise differentiation and verifies the initial conditions and the wave equation.