2022/01/26 by Gröchenig, Karlheinz, Hrycak, Tomasz
#Fourier series #IPRM #inverse methods #pseudospectral methods
paper · doi:10.4230/dagsemproc.08492.6
We generalize the Inverse Polynomial Reconstruction Method (IPRM) for mitigation of the Gibbs phenomenon by reconstructing a function as an algebraic polynomial of degree n-1 from the function's m lowest Fourier coefficients (m ge n). We compute approximate Legendre coefficients of the function by solving a linear least squares problem, and we show that the condition number of the problem does not exceed sqrtfracmm-alpha0 n2, where alpha0 = frac4sqrt2pi2 = 0.573 ldots. Consequently, whenever mboxm ge n2, the convergence rate of the modified IPRM for an analytic function is root exponential on the whole interval of definition. Stability and accuracy of the proposed algorithm are validated with numerical experiments.