Spherical mean theorems for solutions of the Helmholtz equation (Q416931)

The spherical mean theorem for solutions of the Helmholtz equation is discussed. Note that the proof for unbounded domains in \(\mathbb{R}^n\) is much more difficult. In this paper, the author obtains a result for a large class of domains that contain the half-space \(H=\{\vec{x}=\left(x_1,x_2,\ldots,x_n\right)\in \mathbb{R}^n\;:\;x_n>0\}\) under certain constraints in the growth of the function under consideration, thus extending an analogue of the theorem for domains of the form \(\mathbb{R}^n\backslash K\), where \(K\) is a convex compact set.