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Square-integrable function

In mathematics, a square-integrable function is a real- or complex-valued measurable function f defined on a measure space (X, \mathcal{A}, \mu) such that the integral \int_X |f|^2 \, d\mu < \infty. This condition ensures that the function has finite energy in a certain sense, making it suitable for analysis in spaces where norms are defined via this squared integral. Square-integrable functions form the space L^2(X, \mathcal{A}, \mu), which consists of equivalence classes of such functions where two functions are identified if they differ on a set of measure zero. This space is a complete inner product space, specifically a Hilbert space, with the inner product given by \langle f, g \rangle = \int_X f \overline{g} \, d\mu, enabling the application of orthogonal projections, spectral theory, and other tools from functional analysis. The L^2 norm, \|f\|_2 = \sqrt{\int_X |f|^2 \, d\mu}, quantifies the "size" of these functions and satisfies the Cauchy-Schwarz inequality, which bounds inner products and facilitates convergence arguments. These functions are foundational in several areas of mathematics and physics. In Fourier analysis, square-integrable functions on intervals or the real line can be decomposed into orthogonal series or transforms, allowing approximation by trigonometric polynomials in the L^2 norm, which is crucial for signal processing and partial differential equations. In quantum mechanics, wave functions must be square-integrable to ensure finite total probability, as the squared modulus integrates to unity over the space, forming the of states where observables are represented by self-adjoint operators. Additionally, L^2 spaces underpin stochastic processes, such as Gaussian processes in statistics, and appear in the study of integral equations and operator theory.

Definition and basics

Formal definition

A square-integrable function is a measurable function f defined on a measure space (X, \Sigma, \mu), where X is a set, \Sigma is a \sigma-algebra of subsets of X, and \mu is a measure on \Sigma, such that the integral \int_X |f(x)|^2 \, d\mu(x) is finite. This condition ensures that the function's "energy" or squared magnitude is integrable over the space with respect to the measure \mu. For real-valued functions, where f: X \to \mathbb{R}, the condition simplifies to \int_X [f(x)]^2 \, d\mu(x) < \infty. In the complex-valued case, f: X \to \mathbb{C}, the squared absolute value is given by |f(x)|^2 = f(x) \overline{f(x)}, where \overline{f(x)} denotes the complex conjugate, ensuring the integral captures the modulus squared. The integral here is understood in the sense of the Lebesgue integral, which extends the Riemann integral to more general measurable functions and measures. This finiteness condition defines the square of the L^2 norm, \|f\|_2^2 = \int_X |f(x)|^2 \, d\mu(x) < \infty, and the collection of all such functions forms the L^2 space.

Relation to L² spaces

The space L^2(X, \Sigma, \mu) is defined as the set of equivalence classes of square-integrable functions on a measure space (X, \Sigma, \mu), where two measurable functions f and g are identified (i.e., belong to the same equivalence class) if \int_X |f - g|^2 \, d\mu = 0. This condition ensures that functions differing only on sets of \mu-measure zero are treated as identical, which is essential because the Lebesgue integral is unaffected by values on such null sets. The resulting quotient space captures the essential behavior of square-integrable functions while forming a foundational structure in . These equivalence classes endow L^2(X, \Sigma, \mu) with a natural vector space structure over the real or complex numbers. Addition of classes is defined by + = [f + g], where the representative sum f + g is square-integrable if both f and g are, and scalar multiplication by \alpha \in \mathbb{R} (or \mathbb{C}) follows \alpha = [\alpha f]. This structure preserves the square-integrability condition, making L^2 a linear space suitable for applications in partial differential equations and quantum mechanics. In many practical settings, such as the Lebesgue measure on \mathbb{R}^n, the measure space (X, \Sigma, \mu) is \sigma-finite, meaning X can be covered by countably many sets of finite measure. This assumption, which holds for standard Euclidean spaces, enables key results like Fubini's theorem for interchanging integrals and ensures that L^2 functions have \sigma-finite support, simplifying computations and theoretical developments. The notion of L^2 spaces emerged in the early 20th century as part of the broader construction of L^p spaces, with foundational contributions from (1906, on abstract metric spaces), (1902, on integration theory), and (1907, on completeness of square-integrable functions).

Key properties

Norm and inner product

The inner product on the space of square-integrable functions equips it with a structure that allows for notions of angle and length, forming a pre-Hilbert space. For functions f, g in L^2(X, \mathcal{A}, \mu), the inner product is defined by \langle f, g \rangle = \int_X f \, \overline{g} \, d\mu, where the integral is taken with respect to the measure \mu and the bar denotes complex conjugation (which reduces to \langle f, g \rangle = \int_X f g \, d\mu for real-valued functions). This bilinear form is sesquilinear, meaning it is linear in the first argument and conjugate-linear in the second: \langle \alpha f + \beta h, g \rangle = \alpha \langle f, g \rangle + \beta \langle h, g \rangle and \langle f, \alpha g + \beta h \rangle = \overline{\alpha} \langle f, g \rangle + \overline{\beta} \langle f, h \rangle for scalars \alpha, \beta. It is also Hermitian symmetric, satisfying \langle f, g \rangle = \overline{\langle g, f \rangle}, and positive-definite: \langle f, f \rangle \geq 0 with equality if and only if f = 0 almost everywhere. These properties ensure the inner product induces a meaningful geometry on the space. The L^2 norm is derived directly from the inner product as \|f\|_2 = \sqrt{\langle f, f \rangle} = \left( \int_X |f|^2 \, d\mu \right)^{1/2}, which quantifies the "size" of f and aligns with the square-integrability condition \|f\|_2 < \infty. This norm satisfies the standard properties of a vector space norm: positivity (\|f\|_2 \geq 0 with equality precisely when f = 0 almost everywhere), absolute homogeneity (\|\alpha f\|_2 = |\alpha| \|f\|_2 for scalar \alpha), and the triangle inequality \|f + g\|_2 \leq \|f\|_2 + \|g\|_2. The triangle inequality follows from the , a cornerstone result for inner product spaces, which states that |\langle f, g \rangle| \leq \|f\|_2 \|g\|_2, with equality if and only if f and g are linearly dependent (i.e., one is a scalar multiple of the other almost everywhere). To see the connection, note that \|f + g\|_2^2 = \|f\|_2^2 + \|g\|_2^2 + 2 \operatorname{Re} \langle f, g \rangle \leq \|f\|_2^2 + \|g\|_2^2 + 2 \|f\|_2 \|g\|_2 = (\|f\|_2 + \|g\|_2)^2, and taking square roots yields the triangle inequality. Orthogonality in this space is defined via the inner product: two functions f and g are orthogonal if \langle f, g \rangle = 0. This condition implies a geometric perpendicularity, as the Cauchy-Schwarz inequality shows that the "angle" between orthogonal functions is \pi/2, since \cos \theta = \langle f, g \rangle / (\|f\|_2 \|g\|_2) = 0. Orthogonal sets play a key role in decompositions like Fourier series, where basis functions are pairwise orthogonal with respect to the L^2 inner product.

Completeness and Hilbert space aspects

The space L^2(X, \mu) of square-integrable functions, equipped with the L^2 norm \|f\|_2 = \left( \int_X |f|^2 \, d\mu \right)^{1/2}, is a complete metric space. This completeness implies that every Cauchy sequence in L^2(X, \mu) converges to an element within the space, establishing L^2(X, \mu) as a Banach space. To outline the proof of completeness, consider a Cauchy sequence \{f_n\} in L^2(X, \mu). Select a subsequence \{f_{n_k}\} such that \|f_{n_{k+1}} - f_{n_k}\|_2 < 2^{-k}. Define g(x) = \sum_{k=1}^\infty |f_{n_{k+1}}(x) - f_{n_k}(x)|; then \|g\|_2^2 \leq \sum_{k=1}^\infty 2^{-2k} < \infty, so g \in L^2 and the series converges almost everywhere to some f \in L^2(X, \mu) by the Cauchy-Schwarz inequality applied termwise. The subsequence \{f_{n_k}\} converges to f in L^2 norm, and by the Cauchy property of \{f_n\}, the full sequence converges to f in L^2. As a complete inner product space—where the inner product \langle f, g \rangle = \int_X f \overline{g} \, d\mu induces the L^2 norm—L^2(X, \mu) is a . For \sigma-finite measures \mu, L^2(X, \mu) is separable, meaning it admits a countable dense subset, such as the simple functions with rational coefficients on a countable partition of X. This separability ensures the existence of a countable orthonormal basis, allowing representations of elements via Fourier-like series expansions in the basis. The Riesz representation theorem further characterizes the dual space of L^2(X, \mu): every continuous linear functional \Lambda: L^2(X, \mu) \to \mathbb{C} is of the form \Lambda(f) = \langle f, h \rangle for some unique h \in L^2(X, \mu), with \|\Lambda\| = \|h\|_2. This identifies the dual as L^2(X, \mu) itself, underscoring the self-duality of like L^2.

Examples and applications

Square-integrable functions

Square-integrable functions are those for which the integral of the square of their absolute value over the domain is finite, ensuring membership in the L^2 space. Common examples arise on domains like \mathbb{R} or bounded intervals such as [0,1], where the finite L^2 norm distinguishes these functions from others. Continuous functions with compact support, such as bump functions, provide straightforward instances of square-integrable functions on \mathbb{R}. A typical bump function \phi(x) is smooth (C^\infty), positive on a bounded interval like (-1,1), and zero outside, ensuring \int_{-\infty}^\infty |\phi(x)|^2 \, dx < \infty because the support has finite measure and \phi is bounded. Such functions are dense in L^2(\mathbb{R}) and form the basis for test functions in distribution theory. An illustrative example of a function in L^2(\mathbb{R}) but not in L^1(\mathbb{R}) is f(x) = 1/x for x \geq 1 and f(x) = 0 otherwise. Here, \int_1^\infty |f(x)|^2 \, dx = \int_1^\infty x^{-2} \, dx = 1 < \infty, confirming square-integrability, while \int_1^\infty |f(x)| \, dx = \int_1^\infty x^{-1} \, dx = \infty, showing it fails absolute integrability. This highlights how slower decay at infinity can preserve the L^2 norm but violate the L^1 condition on unbounded domains. The Gaussian function g(x) = e^{-x^2/2} on \mathbb{R} is another classic square-integrable example, with its L^2 norm explicitly computable. Specifically, \int_{-\infty}^\infty |g(x)|^2 \, dx = \int_{-\infty}^\infty e^{-x^2} \, dx = \sqrt{\pi}, derived from the known Gaussian integral evaluation, underscoring its rapid decay and central role in probability and analysis. Step functions, particularly indicators of sets with finite measure, also belong to L^2. For a measurable set E \subset \mathbb{R} with \mu(E) < \infty, the indicator function \chi_E(x) = 1 if x \in E and 0 otherwise satisfies \int |\chi_E(x)|^2 \, dx = \mu(E) < \infty. Finite linear combinations of such indicators yield simple functions, which approximate general L^2 elements.

Non-square-integrable functions

Non-square-integrable functions arise when the integral of the square of the function's absolute value diverges over the domain, often due to slow decay at infinity, singularities, unbounded growth, or oscillatory behavior that prevents the integral from converging. Slowly decaying functions at fail to be square-integrable over unbounded domains like \mathbb{R}. For instance, consider f(x) = \frac{1}{|x|^\alpha} for $0 < \alpha \leq \frac{1}{2} on \mathbb{R}. The relevant portion of the squared integral is \int_{|x|>1} |f(x)|^2 \, dx = 2 \int_1^\infty x^{-2\alpha} \, dx, which diverges because the evaluates to \left[ \frac{x^{1-2\alpha}}{1-2\alpha} \right]_1^\infty for $2\alpha < 1, yielding , or logarithmically for \alpha = \frac{1}{2}. A special case is the constant function f(x) = 1, a of degree 0, where \int_\mathbb{R} 1^2 \, dx = \infty due to lack of decay. Singularities cause divergence near points like 0 on bounded intervals such as (0,1). The f(x) = \frac{1}{|x|^\alpha} for \alpha \geq \frac{1}{2} on (0,1) has |f(x)|^2 = x^{-2\alpha}, and \int_0^1 x^{-2\alpha} \, dx diverges since the exponent -2\alpha \leq -1, leading to either or logarithmic at the boundary case \alpha = \frac{1}{2}. Specifically, for \alpha = \frac{1}{2}, f(x) = \frac{1}{\sqrt{|x|}} satisfies \int_{-1}^1 \frac{1}{|x|} \, dx = \infty, confirming it is not square-integrable even locally near 0. Unbounded growth, as seen in polynomials of degree at least 1 on \mathbb{R}, also prevents square-integrability. For example, the linear polynomial f(x) = x yields \int_\mathbb{R} x^2 \, dx = \infty because the integrand grows without bound symmetrically on both sides, with no compensating . Higher-degree polynomials exhibit even faster growth, ensuring . Oscillatory functions with insufficient can lead to through logarithmic accumulation. The function f(x) = \frac{\sin x}{\sqrt{x}} on \mathbb{R}^+ has |f(x)|^2 = \frac{\sin^2 x}{x}, and \int_1^\infty \frac{\sin^2 x}{x} \, dx diverges. This follows from rewriting \sin^2 x = \frac{1 - \cos 2x}{2}, so the integral splits into \frac{1}{2} \int_1^\infty \frac{1}{x} \, dx - \frac{1}{2} \int_1^\infty \frac{\cos 2x}{x} \, dx; the first term diverges harmonically, while the second converges by the Dirichlet test, resulting in overall . In contrast, rapidly decaying functions like Gaussians remain square-integrable.