Pseudospectral Fourier reconstruction with the modified Inverse Polynomial Reconstruction Method

Abstract

We generalize the Inverse Polynomial Reconstruction Method (IPRM) for mitigation of the Gibbs phenomenon by reconstructing a function from its m lowest Fourier coefficients as an algebraic polynomial of degree at most n - 1 ( m ⩾ n ) . We compute approximate Legendre coefficients of the function by solving a linear least squares problem. We show that if m ⩾ n 2 , the condition number of the problem does not exceed 2.39. Consequently, if m ⩾ n 2 , the convergence rate of the modified IPRM for an analytic function is root exponential on the whole interval of definition. Numerical stability and accuracy of the proposed algorithm are validated experimentally.