📊 Fejer's Construction

A Continuous Function with Divergent Fourier Series

Mathematical Insight: This demonstrates Fejer's classical example where a perfectly continuous function f(x) has a Fourier series that diverges at specific points, challenging the intuition that continuous functions always have convergent Fourier series.
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🎮 Controls

View Selection

1 Convergent View
LINEAR lambda=5*nu, weight=1/nu^2
2 Divergent View
EXPONENTIAL lambda=2^(nu+1), weight=1/sqrt(nu)

Parameters

↑ ↓ Adjust Groups (ν)

Animation

SPACE Pause/Resume
R Reset Animation
G Toggle Grid

📖 Understanding the Visualization

Left Panel - f(x): The continuous function over [0, 2Ï€]. Notice it's always bounded and smooth.

Right Panel - Sn(0): Partial Fourier sum values at x=0 as terms are added.

  • View 1 - Convergent (λ=5ν, w=1/ν²): Linear growth with rapid weight decay → partial sums quickly settle down
  • View 2 - Divergent (λ=2ν+1, w=1/√ν): Exponential growth with slower weight decay → growing spikes reveal the "trap"

The "trap": Each group adds positive terms (spike up), then negative terms (spike down). In divergent mode, amplitudes grow unboundedly despite f(x) being continuous!

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Mathematical Construction: Fejér (1910) | C99 Implementation