Fourier Series

The Idea of Expansion into a Trigonometric Series

Jean-Baptiste Joseph Fourier in 1807 proposed a revolutionary idea: any “reasonable” function can be expanded into a series of trigonometric functions. This enabled the solution of the heat equation—a problem Fourier was working on as an engineer.

Fourier’s trigonometric series: $f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx))$, where

$a_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\cos(nx)dx$, $b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\sin(nx)dx$.

Orthogonality of the Trigonometric System

The functions $1$, $\cos x$, $\sin x$, $\cos 2x$, $\sin 2x$, ... are orthogonal in the space $L^2[-\pi, \pi]$:

$\int_{-\pi}^{\pi} \cos(mx)\cos(nx)dx = 0$ for $m \neq n$; $= \pi$ for $m = n > 0$.

It is precisely this orthogonality that allows us to compute the coefficients: we multiply $f(x)$ by $\cos(nx)$ and integrate—all other terms vanish.

Dirichlet’s Theorem

If $f$ is piecewise continuous and piecewise monotonic on $[-\pi, \pi]$, then the Fourier series converges to $f(x)$ at points of continuity and to $\frac{f(x+)+f(x-)}{2}$ at points of discontinuity.

Parseval’s Identity

$\frac{a_0^2}{2} + \sum (a_n^2 + b_n^2) = \frac{1}{\pi} \int_{-\pi}^{\pi} [f(x)]^2dx$.

This is the “Pythagorean theorem” in infinite-dimensional space: the sum of the squares of the coefficients equals the square of the norm of the function. The Fourier series is the best approximation in the sense of the mean-square norm.

Complex Form and the Fourier Transform

In complex notation: $f(x) = \sum c_n e^{inx}$, $c_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx} dx$.

For functions on the whole real line: Fourier transform $\hat{f}(\omega) = \int_{-\infty}^{\infty} f(x) e^{-i\omega x} dx$.

This is a key tool in signal processing, quantum mechanics, and solving differential equations. The Fast Fourier Transform (FFT)—the Cooley–Tukey algorithm—is one of the 10 most important algorithms of the 20th century.

Convergence of the Fourier Series: Subtleties

Dirichlet’s theorem establishes pointwise convergence for “decent” functions. But the general answer is more complicated. Fejér’s theorem (1904): the arithmetic means of the partial sums (Fejér sums) converge uniformly to $f$ for any continuous periodic function. This is weaker than pointwise convergence, but it works for all continuous functions.

The Gibbs Phenomenon: in the neighborhood of a discontinuity of the function, the partial sum of the Fourier series “overshoots”—the maximum deviation is about $9%$ of the jump, regardless of the number of harmonics retained. This fundamentally limits the accuracy of Fourier approximation at discontinuities.

Parseval’s Identity and Signal Energy

Parseval’s identity $\frac{a_0^2}{2} + \sum(a_n^2 + b_n^2) = \frac{1}{\pi}\int_{-\pi}^{\pi} f^2(x) dx$ has a physical meaning: the total energy of the signal equals the sum of the energies of the harmonics. In radio engineering, this is the law of conservation of energy in the spectral representation. For digital audio: in MP3 compression, harmonics with small coefficients $a_n^2$, $b_n^2$ are removed—losing “insignificant” energy gives significant file compression without perceptible loss in sound quality.

Non-Periodic Functions and the Fourier Transform

For functions defined on the entire real line (non-periodic), the Fourier series generalizes to the integral Fourier transform $\hat{f}(\omega) = \int_{-\infty}^{\infty} f(x) e^{-i\omega x} dx$. This is the “limit” of the Fourier series as the period $T \to \infty$: the discrete spectrum ${c_n}$ turns into a continuous $\hat{f}(\omega)$. The inversion formula: $f(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \hat{f}(\omega) e^{i\omega x} d\omega$.

In quantum field mechanics and signal processing, it is precisely the continuous spectrum that is used. The difference between the discrete and continuous cases is the difference between periodic and non-periodic phenomena.

Two-Dimensional Fourier Transform and Image Processing

In two dimensions, the Fourier transform $\hat{f}(u, v) = \iint f(x, y) e^{-2\pi i (ux+vy)} dx dy$ decomposes an image into planar harmonics. Low frequencies (small $u$, $v$) correspond to large details in the image—overall background and contours; high frequencies—to fine details, textures, and edges of objects. Frequency domain filtering: multiplication of $\hat{f}(u,v)$ by a mask $H(u,v)$ and inverse transformation—is a fast way to blur (low-pass filter) or sharpen (high-pass filter) an image. The JPEG algorithm uses the discrete cosine transform (DCT)—a real analogue of Fourier—to blocks of $8\times8$ pixels: at compression, high-frequency coefficients are set to zero. Removing $80–90%$ of the coefficients at JPEG quality $= 75$ yields a visually acceptable result when compressing $5–10$ times.

The Uncertainty Principle in Signal Analysis

The Fourier transform realizes the mathematical analogue of Heisenberg’s uncertainty principle: a signal cannot be simultaneously concentrated both in time and in the frequency domain. The precise formulation: $\sigma_t \cdot \sigma_\omega \geq 1/2$, where $\sigma_t$ and $\sigma_\omega$ are the root mean square deviations of the signal and its Fourier transform. This explains why a short radio pulse has a wide frequency band—a key fact in communication theory and digital signal processing (DSP).

Question for reflection: Why does the Fourier series of a “rectangular” signal (alternating $0$ and $1$) have only odd harmonics? How is the symmetry of the function related to the form of its Fourier expansion?

Fast Fourier Transform and Algorithmic Efficiency

Direct computation of the Fourier coefficients from $N$ points requires $O(N^2)$ multiplications. The FFT algorithm (Cooley–Tukey, 1965) reduces this to $O(N \log N)$—for $N = 10^6$ this is the difference between $10^{12}$ and $2 \cdot 10^7$ operations. The idea: split the signal into even and odd samples, compute the DFT recursively. Applications: audio and image processing (JPEG uses the discrete cosine transform, related to the DFT), multiplication of large numbers, calculation of convolutions in neural networks. FFT is one of the most influential numerical algorithms of the 20th century in terms of real-world applications.