Interpolation for solutions of the Helmholtz equation

We study the interpolation problem for solutions of the two‐dimensional Helmholtz equation, which are sampled along a line. The data are the function values and the normal derivatives at a discrete set of point sensors. A wave transform is used, analogous to the common Fourier transform. The inverse wave transform defines the Hilbert space for oscillatory Helmholtz solutions. We thereby introduce an interpolant that has some advantages over the usual sinc x in the Whittaker–Shannon sampling in one dimension; in particular, coefficients of the two‐dimensional solution are invariant under translations and rotations of the sampling line. The analysis is relevant for the optical sampling problem by sensors on a screen.