Fourier Series & Orthogonal Expansions

A Fourier series does not claim that a complicated waveform is secretly made of physical oscillators. It says that periodic functions can be represented or approximated in an orthogonal coordinate system whose basis functions are sines, cosines, or complex exponentials. Orthogonality makes each coefficient an independent projection and turns approximation error and signal energy into sums of squares.

What coefficients mean on a real waveform #

For a measured periodic signal, the zero-frequency coefficient is its average, the fundamental repeats once per period, and higher harmonics describe shape within the cycle. This representation is useful because a linear system can be analyzed one sinusoidal input at a time and the responses added. Periodic motor torque, switched power, sound, vibration, and heat forcing then become families of small sinusoidal problems instead of one complicated waveform problem.

The coefficients do not prove that hidden oscillators physically generated the signal. They are coordinates. Their usefulness comes from orthogonality and from how linear differential equations act on sinusoidal coordinates.

Fix the convention before calculating #

Let $f$ have period $T=2L$ and fundamental angular frequency $\omega_0=\pi/L=2\pi/T$. The real form is

$$f(x)\sim \frac{a_0}{2}+\sum_{n=1}^{\infty} \left(a_n\cos n\omega_0x+b_n\sin n\omega_0x\right),$$

with

$$a_n=\frac1L\int_{-L}^{L}f(x)\cos(n\omega_0x)\,dx, \qquad b_n=\frac1L\int_{-L}^{L}f(x)\sin(n\omega_0x)\,dx.$$

Here the $a_0$ formula is the same formula at $n=0$, so the DC value is $a_0/2$.

The complex form is

$$f(x)\sim\sum_{n\in\mathbb Z}c_ne^{in\omega_0x}, \qquad c_n=\frac1{2L}\int_{-L}^{L}f(x)e^{-in\omega_0x}\,dx.$$

For $n>0$,

$$c_n=\frac{a_n-ib_n}{2},\qquad c_{-n}=\frac{a_n+ib_n}{2},\qquad c_0=\frac{a_0}{2}.$$

If $f$ is real-valued, $c_{-n}=\overline{c_n}$. Negative frequencies are therefore redundant for reconstructing a real signal, but they are not absent; the pair is what produces a real sinusoid.

Orthogonality derives the coefficients #

On $[-L,L]$, distinct complex modes satisfy

$$\frac1{2L}\int_{-L}^{L}e^{in\omega_0x}\overline{e^{im\omega_0x}}\,dx =\delta_{nm}.$$

Taking the inner product of $f$ with mode $m$ kills every other mode and leaves $c_m$. This is the same coordinate extraction used with orthonormal vectors.

For the partial sum

$$S_Nf=\sum_{|n|\le N}c_ne^{in\omega_0x},$$

orthogonality also gives the projection property

$$\|f-S_Nf\|_2\le\|f-p\|_2$$

for every trigonometric polynomial $p$ using only modes $|n|\le N$. The Fourier partial sum is not merely one approximation; it is the least-squares approximation in that subspace.

Three convergence questions are different #

The notation $f\sim\sum c_ne^{in\omega_0x}$ does not by itself specify how the series converges.

• Mean-square convergence. $\lVert S_Nf-f\rVert_2\to0$. For $f\in L^2$ on a period, the complex exponentials form a complete orthonormal system, so this is the natural baseline.
• Pointwise convergence. For sufficiently regular piecewise-smooth functions, the symmetric partial sums converge at $x$ to $[f(x-)+f(x+)]/2$.
• Uniform convergence. $\sup_x\lvert S_Nf(x)-f(x)\rvert\to0$. This is stronger and cannot hold across a jump because every finite partial sum is continuous.

At a jump, the Fourier series reconstructs the midpoint of the one-sided limits, regardless of the value assigned to $f$ at that single point. Changing finitely many point values also does not change the coefficients because integrals ignore measure-zero changes.

Symmetry is a calculation shortcut, not a new theorem #

On a symmetric interval:

• if $f$ is even, every $b_n=0$ and only cosines remain;
• if $f$ is odd, $a_0=a_n=0$ and only sines remain;
• half-wave symmetry $f(x+L)=-f(x)$ removes even harmonics.

A function given only on $[0,L]$ does not have a unique Fourier series until an extension is chosen. The even extension produces a cosine series, the odd extension a sine series, and a direct periodic extension may create different endpoint jumps. The extension is part of the model.

Parseval turns waveform energy into coefficient energy #

With the convention above, Parseval's identity is

$$\frac1{2L}\int_{-L}^{L}|f(x)|^2\,dx =\sum_{n\in\mathbb Z}|c_n|^2.$$

In real coefficients,

$$\frac1L\int_{-L}^{L}|f(x)|^2\,dx =\frac{|a_0|^2}{2}+\sum_{n=1}^{\infty}\left(|a_n|^2+|b_n|^2\right).$$

This is Pythagoras in an infinite-dimensional Hilbert space. It gives an exact truncation-error budget in mean square:

$$\frac1{2L}\|f-S_Nf\|_2^2=\sum_{|n|>N}|c_n|^2.$$

Smoothness controls decay, with boundary conditions attached #

Integration by parts explains coefficient decay. If $f$ is absolutely continuous over a period and its endpoint values match, then for $n\ne0$,

$$c_n=\frac{1}{in\omega_0}\frac1{2L}\int_{-L}^{L}f'(x)e^{-in\omega_0x}\,dx,$$

so $c_n=o(1/n)$ under standard integrability assumptions. Repeating the argument gives higher inverse powers only when the corresponding derivatives exist, are integrable enough, and the periodic boundary terms continue to match.

A jump in $f$ typically produces $O(1/n)$ coefficients. If $f$ is continuous but $f'$ has jumps, $O(1/n^2)$ is typical. These are patterns under explicit regularity and boundary hypotheses, not a universal rule of the form “the last continuous derivative always determines $1/n^{N+2}$.”

Gibbs behavior is local but does not shrink vertically #

Near a jump, partial sums oscillate and overshoot. As $N$ grows, the oscillatory region narrows, but the peak overshoot approaches a nonzero fraction of the jump rather than vanishing. This does not contradict mean-square convergence: the error is concentrated in a shrinking interval.

Smoother summation methods such as Fejér means average the partial sums and improve convergence behavior. Windowing also changes sidelobes, but it trades resolution for reduced ringing; it is not a free repair.

Worked example: the odd square wave #

Let $f(x)=1$ for $0<x<\pi$, $f(x)=-1$ for $-\pi<x<0$, and extend it with period $2\pi$. The function is odd, so only sine coefficients remain:

$$b_n=\frac1\pi\int_{-\pi}^{\pi}f(x)\sin(nx)\,dx =\frac2\pi\int_{0}^{\pi}\sin(nx)\,dx =\frac{2}{\pi n}\bigl(1-(-1)^n\bigr).$$

Thus $b_n=4/(\pi n)$ for odd $n$ and $b_n=0$ for even $n$:

$$f(x)\sim\frac4\pi\left(\sin x+\frac{\sin3x}{3}+\frac{\sin5x}{5}+\cdots\right).$$

At $x=0$ and $x=\pi$, the series converges to the midpoint of the jump, $0$. Away from jumps it converges to $\pm1$. The $1/n$ decay records the jump discontinuities.

Diagnostic checklist #

• State $T$, $L$, $\omega_0$, and the coefficient convention.
• Check DC separately and use conjugate symmetry only for real signals.
• Decide whether the claim is $L^2$, pointwise, or uniform convergence.
• At a jump, compare with the midpoint of one-sided limits.
• Verify periodic endpoint matching before integrating by parts repeatedly.
• Derive phase from the chosen complex coefficient; a time delay $f(x-x_0)$ contributes $e^{-in\omega_0x_0}$ under this convention.
• Use Parseval to audit normalization: coefficient energy must match waveform energy.