Generalizations of the wave equation

Abstract:

The main result of this paper is a generalization of the property that, for smooth $u$, ${u_{xy}} = 0$ implies $(\ast )$ \[ u(x,y) = a(x) + b(y).\] Any function having generalized unsymmetric mixed partial derivative identically zero is of the form $(\ast )$. There is a function with generalized symmetric mixed partial derivative identically zero not of the form $(\ast )$, but $(\ast )$ does follow here with the additional assumption of continuity. These results connect to the theory of uniqueness for multiple trigonometric series. For example, a double trigonometric series is the ${L^2}$ generalized symmetric mixed partial derivative of its formal $(x,y)$-integral.