Fourier series represent periodic functions as infinite sums of sines and cosines. The convergence of a Fourier series depends on the properties of the function it represents. Here we see a square wave function in red, and its Fourier approximations with different numbers of terms. As we increase the number of terms from 1 to 3 to 9, the approximation gets closer to the original function. However, notice the oscillations near the discontinuities - this is known as the Gibbs phenomenon.
Let's examine different types of convergence for Fourier series. Pointwise convergence means that at each point, the series converges to a specific value. For continuous functions, this value is simply the function value. At jump discontinuities, like the one shown here in yellow, the series converges to the average of the left and right limits. Uniform convergence is stronger - it means the series converges at the same rate across the entire domain. Notice the Gibbs phenomenon near the discontinuity - these oscillations don't disappear even as we increase the number of terms from 5 to 15. This is a fundamental limitation of Fourier series approximation near discontinuities.
L² convergence is a fundamental concept in Fourier analysis. It means that the integral of the squared difference between the function and its Fourier approximation approaches zero as we increase the number of terms. This is visualized here with the green area representing the squared error. For any square-integrable function, L² convergence is guaranteed. The Dirichlet conditions provide sufficient criteria for pointwise convergence of Fourier series. These include periodicity, a finite number of extrema and discontinuities, and boundedness. When these conditions are met, the Fourier series converges pointwise to the function at continuity points and to the average of left and right limits at discontinuities.
The Gibbs phenomenon is a fascinating feature of Fourier series approximations near jump discontinuities. As we zoom in on the discontinuity at x equals zero, we can observe that the Fourier approximations exhibit oscillations near the jump. The key characteristic of the Gibbs phenomenon is that while adding more terms makes the oscillations narrower, the height of the overshoot doesn't diminish. This overshoot approaches approximately 9 percent of the jump height, as indicated by the yellow marker. Notice how the approximations with 5, 20, and 100 terms all show similar maximum overshoot, but the oscillations become narrower and more concentrated around the discontinuity as we increase the number of terms. This phenomenon cannot be eliminated by adding more terms to the Fourier series - it's a fundamental limitation of representing discontinuous functions with continuous sine and cosine functions.
To summarize what we've learned about Fourier series convergence: Fourier series represent periodic functions as infinite sums of sines and cosines. There are different types of convergence to consider, including pointwise, uniform, and L-squared convergence. The Dirichlet conditions provide sufficient criteria for pointwise convergence, requiring periodicity, a finite number of extrema and discontinuities, and boundedness. At points of continuity, the series converges to the function value, while at jump discontinuities, it converges to the average of the left and right limits. The Gibbs phenomenon is a fundamental limitation that causes persistent oscillations near discontinuities, with a characteristic overshoot of approximately 9 percent of the jump height. Finally, L-squared convergence is guaranteed for all square-integrable functions, making Fourier series a powerful tool in mathematical analysis and applications.