Pseudospectral Fourier reconstruction with IPRM

The Inverse Polynomial Reconstruction Method (IPRM) has been recently introduced by J.-H. Jung and B. Shizgal in order to remedy the Gibbs phenomenon, see [2], [3], [4], [5]. Their main idea is to reconstruct a given function from its n Fourier coefficients as an algebraic polynomial of degree n− 1. This leads to an n × n system of linear equations, which is solved to find the Legendre coefficients of the polynomial. This approach is motivated by the classical observation that a smooth, function allows an efficient representation through its Legendre series. In particular, if a function has an analytic extension to a larger domain, its Legendre coefficients decay exponentially. In principle, the function can be efficiently reconstructed from its Fourier data indirectly by first computing its Legendre coefficients. Several fundamental aspects of IPRM are still investigated. A rigorous proof of existence of the reconstruction was published only recently by Michel Krebs, see [1]. A major drawback of IPRM is that the condition number of the underlying n×n matrix grows approximately like O(e), which quickly leads to ill-conditioning. For this reason, IPRM fails to converge in the case of a meromorphic function, whose singularities are located sufficiently close to the interval where the function is defined. This happens because the function’s Legendre series does not converge fast enough to mitigate the exponential growth of the condition numbers. To resolve this problem, we propose a modified version of IPRM, which achieves pseudospectral convergence even for restrictions of meromorphic functions. We reconstruct a function as an algebraic polynomial of degree n−1 from the function’s m lowest Fourier coefficients, as long as m > n. We