Fast sparse nonlinear Fourier expansions of high dimensional functions

The nonlinear Fourier basis has shown its advantages over the classical Fourier basis in the time-frequency analysis. The need of processing large amount of high dimensional data motivates the extension of the methods based upon the nonlinear Fourier basis to high dimensions. We consider the multi-dimensional nonlinear Fourier basis, which is the tensor product of univariate nonlinear Fourier basis. We investigate the convergence order in norm and also the almost everywhere convergence of the nonlinear Fourier expansions. In order to compute fast and efficiently the nonlinear Fourier expansions of d-dimensional functions, we introduce the sparse nonlinear Fourier expansion and develop a fast algorithm for evaluating it. We also prove that the fast sparse nonlinear Fourier expansions enjoy the optimal convergence order and reduce the computational costs to O ( n log 2 d − 1 n ) . Numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed method.