  We analyze and test using Fourier extensions that minimize a Hilbert space
norm for the purpose of solving partial differential equations (PDEs) on
surfaces. In particular, we prove that the approach is arbitrarily high-order
and also show a general result relating boundedness, solvability, and
convergence that can be used to find eigenvalues. The method works by extending
a solution to a surface PDE into a box-shaped domain so that the differential
operators of the extended function agree with the surface differential
operators, as in the Closest Point Method. This differs from approaches that
require a basis for the surface of interest, which may not be available.
Numerical experiments are also provided, demonstrating super-algebraic
convergence. Current high-order methods for surface PDEs are often limited to a
small class of surfaces or use radial basis functions (RBFs). Our approach
offers certain advantages related to conditioning, generality, and ease of
implementation. The method is meshfree and works on arbitrary surfaces (closed
or non-closed) defined by point clouds with minimal conditions.
