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Plancherel theorem

The Plancherel theorem is a fundamental result in stating that the defines an on the L^2(\mathbb{R}) of square-integrable functions, preserving the L^2 inner product and norm. Specifically, for f, g \in L^2(\mathbb{R}), it asserts that \int_{-\infty}^{\infty} f(x) \overline{g(x)} \, dx = \int_{-\infty}^{\infty} \hat{f}(\xi) \overline{\hat{g}(\xi)} \, d\xi, where \hat{f} and \hat{g} denote the transforms, and consequently \|f\|_2 = \|\hat{f}\|_2. This identity serves as the continuous analogue of for on the circle. Proven by Swiss mathematician Michel Plancherel in 1910, the theorem builds on earlier ideas, such as Rayleigh's 1889 application to , and was later refined with precise conditions by E. C. Titchmarsh in 1924. Plancherel, born in 1885 and a professor at from 1920, contributed significantly to , , and , with this theorem as his most renowned achievement. The proof typically involves density arguments, inversion, and Fubini's theorem, extending the transform from L^1 \cap L^2 to all of L^2. Beyond its role in establishing the unitarity of the , the Plancherel theorem underpins numerous applications across mathematics and physics. In partial differential equations, it facilitates in solutions via Fourier methods. In , it ensures that wave function normalization in position space matches that in momentum space, preserving total probability under the position-momentum duality. The theorem also extends to more general settings, such as representations of groups and symmetric spaces, enabling spectral decompositions in .

Introduction

Historical Development

The roots of the Plancherel theorem lie in the late 18th-century study of , where developed early ideas on expanding functions using trigonometric series during his work on vibrating strings and sound propagation in the 1760s and 1770s. These efforts provided foundational concepts for representing functions through sums of sines and cosines in physical contexts, though rigorous convergence issues remained unresolved. Building on this, Marc-Antoine Parseval des Chênes formulated a key identity in 1799 relating the integral of the square of a to the sum of the squares of its Fourier coefficients, establishing an energy-preserving relation for trigonometric expansions; an improved version appeared in 1801. An early application of a related energy preservation identity appeared in 1889, when Lord Rayleigh used it in his investigation of . In the early , the theorem's development advanced with the establishment of rigorous tools. The of 1907, proven independently by and Ernst Sigismund Fischer, demonstrated the completeness of the space of square-integrable functions (L²), providing the framework essential for handling transforms on infinite domains. This result was pivotal as a prerequisite for extending to continuous spectra. Michel Plancherel established the core result for transforms on the real line in his 1910 "Contribution à l'étude de la représentation d'une fonction arbitraire par des intégrales définies," where he proved the identity for square-integrable functions, generalizing to integrals. During the 1910s, contributed significantly to the theory of integrals through works such as his 1911 on 's double integral and , refining properties and analytic techniques that supported Plancherel's framework. In 1924, E. C. Titchmarsh provided refinements, establishing precise conditions under which the theorem holds and extending related results. The identity became widely known as "Plancherel's theorem" in the , particularly through Antoni Zygmund's seminal 1935 monograph Trigonometrical Series, which integrated and popularized the result within the broader context of .

Initial Statement for the Real Line

The Plancherel theorem provides the foundational statement for the on the real line, establishing its preservation of the L^2 norm. For a f \in L^1(\mathbb{R}) \cap L^2(\mathbb{R}), the is defined as \mathcal{F}f(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x \xi} \, dx. This convention places the $2\pi factor within the exponent, ensuring in the inversion formula without additional constants in the transform itself. The theorem asserts that for any f \in L^2(\mathbb{R}), \int_{-\infty}^{\infty} |f(x)|^2 \, dx = \int_{-\infty}^{\infty} |\mathcal{F}f(\xi)|^2 \, d\xi, known as the Plancherel identity, which equates the L^2 norms of f and its Fourier transform. This extends the discrete Parseval identity to the continuous case and implies that the transform preserves inner products: \langle \mathcal{F}f, \mathcal{F}g \rangle_{L^2} = \langle f, g \rangle_{L^2} for all f, g \in L^2(\mathbb{R}). This identity extends the result from the dense subspace L^1(\mathbb{R}) \cap L^2(\mathbb{R}) to all of L^2(\mathbb{R}) via completion. Moreover, the Fourier transform uniquely extends to a unitary operator on L^2(\mathbb{R}) with respect to the L^2 inner product, meaning it preserves inner products: \langle \mathcal{F}f, \mathcal{F}g \rangle_{L^2} = \langle f, g \rangle_{L^2} for all f, g \in L^2(\mathbb{R}). This unitarity underscores the transform's role as an isometry, confirming that the Plancherel identity holds in the Hilbert space sense. Notation for the Fourier transform varies historically and across fields, particularly in the placement of the $2\pi factor. In some mathematical conventions, the forward transform omits the $2\pi in the exponent and places a $1/(2\pi) factor in the inverse, while engineering contexts often favor the symmetric form with $2\pi in the exponent and no prefactors. These variations, dating back to early 20th-century developments, ensure the Plancherel identity adapts accordingly but maintain the underlying norm preservation.

Core Formulation

Plancherel Identity for L²(ℝ)

The Plancherel identity for L^2(\mathbb{R}) asserts that the Fourier transform \mathcal{F} extends to a unitary operator on this Hilbert space, preserving the inner product: for all f, g \in L^2(\mathbb{R}), \langle f, g \rangle_{L^2(\mathbb{R})} = \langle \mathcal{F} f, \mathcal{F} g \rangle_{L^2(\mathbb{R})}. This implies the norm preservation \|\mathcal{F} f\|_{L^2(\mathbb{R})} = \|f\|_{L^2(\mathbb{R})}. Building on the initial statement for the dense subspace L^1(\mathbb{R}) \cap L^2(\mathbb{R}), where the Fourier transform is initially defined and the identity holds, the operator extends uniquely by continuity to all of L^2(\mathbb{R}) as a bounded linear map, and the Plancherel identity follows by density. A key consequence is Parseval's relation, obtained as the special case g = f, which equates the L^2 norms squared and underscores energy conservation in the transform domain. Additionally, the unitarity of \mathcal{F} ensures the existence of an inverse transform, allowing recovery of f from \mathcal{F} f via a similar integral formula. Under the normalization \mathcal{F} f(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i \xi x} \, dx, the identity takes the explicit form \|f\|_2^2 = \int_{-\infty}^{\infty} |\mathcal{F} f(\xi)|^2 \, d\xi.

Extension to ℝⁿ

The multidimensional extends the one-dimensional case to functions on \mathbb{R}^n, defined for f \in L^1(\mathbb{R}^n) by \mathcal{F}f(\xi) = \int_{\mathbb{R}^n} f(x) e^{-2\pi i x \cdot \xi} \, dx, where x \cdot \xi denotes the and the is taken with respect to the on \mathbb{R}^n. This definition applies initially to integrable functions but extends by density to the \mathcal{S}(\mathbb{R}^n) and further to L^2(\mathbb{R}^n). The Plancherel theorem in this setting asserts that the Fourier transform is an on L^2(\mathbb{R}^n): for any f \in L^2(\mathbb{R}^n), \int_{\mathbb{R}^n} |f(x)|^2 \, dx = \int_{\mathbb{R}^n} |\mathcal{F}f(\xi)|^2 \, d\xi, with equality holding with respect to . This identity preserves the L^2 norm and confirms that \mathcal{F} maps L^2(\mathbb{R}^n) unitarily onto itself. The proof proceeds by exploiting the separability of \mathbb{R}^n as a product space. Specifically, L^2(\mathbb{R}^n) is the Hilbert space \bigotimes_{k=1}^n L^2(\mathbb{R}), and the multidimensional decomposes into iterated one-dimensional transforms along each coordinate via Fubini's theorem, which justifies interchanging the . Applying the one-dimensional Plancherel identity to each factor and combining the results using the product structure yields the full identity on the dense subspace of functions, which then extends by continuity to all of L^2(\mathbb{R}^n). Normalization conventions must align with the one-dimensional case for . The form above uses no additional scaling factor, ensuring the directly; however, some texts incorporate a (2\pi)^{-n/2} prefactor in the transform definition to maintain unitarity explicitly across dimensions. This adjustment accounts for the volume growth in higher dimensions while preserving the equality of L^2 norms.

Proof Techniques

Density Argument Using Schwartz Functions

The \mathcal{S}(\mathbb{R}) comprises infinitely differentiable functions f: \mathbb{R} \to \mathbb{C} such that f and all its derivatives decay faster than any polynomial at infinity, i.e., \sup_{x \in \mathbb{R}} |x|^k |f^{(m)}(x)| < \infty for all integers k, m \geq 0. This space forms a dense of L^2(\mathbb{R}) under the L^2 , enabling approximations of arbitrary L^2 functions by elements of \mathcal{S}(\mathbb{R}). The \mathcal{F}: L^1(\mathbb{R}) \cap L^2(\mathbb{R}) \to L^2(\mathbb{R}), defined by \hat{f}(\xi) = \int_{\mathbb{R}} f(x) e^{-2\pi i x \xi} \, dx, restricts to a continuous, bijective map \mathcal{F}: \mathcal{S}(\mathbb{R}) \to \mathcal{S}(\mathbb{R}) with continuous inverse given by the inverse Fourier transform. To establish the Plancherel identity on \mathcal{S}(\mathbb{R}), consider the inner product preservation: for f, g \in \mathcal{S}(\mathbb{R}), yields \langle \hat{f}, \hat{g} \rangle_{L^2(\mathbb{R})} = \langle f, g \rangle_{L^2(\mathbb{R})}, confirming that \mathcal{F} is an on this . A concrete verification arises from the f(x) = e^{-\pi x^2}, which belongs to \mathcal{S}(\mathbb{R}) and satisfies \hat{f} = f, hence \|f\|_{L^2(\mathbb{R})} = \|\hat{f}\|_{L^2(\mathbb{R})}. This example illustrates the norm preservation, extendable to all of \mathcal{S}(\mathbb{R}) via the inner product relation. The extension to all of L^2(\mathbb{R}) proceeds by density. For any f \in L^2(\mathbb{R}), there exists a sequence \{\phi_n\}_{n=1}^\infty \subset \mathcal{S}(\mathbb{R}) such that \|\phi_n - f\|_{L^2(\mathbb{R})} \to 0 as n \to \infty. Since \mathcal{F} is an on the dense \mathcal{S}(\mathbb{R}), the sequence \{\hat{\phi}_n\}_{n=1}^\infty is Cauchy in L^2(\mathbb{R}): \|\hat{\phi}_n - \hat{\phi}_m\|_{L^2(\mathbb{R})} = \|\phi_n - \phi_m\|_{L^2(\mathbb{R})} \to 0 as n, m \to \infty. Thus, \hat{\phi}_n converges in L^2(\mathbb{R}) to some g \in L^2(\mathbb{R}). Defining the extension \tilde{\mathcal{F}} f = g, continuity of this operator on L^2(\mathbb{R}) follows from the density approximation, and the Plancherel identity holds: \|\tilde{\mathcal{F}} f\|_{L^2(\mathbb{R})} = \|f\|_{L^2(\mathbb{R})}, for all f \in L^2(\mathbb{R}), as the isometry on the dense set \mathcal{S}(\mathbb{R}) extends uniquely by . This establishes \tilde{\mathcal{F}} as a on L^2(\mathbb{R}).

Direct Analytic Proof

The direct analytic proof of the Plancherel theorem establishes the identity for functions in L^1(\mathbb{R}) \cap L^2(\mathbb{R}) by leveraging the Fourier inversion theorem and explicit computations involving approximate identities, with complex analysis employed to evaluate key integrals. Consider functions f, g \in L^1(\mathbb{R}) \cap L^2(\mathbb{R}), where the Fourier transform is defined as \hat{f}(\xi) = \int_{\mathbb{R}} f(x) e^{-i \xi x} \, dx. To derive the Parseval identity \int_{\mathbb{R}} f(x) \overline{g(x)} \, dx = \frac{1}{2\pi} \int_{\mathbb{R}} \hat{f}(\xi) \overline{\hat{g}(\xi)} \, d\xi, form the convolution h = f * \tilde{g}, with \tilde{g}(x) = \overline{g(-x)}. Since f, g \in L^1(\mathbb{R}), h is continuous and belongs to L^1(\mathbb{R}), allowing application of the Fourier inversion formula: h(x) = \frac{1}{2\pi} \int_{\mathbb{R}} \hat{h}(\xi) e^{i x \xi} \, d\xi almost everywhere. Evaluating at x = 0 yields h(0) = \int_{\mathbb{R}} f(y) \overline{g(y)} \, dy = \frac{1}{2\pi} \int_{\mathbb{R}} \hat{h}(\xi) \, d\xi. As \hat{h}(\xi) = \hat{f}(\xi) \overline{\hat{g}(\xi)} by the convolution theorem, the identity follows upon interchanging the order of integration, justified by Fubini's theorem since \hat{f}, \hat{g} \in L^2(\mathbb{R}). For the norm equality \|f\|_2^2 = \frac{1}{2\pi} \|\hat{f}\|_2^2, apply the Parseval identity with g = f, or use polarization. To verify the constant and extend within L^1(\mathbb{R}) \cap L^2(\mathbb{R}), approximate f in the L^2-norm by convolutions with approximate identities, such as rectangular kernels k_\lambda(x) = \frac{1}{2\lambda} \chi_{[-\lambda, \lambda]}(x), which converge to the Dirac delta as \lambda \to \infty. The Fourier transform of k_\lambda is the sinc function \hat{k}_\lambda(\xi) = \frac{\sin(\lambda \xi)}{\lambda \xi}, and the L^2-convergence follows from the density of such approximations in L^2(\mathbb{R}). The Plancherel constant is determined by direct computation for the rectangular function itself: \|\chi_{[-a,a]}\|_2^2 = 2a, while \|\widehat{\chi_{[-a,a]}}\|_2^2 = 4 \int_{-\infty}^{\infty} \frac{\sin^2(a \xi)}{\xi^2} \, d\xi = 4\pi a, confirming the factor after rescaling since \frac{1}{2\pi} \cdot 4\pi a = 2a. The integral \int_{-\infty}^\infty \frac{\sin^2 \xi}{\xi^2} \, d\xi = \pi is evaluated using contour integration in the complex plane, for example by considering the function (1 - e^{i z})/z^2 over a suitable indented semicircular contour in the upper half-plane and applying the residue theorem, or alternatively by differentiation under the integral sign applied to the Dirichlet integral \int_{-\infty}^{\infty} \frac{\sin \xi}{\xi} \, d\xi = \pi. This confirms the norm equality holds. This approach is limited to the real line \mathbb{R} and functions in L^1(\mathbb{R}) \cap L^2(\mathbb{R}); extension to the full L^2(\mathbb{R}) requires a density argument, such as approximating general L^2 functions by those in L^1 \cap L^2 using the completeness of the , highlighting the need for complementary abstract techniques.

Generalizations

Abelian Locally Compact Groups

In the context of , the Plancherel theorem generalizes to arbitrary locally compact abelian (LCA) groups, abstracting the and its L²-preserving properties from the Euclidean setting. Let G be an LCA group equipped with a left \mu, which is a non-zero, positive, σ-finite satisfying \mu(aE) = \mu(E) for all a \in G and Borel sets E \subseteq G. The dual group \hat{G} consists of all continuous group homomorphisms \chi: G \to \mathbb{T}, where \mathbb{T} is the circle group of complex numbers with modulus 1; these are the characters of G. By the theorem, \hat{G} is also an LCA group, and the duality map G \to \hat{\hat{G}} is a topological . The on G is initially defined for functions f \in L^1(G, \mu) \cap L^2(G, \mu) by \mathcal{F}f(\chi) = \int_G f(g) \overline{\chi(g)} \, d\mu(g), \quad \chi \in \hat{G}. This operator extends uniquely to a bounded on all of L^2(G, \mu). There exists a unique (up to positive scalar multiple) Haar measure \nu on \hat{G}, known as the Plancherel measure, such that \mathcal{F} becomes a unitary operator from L^2(G, \mu) onto L^2(\hat{G}, \nu). The Plancherel theorem then asserts the identity \int_G |f(g)|^2 \, d\mu(g) = \int_{\hat{G}} |\mathcal{F}f(\chi)|^2 \, d\nu(\chi) for all f \in L^2(G, \mu), with the normalization chosen so that the constant is 1. This result, establishing the unitarity of the Fourier transform, was proved independently by several authors in the 1940s using integral representations and properties of positive definite functions. The Plancherel measure \nu plays a central role, as it ensures the inversion formula and convolution theorems hold in the L² sense, mirroring the classical case. For compact G, \hat{G} is discrete, and \nu is counting measure, reducing the theorem to the Peter-Weyl theorem for finite-dimensional representations. For discrete G, \hat{G} is compact, and \mu is counting measure. Specific examples illustrate the abstraction: when G = \mathbb{R}, \hat{G} = \mathbb{R}, and \nu(d\xi) = d\xi / (2\pi) (Lebesgue measure scaled), recovering the standard Plancherel identity for the real line. Similarly, for G = \mathbb{Z}, \hat{G} = \mathbb{T} (the torus), \nu is normalized Haar measure on \mathbb{T}, and the theorem yields the Plancherel identity for the discrete Fourier transform on \ell^2(\mathbb{Z}). These cases highlight how the general framework unifies classical Fourier analysis across different group structures.

Non-Abelian Locally Compact Groups

The Plancherel theorem extends to non-abelian locally compact groups G, assuming G is unimodular, second countable, and of type I, through the lens of its irreducible unitary representations. The unitary dual \widehat{G} consists of equivalence classes of irreducible unitary representations \pi: G \to U(\mathcal{H}_\pi), where \mathcal{H}_\pi is the of \pi. The left of G on L^2(G) decomposes as a direct over \widehat{G}: L^2(G) \cong \int^\oplus_{\widehat{G}} \mathcal{H}_\pi \otimes \overline{\mathcal{H}_\pi} \, d\mu(\pi), where \mu is the Plancherel measure on \widehat{G} and \overline{\mathcal{H}_\pi} denotes the conjugate space, equivalent to the space of Hilbert-Schmidt operators on \mathcal{H}_\pi. The for functions f \in L^1(G) \cap L^2(G) is defined operator-valuedly by \mathcal{F}f(\pi) = \pi(f) = \int_G f(g) \pi(g) \, dg, where the is understood in the weak topology and yields a Hilbert-Schmidt operator on \mathcal{H}_\pi. The Plancherel theorem asserts that \|f\|_{L^2(G)}^2 = \int_{\widehat{G}} \|\pi(f)\|_{\mathrm{HS}}^2 \, d\mu(\pi), with \|\cdot\|_{\mathrm{HS}} the Hilbert-Schmidt norm, \|\pi(f)\|_{\mathrm{HS}}^2 = \mathrm{Tr}(|\pi(f)|^2). This identity preserves the L^2-norm under the and identifies \mu uniquely up to scalar multiple as the measure making the decomposition unitary. This formulation, central to on non-commutative groups, recovers the abelian case upon restriction to one-dimensional representations, where \mathcal{H}_\pi \cong \mathbb{C} and the Hilbert-Schmidt norm reduces to the modulus squared. The key result was established by Jacques Dixmier in the 1950s, building on earlier work by Gelfand and Naimark, with a comprehensive treatment in his monograph on C^*-algebras.

Applications and Implications

In Harmonic Analysis

In harmonic analysis, the Plancherel theorem provides a foundational of the L^2(G) for a unimodular G, expressed as a direct over the unitary \hat{G}. This decomposition integrates irreducible unitary representations \pi \in \hat{G} with respect to the Plancherel measure \mu, allowing functions on G to be broken down into components corresponding to these representations. Such a structure underpins on groups, facilitating the analysis of operators and functions through their irreducible constituents, much like eigenvalue decompositions in linear algebra. A significant application arises in the uniqueness of the on G. The Plancherel measure \mu is canonically tied to the left dg on G, such that rescaling dg by a positive constant \lambda scales \mu by \lambda^{-1}; this reciprocity ensures that the Plancherel theorem determines the up to a unique normalization, confirming its role in standardizing measures for . Furthermore, the theorem supports inversion formulas for convolutions in the group algebra L^1(G). By defining the \hat{f}(\pi) via the action of f \in L^1(G) on irreducible representations, convolution f * g maps to pointwise multiplication \hat{f} \cdot \hat{g} on \hat{G}, enabling recovery of the original convolution through the inverse transform integrated against the Plancherel measure. The Plancherel theorem also extends classical results like Wiener's tauberian theorems to broader settings in non-abelian . These extensions leverage the spectral support in \hat{G} to characterize ideals and approximate identities in the convolution algebra, determining when dense subalgebras coincide with L^1(G) based on the Plancherel measure's properties. For compact groups, the Plancherel theorem specializes to the Peter-Weyl theorem, where L^2(G) decomposes as the over all irreducible representations \pi of \dim \pi copies of the representation space \mathcal{H}_\pi, with the matrix coefficients of these representations forming a complete . On discrete groups, it manifests as the \ell^2- of the matrix coefficients of irreducible representations, yielding a discrete analog of the where the integrates these coefficients with multiplicities given by the dimensions. An important implication concerns the convolution algebra structure: the Plancherel theorem ensures that the left convolution action of L^1(G) on L^2(G) is bounded, with the operator norm of f \in L^1(G) equal to the essential supremum over \pi \in \hat{G} of the operator norm \|\pi(f)\| with respect to the Plancherel measure, thereby controlling L^1(G) * L^1(G) \subset L^1(G) through spectral bounds on the group .

In Quantum Mechanics and PDEs

In , wave functions describing the state of particles are elements of the L^2(\mathbb{R}^3), where the Plancherel theorem guarantees that the L^2 norm is preserved under the , establishing an isometric correspondence between position space and space representations. This preservation ensures that the total probability density integrates to unity in both domains, linking observables like position and directly. Specifically, the operator in position space, given by \int_{\mathbb{R}^3} |\nabla \psi(x)|^2 \, dx (up to scaling factors involving \hbar and mass), transforms via Plancherel to \int_{\mathbb{R}^3} |\xi|^2 |\hat{\psi}(\xi)|^2 \, d\xi in space, where \hat{\psi} denotes the of \psi, thereby equating the values of across representations. The Plancherel theorem also plays a crucial role in partial differential equations (PDEs), particularly in solving evolution equations through methods while maintaining L^2 control. For the u_t = \Delta u on \mathbb{R}^n with initial data u(0) = \phi \in L^2(\mathbb{R}^n), the converts the PDE into \partial_t \hat{u}(t, \xi) = -|\xi|^2 \hat{u}(t, \xi), yielding the solution \hat{u}(t, \xi) = e^{-t |\xi|^2} \hat{\phi}(\xi). Plancherel then implies that the L^2 of the solution decays monotonically, \|u(t)\|_{L^2} = \left( \int_{\mathbb{R}^n} e^{-2t |\xi|^2} |\hat{\phi}(\xi)|^2 \, d\xi \right)^{1/2} \leq \|\phi\|_{L^2}, quantifying the dissipative nature of heat flow without altering the underlying preservation. This framework extends to other parabolic PDEs, ensuring stability in L^2 settings. A key implication in quantum mechanics is the derivation of the Heisenberg uncertainty principle from the unitarity of the Fourier transform, as affirmed by Plancherel. For a wave function f \in L^2(\mathbb{R}) with finite position and frequency variances, the principle yields \|x f\|_{L^2} \|\xi \hat{f}\|_{L^2} \geq \frac{1}{2} \|f\|_{L^2}^2 (in the convention where the Fourier transform is unitary and \xi is angular frequency). In the time-dependent Schrödinger equation i \partial_t \psi = -\Delta \psi + V \psi (for potential V), the free evolution (V=0) is unitary in L^2, and Plancherel confirms that the representation \hat{\psi}(t, \xi) = e^{-i t |\xi|^2} \hat{\psi}(0, \xi) preserves the L^2 at all times, reflecting conservation of probability. This duality facilitates of dispersive propagation. In theory, Plancherel underpins the completeness of wave operators, ensuring that asymptotic states in momentum space recover the full L^2 of incoming , essential for computing cross-sections in potential problems.