On the convergence of the formal Fourier solution of the wave equation with a summable potential

The convergence of the formal Fourier solution to a mixed problem for the wave equation with a summable potential is analyzed under weaker assumptions imposed on the initial position $u(x,0)=\varphi(x)$ than those required for a classical solution up to the case $\varphi(x)\in L_p[0,1]$ for $p>1$. It is shown that the formal solution series always converges and represents a weak solution of the mixed problem.