In mathematics, a square-integrable function (also called a quadratically integrable function or $L^2$ function) is a measurable function for which the integral of the square of its absolute value is finite. This concept is fundamental in functional analysis, quantum mechanics, and signal processing.
Definition
Let $(X, \Sigma, \mu)$ be a measure space. A measurable function $f: X \to \mathbb{C}$ (or $\mathbb{R}$) is square-integrable if $$ \int_X |f(x)|^2 , d\mu(x) < \infty. $$ The set of all equivalence classes of square-integrable functions (where functions equal almost everywhere are identified) forms the $L^2$ space, denoted $L^2(X, \mu)$ or simply $L^2$.
Properties
- Hilbert Space Structure: $L^2$ is a Hilbert space under the inner product $$ \langle f, g \rangle = \int_X f(x) \overline{g(x)} , d\mu(x). $$ The induced norm is $|f|_2 = \left( \int_X |f(x)|^2 , d\mu(x) \right)^{1/2}$.
- Completeness: $L^2$ is complete with respect to this norm (Riesz-Fischer theorem), meaning every Cauchy sequence converges to a limit within the space.
- Dual Space: The dual space of $L^2$ is isometrically isomorphic to $L^2$ itself (Riesz representation theorem).
- Inclusion Relations: On spaces of finite measure (e.g., a bounded interval $[a,b]$), $L^2 \subset L^1$ by Hölder's inequality. On spaces of infinite measure (e.g., $\mathbb{R}$), neither $L^1 \subset L^2$ nor $L^2 \subset L^1$ holds generally.
Examples
- Continuous functions on a closed interval: Any continuous function $f: [a, b] \to \mathbb{R}$ is square-integrable with respect to Lebesgue measure.
- Gaussian functions: $f(x) = e^{-x^2}$ on $\mathbb{R}$ is square-integrable.
- Compactly supported functions: Any bounded function with compact support is square-integrable.
- Non-examples: The constant function $f(x) = 1$ on $\mathbb{R}$ is not square-integrable. The function $f(x) = 1/\sqrt{x}$ on $(0,1]$ is not square-integrable (though it is in $L^1$).
Applications
- Quantum Mechanics: Wave functions describing physical states must be square-integrable (normalizable) to satisfy the Born rule probability interpretation. The space of states is the Hilbert space $L^2(\mathbb{R}^3)$.
- Fourier Analysis: The Fourier transform is a unitary operator on $L^2(\mathbb{R}^n)$ (Plancherel theorem). Parseval's identity holds for square-integrable functions.
- Signal Processing: Finite-energy signals are modeled as elements of $L^2$. The energy of a signal $f(t)$ is $|f|_2^2$.
Generalizations
The concept extends to vector-valued functions (Bochner integrals) and sections of vector bundles, playing a central role in the theory of Sobolev spaces $W^{k,2} = H^k$ and the study of partial differential equations.
