Generalizations of the Wave Equation

The main result of this paper is a generalization of the property that, for smooth u, uxy = 0 implies (*) u(x, y) = a(x) + b(y). Any function having generalized unsymmetric mixed partial derivative identically zero is of the form (*). There is a function with generalized symmetric mixed partial derivative identically zero not of the form (*), but (*) 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 L2 generalized symmetric mixed partial derivative of its formal (x, >>)-integral. In this paper we introduce some generalized versions of the operator d2/dxdy and use them to extend the fact that for smooth real-valued functions on R2, d2f/dxdy = 0 implies that f(x, y) = a(x) + b(y). In §1 we outline our results and in §5 we explain the connection of this material to the theory of uniqueness for multiple trigonometric series.