Section 25.7
Convergence of Fourier Series
To what value does the infinite series actually converge?
1
Introduction
Most of the time, the series equals the function.
At points of discontinuity (jumps), it converges to the average of the jump.
2
Convergence Theorem
Dirichlet Conditions
If and are piecewise continuous on , then the Fourier series converges to:
At points where f is continuous, this is just .
3
Gibbs Phenomenon
Near a jump discontinuity, the partial sums "overshoot" the function by about 9%.
As , the overshoot moves closer to the jump but does not disappear in amplitude.
4
Worked Examples
Example 1: Continuous Point
on .
At , it converges to .
At endpoints : .
Since boundaries match, it converges to .
Example 2: Jump
for , for .
At , left limit is -1, right limit is 1.
Series converges to .
This is why Fourier Series for signum function is 0 at the origin.
Example 3: Visual Gibbs
Zoom in near zero to see the overshoot!
4
Practice Quiz
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