Some isometrical identities in the wave equation (Q1061929)

The wave equation \(\partial^ 2u(x,t)/\partial t^ 2=c^ 2(\partial^ 2u(x,t)/\partial x^ 2),\) (c: \(const.>0)\), is considered satisfying one of the conditions: (1) on the space \(x\geq 0\), \(u(0,t)=F(t)\) and \(u(x,0)=0\), (2) on the space \(-\infty <x<\infty\), \(u_ t(x,t)|_{t=0}=F(x)\) and \(u(x,0)=0\) and (3) on the space \(-\infty <x<\infty\), \(u(x,0)=F(x)\) and \(u_ t(x,t)|_{t=0}=0\). In each case, an isometrical identity and inverse formula between u(x,t) and F are established.