The Fourier Transform on L²: The Plancherel Theorem

Precise statement: what Plancherel claims

Define the Fourier transform on L¹(ℝ) by f̂(ξ) = ∫_ℝ f(x) e^(−2πixξ) dx. For f ∈ L¹ ∩ L²(ℝ) one has the Plancherel identity ‖f̂‖₂ = ‖f‖₂, and more generally the Parseval identity ⟨f̂, ĝ⟩ = ⟨f, g⟩. Since L¹ ∩ L² is dense in L²(ℝ) and L²(ℝ) is complete, the map 𝓕: f ↦ f̂ extends uniquely to a bounded linear operator on all of L²(ℝ).

The full theorem (Plancherel, 1910) states: this extension 𝓕 is a unitary isomorphism of L²(ℝ) onto itself. That is, 𝓕 is linear, bijective, norm-preserving (‖𝓕f‖₂ = ‖f‖₂ for every f ∈ L²), and its inverse equals its Hilbert-space adjoint, 𝓕⁻¹ = 𝓕*, with (𝓕*g)(x) = ∫ g(ξ) e^(+2πixξ) dξ interpreted as an L² limit. For general f ∈ L², f̂ = lim_{N→∞} ∫_{−N}^{N} f(x) e^(−2πixξ) dx, the limit taken in L²-norm, not pointwise.

The picture: Fourier transform as a rotation

Think of L²(ℝ) as an infinite-dimensional Euclidean space, with inner product ⟨f, g⟩ = ∫ f ḡ and length ‖f‖₂ = (∫|f|²)^{1/2}. A unitary operator is precisely the infinite-dimensional analogue of an orthogonal rotation: it moves vectors around but never stretches, shrinks, or shears them. Plancherel says 𝓕 is such a rotation.

The physical reading is conservation of energy. Whether you measure a signal's energy in the time domain (∫|f(x)|² dx) or in the frequency domain (∫|f̂(ξ)|² dξ), you get the same number. The transform merely repackages the information — redistributing it across frequencies — without creating or destroying any of it. Angles are preserved too (Parseval): orthogonal signals have orthogonal spectra. This is why an orthonormal basis of L² maps to another orthonormal basis, and why 𝓕 has a clean spectral structure: its eigenvalues are the fourth roots of unity {1, i, −1, −i}, since 𝓕⁴ = I.

Key idea of the proof: extend an isometry by density

The engine is the bounded linear transformation (BLT) theorem: a bounded operator defined on a dense subspace of a Banach space, with values in a complete space, extends uniquely to the whole space with the same norm. So everything reduces to proving the identity ‖f̂‖₂ = ‖f‖₂ on the dense subspace L¹ ∩ L².

The cleanest route is via a Gaussian or via the multiplication formula ∫ f̂ g = ∫ f ĝ combined with self-duality of the Gaussian. One standard argument: set h = f ∗ f̃ where f̃(x) = f̄(−x); then h ∈ L¹ is continuous with ĥ = |f̂|² ≥ 0. Evaluating the inversion formula for h at 0 gives h(0) = ∫ ĥ = ∫ |f̂|², while directly h(0) = ∫ f(x) f̄(x) dx = ‖f‖₂². Hence ‖f̂‖₂² = ‖f‖₂². Surjectivity then follows because 𝓕 has dense range (it contains all Schwartz functions, since 𝓕 maps the Schwartz space 𝒮 bijectively onto itself) and, being an isometry, has closed range — a closed dense subspace is everything.

Canonical example: the Gaussian and the sinc/box pair

The Gaussian is the eigenfunction that anchors the theory: with g(x) = e^(−πx²), one computes ĝ(ξ) = e^(−πξ²) = g(ξ). It is a fixed point of 𝓕 (eigenvalue 1), and ∫ e^(−2πx²) dx appears identically on both sides, confirming ‖ĝ‖₂ = ‖g‖₂ by hand.

A more revealing example is the box function f = 𝟙_{[−1/2, 1/2]}, whose transform is the sinc: f̂(ξ) = sin(πξ)/(πξ). Here f ∈ L¹ ∩ L², but f̂ ∉ L¹ (it decays like 1/|ξ|, not integrable), so the inversion integral ∫ f̂ e^(2πixξ) dξ does not converge absolutely — yet it converges in L² to f. Plancherel gives ∫_{−1/2}^{1/2} 1 dx = 1 = ∫_ℝ sin²(πξ)/(πξ)² dξ. This last integral equals 1 is a nontrivial fact you get for free — a small taste of how the theorem converts geometry into concrete evaluations.

Why the hypotheses matter — and what breaks

Completeness of L² is essential. The whole extension argument rests on Cauchy sequences of truncated integrals converging; on a non-complete inner-product space (e.g. the continuous functions with the L² norm), the limit f̂ need not exist in the space, and surjectivity fails. The Riesz–Fischer completeness of L² is the load-bearing beam.

Square-integrability is the right regularity. On L¹ alone, norm equality is false: the transform of an L¹ function is merely continuous and vanishing at infinity (Riemann–Lebesgue), 𝓕 is not onto L¹, and no isometry holds — 𝟙_{[0,1]} has ‖f‖₁ = 1 but ‖f̂‖_∞ = 1 with f̂ ∉ L¹. For 1 < p < 2, only the one-sided Hausdorff–Young inequality ‖f̂‖_{p′} ≤ ‖f‖_p survives, and it is generally strict (Babenko–Beckner give the sharp constant). L² = L^{p} with p = p′ = 2 is the unique self-dual case where equality reigns. This exactness is what makes L² the natural home of Fourier analysis.

Significance: what Plancherel unlocks

Plancherel turns the Fourier transform into a rigorous tool of Hilbert-space geometry, and a great deal of modern analysis is downstream. It underwrites the spectral theorem for self-adjoint operators with continuous spectrum: differentiation d/dx becomes multiplication by 2πiξ, diagonalizing constant-coefficient differential operators — the foundation of PDE theory, the heat and Schrödinger equations, and Sobolev space definitions (Hˢ via ∫(1+|ξ|²)ˢ|f̂|² dξ).

In probability it is the Hilbert-space backbone behind characteristic functions and the spectral representation of stationary processes. In signal processing it is Parseval's energy theorem, guaranteeing that no information is lost in the spectrum. The result generalizes sweepingly: the Plancherel theorem for locally compact abelian groups (Pontryagin duality) gives a unitary L²(G) ≅ L²(Ĝ), and the non-commutative Plancherel theorem decomposes L²(G) over the unitary dual via the Plancherel measure — the gateway to representation theory and the Langlands program.

The Fourier transform on different function spaces: what the integral does and what is preserved

Space	Does ∫f e^(−2πixξ)dx converge?	Target / regularity	What holds
L¹(ℝ)	Yes, absolutely	f̂ continuous, →0 at ∞ (Riemann–Lebesgue)	‖f̂‖_∞ ≤ ‖f‖₁; not onto, no norm equality
L¹∩L²(ℝ)	Yes	f̂ ∈ L²	‖f̂‖₂ = ‖f‖₂ (Plancherel identity, the seed)
L²(ℝ)	Not in general	f̂ ∈ L² (defined by L² limit)	𝓕 unitary, onto, ‖f̂‖₂ = ‖f‖₂, 𝓕⁻¹ = 𝓕*
L^p, 1<p<2	No (only as principal value)	f̂ ∈ L^p′, 1/p+1/p′=1	Hausdorff–Young: ‖f̂‖_{p′} ≤ ‖f‖_p (not equality)