Since Fourier introduced the Fourier series to solve the heat equation, the Fourier or polynomial approximation has served as a useful tool in solving various problems arising in industrial applications. If the function to approximate with the finite Fourier series is smooth enough, the error between the function and the approximation decays uniformly. If, however, the function is nonperiodic or has a jump discontinuity, the approximation becomes oscillatory near the jump discontinuity and the error does not decay uniformly anymore. This is known as the Gibbs-Wilbraham phenomenon. The Gibbs phenomenon is a theoretically well-understood simple phenomenon, but its resolution is not and thus has continuously inspired researchers to develop theories on its resolution. Resolving the Gibbs phenomenon involves recovering the uniform convergence of the error while the Gibbs oscillations are well suppressed. This talk explains recent progresses on the resolution of the Gibbs phenomenon focusing on the discussion of how to recover the uniform convergence from the Fourier partial sum and its numerical implementation. There is no best methodology on the resolution of the Gibbs phenomenon and each methodology has its own merits with differences demonstrated when implemented. This talk also explains possible issues when the methodology is implemented numerically. The talk is intended for a general audience.
