Intuitive Understanding of Convergence
When does a Fourier series converge to the original function?
Introduction (High School Level)
Introduction
When we truncate a Fourier series to a finite number of terms, how close is it to the original function? Does adding more terms really make it converge to the original function? This chapter builds an intuitive understanding of convergence.
Approximation by Partial Sums
The sum of the first $N$ terms of a Fourier series is called a partial sum:
$$S_N(x) = \dfrac{a_0}{2} + \displaystyle\sum_{n=1}^{N}\left(a_n\cos nx + b_n\sin nx\right)$$
The case of the square wave
- $N = 1$: $S_1(x) = \dfrac{4}{\pi}\sin x$ (a smooth sine wave)
- $N = 3$: $S_3(x) = \dfrac{4}{\pi}\left(\sin x + \dfrac{\sin 3x}{3}\right)$ (slightly more angular)
- $N = 10$: gets quite close to the square wave
- $N \to \infty$: converges to the square wave
The larger $N$ is, the more high-frequency components are added and the finer the details that are reproduced.
Smoothness and Convergence
The speed of convergence depends on the smoothness of the function.
Smooth functions
- Fourier coefficients decay rapidly (as $1/n^k$ with large $k$)
- A good approximation is obtained with few terms
- Example: $\sin x$, $\cos x$ (which can be represented exactly with a single term)
Discontinuous functions
- Coefficients decay slowly (roughly as $1/n$)
- Many terms are needed
- Example: square wave, sawtooth wave
Intuitive understanding
Representing abrupt changes (discontinuities or corners) requires high-frequency components. A smooth function can be represented well with only low-frequency components. In general, the more times a function can be differentiated, the faster its Fourier coefficients decay.
The Gibbs Phenomenon
Approximating a function with a discontinuity by a Fourier series produces a curious phenomenon.
Description of the phenomenon
Near a discontinuity such as that of a square wave, the partial sum "overshoots": it temporarily exceeds the value of the original function by about 9% of the jump (the height of the discontinuity).
- Even as $N$ increases, this overshoot does not disappear
- The width of the overshoot narrows, but its height stays at about 9% of the jump (precisely about 8.95%)
This is called the Gibbs phenomenon. It was discovered in 1899 by Josiah Willard Gibbs.
Why it happens
Trigonometric functions are smooth, so no matter how many of them we superpose, they can never perfectly reproduce an "instantaneous jump." The ripples near a discontinuity are the manifestation of the "strain" of trying to represent an abrupt change with finitely many terms.
Where Does It Converge?
Here we consider convergence at each point (pointwise convergence). For other modes such as uniform and $L^2$ convergence, see the FAQ at the end of the chapter.
Convergence at points of continuity
If $f(x)$ is continuous at a point $x$, the Fourier series converges to $f(x)$.
Convergence at points of discontinuity
If $f(x)$ is discontinuous at a point $x$, the Fourier series converges to the average of the left and right limits:
$$S(x) = \dfrac{f(x^+) + f(x^-)}{2}$$
where $f(x^+) = \lim_{t \to x^+}f(t)$ and $f(x^-) = \lim_{t \to x^-}f(t)$.
Example: the discontinuity of the square wave
At $x = 0$ we have $f(0^+) = 1$ and $f(0^-) = -1$, so:
$$S(0) = \dfrac{1 + (-1)}{2} = 0$$
Indeed, substituting $x = 0$ into the Fourier series of the square wave makes every $\sin$ term $0$, so $S(0) = 0$.
Conditions for Convergence
A sufficient condition for a Fourier series to converge is given by the following Dirichlet conditions:
- $f(x)$ is periodic
- $f(x)$ is piecewise continuous (it may have finitely many discontinuities)
- $f(x)$ is piecewise smooth (it may have finitely many corners)
Most functions encountered in practice satisfy these conditions, so Fourier series are widely applicable.
Summary
- The more terms in the partial sum, the closer it gets to the original function
- The smoother the function, the faster the convergence
- At a discontinuity the "Gibbs phenomenon" occurs (about 9% overshoot)
- At a discontinuity the Fourier series converges to the average of the left and right values
- A Fourier series applies to any function satisfying the Dirichlet conditions
Next Steps
In the introduction we have learned the basic ideas of Fourier series and built an intuitive understanding. At the next level we will study more rigorous definitions, detailed calculation methods, and important theorems such as Parseval's identity.
Frequently Asked Questions
Q1: What is the intuitive picture of Fourier series convergence?
A: Starting from the low notes (low frequencies) and stacking them up, the waveform gets closer and closer to the original. The partial sum up to $N$ terms $S_N(x) = \frac{a_0}{2} + \sum_{n=1}^N (a_n \cos nx + b_n \sin nx)$ approaches $f(x)$ as $N \to \infty$. The smoother the function, the smaller its high-frequency components and the faster it converges.
Q2: What are the differences between the various types of convergence of a Fourier series?
A: Pointwise convergence: $S_N(x) \to f(x)$ at each $x$. Uniform convergence: $\sup_x |S_N(x) - f(x)| \to 0$ (a strong mode of convergence, uniform over the whole interval). $L^2$ convergence: $\int |S_N(x) - f(x)|^2 dx \to 0$ (the most general, with physical meaning). Absolute convergence: $\sum |a_n| + |b_n| < \infty$ (a strong condition).
Q3: What is Parseval's identity?
A: When $f$ belongs to $L^2[-\pi, \pi]$, the identity $\dfrac{1}{\pi}\int_{-\pi}^{\pi} |f(x)|^2\,dx = \dfrac{|a_0|^2}{2} + \displaystyle\sum_{n=1}^\infty (|a_n|^2 + |b_n|^2)$ holds. It expresses that the energy of the function equals the total energy of its frequency components, and it is the theoretical foundation of signal processing.