defined as above and below the real axis and F is the distribution defined on the rectangle by the formula then F equals off the real axis and the distribution is induced by the distribution on the real axis. In particular if the hypotheses of the edge-of-the-wedge theorem apply, i.e. , then By elliptic regularity it then follows that the function F is holomorphic in . In this case elliptic regularity can be deduced directly from the fact that is known to provide a fundamental solution for the Cauchy–Riemann operator . Using the Cayley transform between the circle and the real line, this argument can be rephrased in a standard way in terms of Fourier series and Sobolev spaces on the circle. Indeed, let and be holomorphic functions defined exterior and interior to some arc on the unit circle such that locally they have radial limits in some Sobolev space, Then, letting the equations can be solved locally in such a way that the radial limits of G and F tend locally to the same function in a higher Sobolev space. For k large enough, this convergence is uniform by the Sobolev embedding theorem. By the argument for continuous functions, F and G therefore patch to give a holomorphic function near the arc and hence so do f and g. The general case A wedge is a product of a cone with some set. Let be an open cone in the real vector space , with vertex at the origin. Let E be an open subset of , called the edge. Write W for the wedge in the complex vector space , 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