Plancherel Theorem

Statement

The Plancherel theorem states that the Fourier transform is a unitary isomorphism on $L^2(\mathbb{R})$ (see Lp_space).

This means the Fourier transform preserves inner products:

$$ \langle f, g \rangle = \langle \mathcal{F}\{f\}, \mathcal{F}\{g\} \rangle $$

or explicitly:

$$ \int_{-\infty}^{\infty} f(x) \overline{g(x)} \, dx = \int_{-\infty}^{\infty} F(\nu) \overline{G(\nu)} \, d\nu $$

Consequences

Since inner products are preserved, so are:

Property	Formula
Norms (energy)	$\|f\|_2 = \|\mathcal{F}\{f\}\|_2$
Orthogonality	$\langle f, g \rangle = 0 \Leftrightarrow \langle F, G \rangle = 0$
Angles	The “angle” between functions is unchanged

Relation to Parseval’s Theorem

Parseval’s theorem is the special case where $f = g$:

$$ \int_{-\infty}^{\infty} |f(x)|^2 \, dx = \int_{-\infty}^{\infty} |F(\nu)|^2 \, d\nu $$

This says energy is conserved — the total energy in the spatial domain equals the total energy in the frequency domain.

Plancherel is more general: it says not just energy, but all geometric structure (inner products) is preserved.

Why This Matters

The Plancherel theorem tells us that $L^2(\mathbb{R})$ in the spatial domain and $L^2(\mathbb{R})$ in the frequency domain are the same space — not just containing the same information, but having identical geometric structure.

The Fourier transform is like a rotation: it rearranges the coordinates but doesn’t distort the space.

Technical Note

The Fourier transform is initially defined for $L^1$ functions (where the integral converges absolutely). For $L^2$ functions, it’s extended via a limiting process. The Plancherel theorem guarantees this extension is well-defined and unitary.

Related Concepts

• Parseval_theorem — the special case for norms
• Fourier_transform — the map that Plancherel characterizes
• unitary operator — what the Fourier transform is on $L^2$
• L² space — the domain where Plancherel applies