Fejér Fourier Averaging

The Fourier series of a piecewise smooth function converges pointwise to the function value at points of continuity, and to the midpoint at jump discontinuities. However, near these jumps, truncating the series at degree N exhibits an overshoot known as theGibbs phenomenon, which does not vanish as N increases.

Experiment

In the interactive module, observe how changing the degree affects the approximation:

1. Set Degree N to 1 and Probe fraction of pi to 1/4. Notice the significant error in both the Fourier partial sum and the Fejér mean due to underfitting.
2. Increase Degree N to 7. The Fourier partial sum becomes closer to 1 (error ≈ 0.108), but still suffers from oscillations near the jump at x = 0. The Fejér mean smooths out these oscillations but approaches the true value more slowly.
3. Set Probe fraction of pi to 0, exactly at the jump discontinuity. Observe that both approximations correctly evaluate to 0.0, matching the midpoint limit.