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.

Fejér's theorem instead considers the arithmetic mean of the first N Fourier partial sums. This averaging process strongly dampens higher-frequency oscillations. As a result, the Fejér means do not exhibit the Gibbs overshoot and uniformly converge for continuous periodic functions.

Worked Example: N = 1

Consider the square wave model f(x) = sign(sin x) evaluated at x = π/4.

The pointwise limit for x = π/4 is 1.
The degree N=0 partial sum is S₀ = 0.
The degree N=1 partial sum includes only the first mode:S₁ = S₀ + 4 sin(π/4) / π ≈ 0.900316.
The Fejér mean is the average of the partial sums up to N:σ₁ = (S₀ + S₁) / 2 = (0 + 0.900316) / 2 ≈ 0.450158.

Experiment

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

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.
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.
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.

Limitations

Although Fejér averaging improves convergence properties in some respects, it has important limitations to keep in mind when applying it to approximations of periodic functions like the square wave.

While Fejér means eliminate the Gibbs overshoot, they converge much more slowly to the true function away from the discontinuities compared to the raw Fourier series.
This module evaluates the specific square wave f(x) = sign(sin x). It does not act as a general Fourier or Fejér series solver for arbitrary functions.
Uniform convergence of the Fejér means applies only to continuous periodic functions. The square wave is discontinuous, so uniform convergence does not hold across the entire interval, even for the averaged sums.

References

The following reference material provides foundational definitions of Fourier series, partial sums, and convergence properties used in constructing the models presented in this module.

MIT Math 103 formulas midterm: Includes fundamental definitions of Fourier series and convergence properties. formulasmidterm.pdf

Fejér Fourier Averaging