Nonlinear approximation of functions in two dimensions by sums of exponential functions

We consider the problem of approximating a given function in two dimensions by a sum of exponential functions, with complex-valued exponents and coefficients. In contrast to Fourier representations where the exponentials are fixed, we consider the nonlinear problem of choosing both the exponents and coefficients. In this way we obtain accurate approximations with only few terms. Our approach is built on recent work done by G. Beylkin and L. Monzón in the one-dimensional case. We provide constructive methods for how to find the exponents and the coefficients, and provide error estimates. We also provide numerical simulations where the method produces sparse approximations with substantially fewer terms than what a Fourier representation produces for the same accuracy.