Fourier Transform and Plancherel's Theorem

Idea: Decompose a signal into frequencies

Fourier series decompose a periodic function into frequencies 1/T, 2/T, 3/T, ... But what should we do with a non-periodic signal — for example, a finite pulse or a function on the entire real line? A step toward the non-periodic case: increase the period T → ∞. The discrete set of frequencies n/T becomes a continuous spectrum, and the sum in the Fourier series turns into an integral. This is how the Fourier transform is born.

It is one of the most universal tools in mathematics — it is simultaneously used in number theory, quantum mechanics, partial differential equation theory, statistics, signal processing, and machine learning.

Fourier Transform on L¹(ℝ)

For f ∈ L¹(ℝ) (absolutely integrable function), the Fourier transform is defined by:

f̂(ω) = ℱf = ∫_{-∞}^{∞} f(x) e^{-iωx} dx.

(The convention e^{-iωx} is used here; physics often uses e^{-2πiξx}, engineering — e^{-jωt}.)

Meaning: f̂(ω) is the “weight” of the frequency ω in the signal f. If f is a sound, then f̂ is the spectrum: the amplitude of each frequency. A large |f̂(ω)| means that the frequency ω is strongly represented in the signal.

Elementary properties:

Linearity: (αf + βg)^ = αf̂ + βĝ.

Shift in x: (f(x − a))^(ω) = e^{−iaω} f̂(ω). Shifting the signal changes the phase but not the amplitude of the spectrum.

Shift in ω: (e^{ibx} f(x))^(ω) = f̂(ω − b). Modulating the signal shifts the spectrum.

Scaling: (f(ax))^(ω) = (1/|a|) f̂(ω/a). Compressing the signal expands the spectrum.

Differentiation: (f^{(n)}(x))^(ω) = (iω)ⁿ f̂(ω). Differentiation → multiplication by iω!

Convolution: (f * g)^(ω) = f̂(ω) · ĝ(ω). Convolution in space = multiplication in the spectrum.

Inversion Formula and Schwartz Space

Inversion formula: Under appropriate conditions for f:

f(x) = (1/2π) ∫_{-∞}^{∞} f̂(ω) e^{iωx} dω.

To ensure everything works without technical caveats, we introduce the Schwartz space S(ℝ) — smooth functions rapidly decreasing along with all derivatives: φ ∈ S if and only if |xⁿ φ^{(k)}(x)| → 0 as |x| → ∞ for all n, k ≥ 0. These are “ideal” functions: e^{-x²}, functions with compact support (from class C∞).

The Fourier transform is an isomorphism S(ℝ) → S(ℝ). All properties (differentiation, inversion, convolution) work perfectly on S.

Calculation of Examples

Rectangular impulse. f(x) = 1 for |x| ≤ a, and 0 otherwise.

f̂(ω) = ∫{-a}^{a} e^{-iωx} dx = [e^{-iωx}/(−iω)]{-a}^{a} = 2 sin(aω)/ω = 2a sinc(aω/π).

The spectrum is the sinc function: the main lobe of width 2π/a, decaying side lobes. A narrower rectangle → a broader spectrum. This is a manifestation of the uncertainty principle.

Gaussian function. f(x) = e^{-x²/2}.

f̂(ω) = √(2π) e^{-ω²/2}.

The Gaussian maps to Gaussian! This is the unique function (up to scaling) that is an eigenfunction of the Fourier transform.

Delta function. δ(x) is a “function” equal to 0 everywhere except x = 0, with ∫δ(x) dx = 1. In the sense of generalized functions: δ̂(ω) = 1 (the spectrum is constant over all frequencies — “white noise”). Conversely: 1̂(ω) = 2πδ(ω).

Plancherel's Theorem: Extension to L²

The Fourier transform is defined on L¹, but many functions from L² are not in L¹ (for example, f(x) = sin(x)/x). How to define ℱ on L²?

Plancherel's Theorem (1910): The Fourier transform, first defined on L¹ ∩ L², extends to a unitary operator ℱ: L²(ℝ) → L²(ℝ), such that:

‖f̂‖{L²} = √(2π) ‖f‖{L²} (the Parseval–Plancherel equality).

With normalization by the coefficient 1/√(2π) in front of the integral, this turns into ‖f̂‖ = ‖f‖: Fourier is an isometry, preserving the L² norm (“energy”).

“Unitarity” means: ℱℱ = I (ℱ⁻¹ = ℱ, where ℱ* is the adjoint operator). In physics: energy of the signal = energy of its spectrum.

Heisenberg Uncertainty Principle

Let f ∈ L²(ℝ) be normalized: ‖f‖₂ = 1. Define the “spread” of the function in x and ω:

Δx = ‖(x − ⟨x⟩) f‖₂, Δω = ‖(ω − ⟨ω⟩) f̂/√(2π)‖₂.

Theorem (Heisenberg inequality): Δx · Δω ≥ 1/2.

Equality is achieved only for Gaussian functions. The more precisely the signal is localized in time — the wider its spectrum, and vice versa.

In quantum mechanics: x̂ is the coordinate operator, p̂ = −iℏ d/dx is the momentum operator. The uncertainty principle: σ_x · σ_p ≥ ℏ/2. Mathematically, this is the same Heisenberg inequality, rewritten in the language of physics.

Solving PDEs by the Fourier Method

Heat equation: ∂u/∂t = k ∂²u/∂x², u(x,0) = f(x), x ∈ ℝ, t > 0.

Apply ℱ in x: ∂û/∂t = k(iω)² û = −kω² û. This is an ODE in t with initial condition û(ω,0) = f̂(ω):

û(ω,t) = f̂(ω) e^{-kω²t}.

Inverse Fourier: u(x,t) = (f * G_t)(x), where the Gaussian kernel G_t(x) = 1/√(4πkt) · e^{-x²/(4kt)}.

Meaning: the initial temperature profile is smoothed by Gaussian “spreading” with width √(4kt). This matches physics: heat “spreads out” over time.

Fast Fourier Transform (FFT)

The discrete Fourier transform (DFT) for the vector (x₀,...,xₙ₋₁): Xₖ = Σₙ₌₀^{N-1} xₙ e^{-2πi nk/N}.

Direct computation: O(N²) operations. Cooley–Tukey algorithm (FFT, 1965): O(N log N), based on recursive decomposition N = 2^m.

For N = 10⁶, direct calculation: 10¹² operations; FFT: ≈ 2·10⁷ operations — 50,000 times faster! FFT entered the list of the 10 most important algorithms of the 20th century (Computing in Science & Engineering magazine, 2000).

Applications of FFT: audio compression (MP3: discarding frequencies with low |f̂|), image compression (JPEG: discrete cosine transform — a relative of DFT), multiplication of large numbers (via convolution), analysis of time series, medical tomography (MRI — reconstructing images from Fourier slices).

Question for thought: why does discarding high-frequency components f̂(ω) at large |ω| during MP3 compression cause a small audible effect, whereas in JPEG compression, removing high-frequency components of the 2D Fourier transform creates characteristic “blocky” artifacts? How does the Heisenberg uncertainty principle explain the trade-off between compression and quality?