Rational Approximation and Coefficient Recovery

Abstract

We present a new framework for recovering Chebyshev coefficients of non-periodic functions using rational approximation. Building on the AAA algorithm, we construct a type (m−1, m) rational approximant whose poles and residues can be efficiently computed. Using the change of variable x = cos θ, the approximant is transformed into a rational trigonometric function in θ, whose Fourier coefficients can be expressed as short exponential sums via Fourier inversion. Mapping back, these exponentials provide a reconstruction of the original function’s Chebyshev expansion.We demonstrate that our method (i) achieves high-accuracy Chebyshev coefficient recovery from both equally spaced and clustered data, (ii) outperforms classical polynomial-based approaches for functions exhibiting slow coefficient decay or singular behavior, and (iii) supports fundamental arithmetic operations and differentiation directly in the coefficient domain.