Edge-of-the-wedge Theorem - The General Case

The General Case

A wedge is a product of a cone with some set.

Let C be an open cone in the real vector space Rn, with vertex at the origin. Let E be an open subset of Rn, called the edge. Write W for the wedge in the complex vector space Cn, and write W' for the opposite wedge . Then the two wedges W and W' meet at the edge E, where we identify E with the product of E with the tip of the cone.

• Suppose that f is a continuous function on the union that is holomorphic on both the wedges W and W' . Then the edge-of-the-wedge theorem says that f is also holomorphic on E (or more precisely, it can be extended to a holomorphic function on a neighborhood of E).

The conditions for the theorem to be true can be weakened. It is not necessary to assume that f is defined on the whole of the wedges: it is enough to assume that it is defined near the edge. It is also not necessary to assume that f is defined or continuous on the edge: it is sufficient to assume that the functions defined on either of the wedges have the same distributional boundary values on the edge.