Two algorithms for periodic extension on uniform grids

Given function values on a uniform grid in a domain Ω in ℝ^d, one is often interested in extending the values to a larger grid on a box B containing Ω. In particular, we are interested in “periodic extensions.” For such extensions the discrete Fourier transform (DFT) of the resulting grid values on B is expected to provide good efficient approximation to the underlying function on Ω. This paper presents two different extension algorithms. The first method is a natural approach to this problem, aiming at achieving the fastest decay of the DFT coefficients of the extended data.The second is a fast algorithm which is appropriate for the univariate case and for limited cases of multivariate scenarios. It is shown that if a “good” periodic extension exists, the proposed method will find an extension with similar properties.