Computing the Sparse Solution to Fourier Extension of Nonperiodic Function

For a given basis, the approximate solution to Fourier extension of Nonperiodic function computed with numerical least squares methods including projection methods and collocation methods are successful but not sparse. To solve the linear system equations built on projection methods by using Orthogonal Matching Pursuit algorithm, we extract some items from the original basis. We do optimization based on the low-dimensional basis generated by those items extracted from the original basis, thus we find the sparse solution to Fourier extension of Nonperiodic function without loss of approximation accuracy and unbounded Fourier coefficients.