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Parseval's theorem

Parseval's theorem is a fundamental result in Fourier analysis that equates the energy (measured by the squared L² norm) of a periodic function to the sum of the squared magnitudes of its Fourier coefficients, serving as a Pythagorean theorem for infinite-dimensional Hilbert spaces.^[1] Specifically, for a 2π-periodic square-integrable function f on [-\pi, \pi] with Fourier series \frac{a_0}{\sqrt{2}} + \sum_{n=1}^\infty (a_n \cos(nx) + b_n \sin(nx)), where the coefficients are given by a_0 = \frac{1}{\pi} \int_{-\pi}^\pi f(x) \frac{dx}{\sqrt{2}}, a_n = \frac{1}{\pi} \int_{-\pi}^\pi f(x) \cos(nx) \, dx, and b_n = \frac{1}{\pi} \int_{-\pi}^\pi f(x) \sin(nx) \, dx, the theorem states that \|f\|^2 = a_0^2 + \sum_{n=1}^\infty (a_n^2 + b_n^2), with \|f\|^2 = \frac{1}{\pi} \int_{-\pi}^\pi f(x)^2 \, dx.^[1] Named after the French mathematician Marc-Antoine Parseval (1755–1836), the theorem originated in his 1799 memoir on summing series, which was later adapted to Fourier series by Joseph Fourier and others in the early 19th century.^[2]^[3] Parseval published only a few works, but this identity highlighted the orthogonality of trigonometric functions and laid groundwork for modern harmonic analysis.^[4] A related form, often called Parseval's identity or Plancherel's theorem in the continuous case, extends to Fourier transforms on the real line: for a function f \in L^2(\mathbb{R}) and its Fourier transform F(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(x) e^{ikx} \, dx, the theorem asserts that \int_{-\infty}^\infty |f(x)|^2 \, dx = \int_{-\infty}^\infty |F(k)|^2 \, dk.^[5] This version preserves the total energy between the time (or spatial) domain and the frequency domain, making it essential for applications in signal processing, where it ensures that operations in the Fourier domain do not alter the overall power of a signal.^[6]^[7] In broader functional analysis, Parseval's theorem generalizes to any orthonormal basis in a separable Hilbert space, stating that for an element f expanded as f = \sum c_n e_n, the norm satisfies \|f\|^2 = \sum |c_n|^2.^[2] This principle underpins concepts like frame theory and wavelet analysis, and it has profound implications in physics, such as in quantum mechanics for preserving probabilities via unitary transformations.^[8] The theorem's emphasis on energy conservation distinguishes it as a cornerstone of linear systems theory and spectral methods.^[9]

History and Overview

Historical Development

Parseval's theorem traces its origins to the work of French mathematician Marc-Antoine Parseval des Chênes, who presented a memoir to the Académie des Sciences on April 5, 1799, titled Mémoire sur les séries et sur l'intégration complète d'une équation aux différences partielles linéaires du second ordre, à coefficients constants. In this paper, Parseval addressed the summation of infinite series derived from expansions involving powers of complex numbers on the unit circle, using de Moivre's formula; he stated the key identity without proof, deeming it self-evident, and focused on its application to solving partial difference equations rather than trigonometric expansions. The result appeared in print only in 1806 as part of the academy's proceedings, after revisions in a 1801 note that resolved issues with imaginary components by considering real parts of the series.^[10] Although Parseval's identity initially stood apart from trigonometric analysis, it found application to Fourier series through the efforts of subsequent mathematicians in the early 19th century, building on Joseph Fourier's foundational ideas. Fourier introduced the concept of representing arbitrary functions as trigonometric series in his 1807 prize memoir and elaborated it in Théorie analytique de la chaleur (1822), where he employed similar summation relations for heat conduction solutions, implicitly aligning with Parseval's framework without direct citation. The explicit connection to Fourier coefficients was formalized by Augustin-Louis Cauchy in his 1826 analysis of trigonometric series convergence, marking the theorem's adaptation as a tool for verifying the completeness of Fourier expansions.^[10]^[11] In the late 19th century, the theorem gained prominence in physics through John William Strutt, Lord Rayleigh, who linked it to energy conservation principles in wave phenomena. Rayleigh's investigations into sound and vibration, particularly in his 1889 paper "On the character of the complete radiation at a given temperature" published in the Philosophical Magazine, demonstrated that the theorem equates the total energy of a waveform to the sum of energies in its harmonic components, earning it the alias "Rayleigh's energy theorem." This interpretation, spanning Rayleigh's broader works from the 1870s to 1900, underscored its utility in acoustics and optics.^[2] By the 20th century, Parseval's theorem evolved into a foundational element of functional analysis, integrated into the study of infinite-dimensional spaces. David Hilbert incorporated it in his 1904–1906 series of papers on integral equations, proving a general form relating the L² norm of a function to the sum of squares of its projections onto an orthonormal basis, as detailed in Grundzüge einer allgemeinen Theorie der Integralgleichungen. Erhard Schmidt extended this in 1905 by relaxing function class restrictions, while Ernst Fischer and Frigyes Riesz independently established the Riesz-Fischer theorem in 1907, confirming the isomorphism between l² sequences and L² functions via Parseval's identity. These developments, culminating in the abstract Hilbert space framework by the 1920s, positioned the theorem as a cornerstone for spectral theory and operator algebras in modern mathematics.^[12]

Core Concept and Importance

Parseval's theorem establishes a fundamental equality between the energy of a function, quantified as the square of its L² norm, and the energy of its representation in the Fourier domain. This principle asserts that the total energy remains invariant under the Fourier transform, meaning the integrated squared magnitude of the original function equals the corresponding sum or integral over its Fourier coefficients or transform values. This energy preservation underscores the theorem's role as a cornerstone of Fourier analysis, ensuring that transformations between time and frequency domains do not alter the intrinsic power or norm of the signal.^[13] The theorem's significance lies in revealing the unitary nature of Fourier analysis, which preserves not only norms but also inner products between functions. This unitarity implies that the Fourier basis functions, such as complex exponentials, form an orthonormal set, allowing the transform to act as an isometry in the L² space. Intuitively, Parseval's theorem extends the Pythagorean theorem to infinite-dimensional Hilbert spaces, where the energy of a function is the sum of the squared magnitudes of its projections onto orthogonal basis elements, much like the length of a vector is the root sum of squares of its components. This geometric insight validates the completeness and orthogonality of the Fourier basis, making it a powerful tool for decomposition and reconstruction.^[13]^[14] By confirming these properties, Parseval's theorem bolsters the reliability of Fourier methods for function approximation and signal decomposition, ensuring that partial sums of Fourier series provide optimal L² approximations. It plays a crucial role in theoretical analysis by enabling the evaluation of difficult integrals through simpler sums of coefficient magnitudes, which is essential for proving convergence and stability in expansions. In numerical contexts, this facilitates efficient computations in areas like spectral methods and filtering, where energy conservation guarantees the accuracy of domain transformations without information loss. Overall, the theorem's emphasis on preservation and equivalence drives its broad impact across mathematics, physics, and engineering.^[13]^[5]

Mathematical Formulations

For Fourier Series

Parseval's theorem for Fourier series applies to periodic functions and equates the integral of the square of the function over one period to a sum involving the squares of its Fourier coefficients. For a 2π-periodic function f on the interval [-\pi, \pi], the theorem states that

\frac{1}{\pi} \int_{-\pi}^{\pi} |f(x)|^2 \, dx = \frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2),

where the Fourier coefficients are defined as a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \, dx, a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx for n \geq 1, and b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx for n \geq 1.^[15] This formulation generalizes to functions with arbitrary period $2L. For a $2L-periodic function f on [-L, L], Parseval's theorem gives

\frac{1}{L} \int_{-L}^{L} |f(x)|^2 \, dx = \frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2),

with coefficients a_0 = \frac{1}{L} \int_{-L}^{L} f(x) \, dx, a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) \, dx for n \geq 1, and b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) \, dx for n \geq 1.^[13] The theorem holds under the assumption that f is square-integrable over the period, meaning \int |f(x)|^2 \, dx < \infty, and periodic with the specified period; the Fourier series converges to f in the L^2 sense.^[15] An equivalent complex form uses exponential basis functions. For a $2\pi-periodic square-integrable f on [-\pi, \pi],

\frac{1}{2\pi} \int_{-\pi}^{\pi} |f(x)|^2 \, dx = \sum_{n=-\infty}^{\infty} |c_n|^2,

where the complex coefficients are c_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx} \, dx. This generalizes to period $2L as

\frac{1}{2L} \int_{-L}^{L} |f(x)|^2 \, dx = \sum_{n=-\infty}^{\infty} |c_n|^2,

with c_n = \frac{1}{2L} \int_{-L}^{L} f(x) e^{-i n \pi x / L} \, dx.^[13]^[15] In the real form, the coefficients a_n and b_n correspond to the real and imaginary parts of the complex coefficients via Euler's formula, relating cosine and sine terms to exponentials e^{\pm i n x}; for real-valued f, c_{-n} = \overline{c_n}, ensuring a_n = 2 \operatorname{Re}(c_n) and b_n = -2 \operatorname{Im}(c_n) for n \geq 1, with a_0 = 2 c_0.^[15]

For Continuous Fourier Transforms

Parseval's theorem for the continuous Fourier transform addresses non-periodic, square-integrable functions on the real line. For f \in L^2(\mathbb{R}), the theorem states that the energy of the function equals that of its Fourier transform, scaled appropriately by the normalization convention:

\int_{-\infty}^{\infty} |f(t)|^2 \, dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |F(\omega)|^2 \, d\omega,

where F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} \, dt denotes the Fourier transform in terms of angular frequency \omega. This relation holds under the assumption that f is square-integrable, meaning \|f\|_{L^2(\mathbb{R})} < \infty, ensuring the integrals converge.^[16] The rigorous formulation in L^2(\mathbb{R}) is known as the Plancherel theorem, which establishes that the Fourier transform extends to a bounded linear operator on this Hilbert space and acts as an isometry up to a constant factor, preserving the L^2-norm structure. In particular, it guarantees that the transform maps L^2(\mathbb{R}) onto itself, maintaining the inner product equality \langle F, G \rangle_{L^2} = 2\pi \langle f, g \rangle_{L^2} for f, g \in L^2(\mathbb{R}). This isometry property is crucial for applications requiring norm preservation in function spaces.^[17] Different normalizations lead to variations in the theorem's statement. In the unitary convention, the Fourier transform is defined as \hat{f}(\xi) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(t) e^{-i \xi t} \, dt, yielding the isometry directly:

\int_{-\infty}^{\infty} |f(t)|^2 \, dt = \int_{-\infty}^{\infty} |\hat{f}(\xi)|^2 \, d\xi,

without additional scaling, as the operator is unitary on L^2(\mathbb{R}). Another common variation uses ordinary frequency \nu = \omega / (2\pi), with \hat{f}(\nu) = \int_{-\infty}^{\infty} f(t) e^{-i 2\pi \nu t} \, dt, resulting in the unscaled form \int |f|^2 \, dt = \int |\hat{f}|^2 \, d\nu. These adjustments ensure consistency across physics, engineering, and mathematics contexts, where the choice of frequency variable and exponential factor affects the scaling.^[18] The continuous Fourier transform arises as the limiting case of the Fourier series when the period tends to infinity, transitioning from discrete sums over periodic functions to integrals over aperiodic ones.^[13]

For Discrete Fourier Transforms

Parseval's theorem for the discrete Fourier transform (DFT) states that for a finite-length sequence x of length N, the energy in the time domain equals the scaled energy in the frequency domain:

\sum_{n=0}^{N-1} |x|^2 = \frac{1}{N} \sum_{k=0}^{N-1} |X|^2,

where X denotes the N-point DFT of x.^[19] This relation preserves the total energy of the signal through the transform, interpreting the DFT as an orthogonal transformation that redistributes but does not alter the signal's \ell^2-norm.^[20] The theorem assumes the sequence is finite or treated as one period of a periodic extension, ensuring the DFT captures the full energy without aliasing from infinite tails.^[21] For the discrete-time Fourier transform (DTFT), which applies to infinite or aperiodic discrete sequences, Parseval's theorem takes an integral form:

\sum_{n=-\infty}^{\infty} |x|^2 = \frac{1}{2\pi} \int_{-\pi}^{\pi} |X(e^{j\omega})|^2 \, d\omega,

where X(e^{j\omega}) is the DTFT of x. This version equates the total energy of the discrete-time signal to the average power over one period of the frequency response, assuming the sequence has finite energy (i.e., is in \ell^2(\mathbb{Z})).^[22] In digital signal processing, this relation links time-domain energy directly to the integral of the squared magnitude of the frequency-domain representation, facilitating energy computations for sampled signals.^[23] The scaling factors in these formulations—$1/N for the DFT and $1/2\pi for the DTFT—arise from the normalization conventions of the transforms, ensuring unitarity up to a constant.^[19] These factors have key implications for power spectral density (PSD) estimation: in the DFT case, the PSD is often approximated as |X|^2 / N, so Parseval's theorem confirms that integrating the PSD over frequency yields the total signal power.^[20] Similarly, for the DTFT, the PSD S_{xx}(e^{j\omega}) = |X(e^{j\omega})|^2 integrates to the signal energy when scaled by $1/2\pi, providing a foundation for analyzing power distribution in discrete signals. This discrete adaptation mirrors the energy conservation in continuous Fourier transforms but uses sums and integrals suited to sampled data.^[22]

Proofs and Derivations

Derivation for Fourier Series

Consider a function f(x) that is integrable over the interval [-\pi, \pi] and periodic with period $2\pi. Its Fourier series expansion is given by

f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty \left( a_n \cos(nx) + b_n \sin(nx) \right),

where the coefficients are

a_0 = \frac{1}{\pi} \int_{-\pi}^\pi f(x) \, dx, \quad a_n = \frac{1}{\pi} \int_{-\pi}^\pi f(x) \cos(nx) \, dx, \quad b_n = \frac{1}{\pi} \int_{-\pi}^\pi f(x) \sin(nx) \, dx

for n \geq 1.^[24] To derive Parseval's theorem, compute the L^2 norm squared of f(x), defined as

\|f\|^2 = \frac{1}{\pi} \int_{-\pi}^\pi |f(x)|^2 \, dx.

Assuming f belongs to the L^2[-\pi, \pi] space, the Fourier series converges to f in the L^2 sense, meaning the partial sums approach f in the L^2 norm.^[1] Substitute the Fourier series expansion into the integral:

\int_{-\pi}^\pi |f(x)|^2 \, dx = \int_{-\pi}^\pi \left( \frac{a_0}{2} + \sum_{n=1}^\infty \left( a_n \cos(nx) + b_n \sin(nx) \right) \right)^2 dx.

Expanding the square yields the square of the constant term, the squares of the individual series terms, and all cross terms between different basis functions. The integral of the squared constant term is

\int_{-\pi}^\pi \left( \frac{a_0}{2} \right)^2 dx = \left( \frac{a_0}{2} \right)^2 \cdot 2\pi = \frac{\pi a_0^2}{2}.

For the squared cosine terms,

\int_{-\pi}^\pi a_n^2 \cos^2(nx) \, dx = a_n^2 \int_{-\pi}^\pi \frac{1 + \cos(2nx)}{2} \, dx = a_n^2 \cdot \pi

for each n \geq 1, since the \cos(2nx) integral vanishes over the interval. Similarly, for sine terms,

\int_{-\pi}^\pi b_n^2 \sin^2(nx) \, dx = b_n^2 \cdot \pi.

Thus, the sum of these squared terms contributes

\sum_{n=1}^\infty \left( a_n^2 \pi + b_n^2 \pi \right) = \pi \sum_{n=1}^\infty (a_n^2 + b_n^2). $$<sup></sup>[](https://math.mit.edu/~gs/cse/websections/cse41.pdf) The cross terms vanish due to the orthogonality of the trigonometric basis functions over $[-\pi, \pi]$. Specifically,

\int_{-\pi}^\pi \cos(mx) \cos(nx) , dx = \pi \delta_{mn} \quad (m, n \geq 1),

\int_{-\pi}^\pi \sin(mx) \sin(nx) , dx = \pi \delta_{mn} \quad (m, n \geq 1),

\int_{-\pi}^\pi \cos(mx) \sin(nx) , dx = 0 \quad (m, n \geq 1),

and integrals involving the constant term with $\cos(nx)$ or $\sin(nx)$ are also zero because $\int_{-\pi}^\pi \cos(nx) \, dx = 0$ and $\int_{-\pi}^\pi \sin(nx) \, dx = 0$ for $n \geq 1$.<sup></sup>[](https://mathworld.wolfram.com/ParsevalsTheorem.html) Combining these results,

\int_{-\pi}^\pi |f(x)|^2 , dx = \frac{\pi a_0^2}{2} + \pi \sum_{n=1}^\infty (a_n^2 + b_n^2).

Dividing by $\pi$ yields Parseval's theorem:

\frac{1}{\pi} \int_{-\pi}^\pi |f(x)|^2 , dx = \frac{a_0^2}{2} + \sum_{n=1}^\infty (a_n^2 + b_n^2).

### Derivation for Continuous Fourier Transforms The derivation of Parseval's theorem for the continuous Fourier transform relies on the preservation of the inner product under the transform, leveraging the Fourier inversion formula. Consider two square-integrable functions $f(t)$ and $g(t)$ with Fourier transforms $F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j \omega t} \, dt$ and $G(\omega) = \int_{-\infty}^{\infty} g(t) e^{-j \omega t} \, dt$, assuming the integrals exist in the appropriate sense. The inversion formula recovers the original functions as $f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{j \omega t} \, d\omega$ and similarly for $g(t)$.[](https://neuron.eng.wayne.edu/auth/ece4330/lectures/lecture_17_ece4330t.pdf) To establish the inner product relation, substitute the inversion for $f(t)$ into the time-domain inner product $\int_{-\infty}^{\infty} f(t) \overline{g(t)} \, dt$:

\int_{-\infty}^{\infty} f(t) \overline{g(t)} , dt = \int_{-\infty}^{\infty} \left( \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{j \omega t} , d\omega \right) \overline{g(t)} , dt.

Assuming the integrals converge absolutely (as holds for suitable test functions), interchange the order of integration:

\int_{-\infty}^{\infty} f(t) \overline{g(t)} , dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) \left( \int_{-\infty}^{\infty} e^{j \omega t} \overline{g(t)} , dt \right) d\omega.

The inner integral is the Fourier transform of $\overline{g(t)}$ evaluated at -$\omega$, which equals $\overline{G(\omega)}$. Thus,

\int_{-\infty}^{\infty} f(t) \overline{g(t)} , dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) \overline{G(\omega)} , d\omega.

This confirms that the Fourier transform preserves the inner product up to the normalization factor.[](https://neuron.eng.wayne.edu/auth/ece4330/lectures/lecture_17_ece4330t.pdf) Specializing to the case $g(t) = f(t)$, the relation yields Parseval's theorem in its energy-preserving form:

\int_{-\infty}^{\infty} |f(t)|^2 , dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |F(\omega)|^2 , d\omega.

The factor of $1/2\pi$ arises directly from the asymmetric placement of the normalization constant in the inversion formula, ensuring consistency with the angular frequency $\omega$. This form equates the total energy in the time domain to a scaled version of the energy in the frequency domain.[](https://neuron.eng.wayne.edu/auth/ece4330/lectures/lecture_17_ece4330t.pdf) For a rigorous justification in the $L^2(\mathbb{R})$ setting, known as the Plancherel theorem, the result is first established for the dense subspace of Schwartz functions $\mathcal{S}(\mathbb{R})$—infinitely differentiable functions with rapid decay and all derivatives—where the Fourier transform is well-defined and the integrals converge absolutely, allowing the interchange without issue. The Fourier transform extends to a bounded operator on $L^2(\mathbb{R})$ by continuity, since $\mathcal{S}(\mathbb{R})$ is dense in $L^2(\mathbb{R})$ with respect to the $L^2$-norm, and the isometry property holds by approximation: for any $f \in L^2(\mathbb{R})$, there exists a sequence of Schwartz functions $f_n \to f$ in $L^2$, and the equality follows in the limit. The normalization constant is handled analogously in this convention, preserving the $L^2$-norm up to the $1/2\pi$ factor.[](https://www-users.cse.umn.edu/~garrett/m/fun/intro_plancherel.pdf) ### Derivation for Discrete Fourier Transforms The discrete Fourier transform (DFT) of a finite-length sequence $x$, $n = 0, 1, \dots, N-1$, is defined as

X = \sum_{n=0}^{N-1} x e^{-j 2\pi k n / N}, \quad k = 0, 1, \dots, N-1.

This operation can be represented in matrix form as $\mathbf{X} = \mathbf{W} \mathbf{x}$, where $\mathbf{x}$ and $\mathbf{X}$ are column vectors containing the sequences $x$ and $X$, respectively, and $\mathbf{W}$ is the $N \times N$ DFT matrix with entries $W_{k,n} = e^{-j 2\pi k n / N}$.[](https://web.mit.edu/~gari/teaching/6.555/lectures/ch_DFT.pdf) To derive Parseval's theorem, consider the conjugate transpose $\mathbf{W}^*$, whose entries are $(W^*)_{n,k} = e^{j 2\pi k n / N}$. The product $\mathbf{W}^* \mathbf{W}$ has elements

(\mathbf{W}^* \mathbf{W}){m,n} = \sum{k=0}^{N-1} e^{j 2\pi k m / N} e^{-j 2\pi k n / N} = \sum_{k=0}^{N-1} e^{-j 2\pi k (n - m) / N}.

The summation is the geometric series summing the $N$th roots of unity, which equals $N$ if $n \equiv m \pmod{N}$ and 0 otherwise, yielding $\mathbf{W}^* \mathbf{W} = N \mathbf{I}$, where $\mathbf{I}$ is the identity matrix. This shows that $\mathbf{W}$ is unitary up to a scaling factor of $N$. Taking the inner product, $\mathbf{x}^* \mathbf{x} = \mathbf{x}^* \mathbf{W}^* \mathbf{W} \mathbf{x} / N = \mathbf{X}^* \mathbf{X} / N$, or equivalently,

\sum_{n=0}^{N-1} |x|^2 = \frac{1}{N} \sum_{k=0}^{N-1} |X|^2.

This derivation assumes the sequence $x$ is periodic with period $N$, consistent with the finite support over one period in the DFT definition.[](https://www.cs.princeton.edu/~ken/Eigenvectors82.pdf) An alternative derivation uses direct summation and the inverse DFT, which is

x = \frac{1}{N} \sum_{k=0}^{N-1} X e^{j 2\pi k n / N}.

The complex conjugate is $\overline{x} = \frac{1}{N} \sum_{l=0}^{N-1} \overline{X} e^{-j 2\pi l m / N}$. Then,

\sum_{n=0}^{N-1} x \overline{x} = \frac{1}{N^2} \sum_{n=0}^{N-1} \sum_{k=0}^{N-1} \sum_{l=0}^{N-1} X \overline{X} e^{j 2\pi (k - l) n / N}.

Interchanging the order of summation gives

\sum_{n=0}^{N-1} x \overline{x} = \frac{1}{N^2} \sum_{k=0}^{N-1} \sum_{l=0}^{N-1} X \overline{X} \sum_{n=0}^{N-1} e^{j 2\pi (k - l) n / N}.

The inner sum over $n$ again evaluates to $N \delta_{k l}$ due to the orthogonality of the complex exponentials, simplifying to

\sum_{n=0}^{N-1} |x|^2 = \frac{1}{N} \sum_{k=0}^{N-1} |X|^2.

This approach highlights the role of exponential orthogonality in finite dimensions.[](https://math.constructor.university/oliver/teaching/iub/spring2007/cps102/handouts/dft.pdf) For the discrete-time Fourier transform (DTFT), which extends the DFT to infinite sequences with continuous frequency, Parseval's theorem takes the form $\sum_{n=-\infty}^{\infty} |x|^2 = \frac{1}{2\pi} \int_{-\pi}^{\pi} |X(e^{j\omega})|^2 d\omega$. The proof follows similarly by substituting the inverse DTFT $x = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\omega}) e^{j \omega n} d\omega$ into the left-hand side, interchanging sum and integral, and applying the orthogonality relation $\frac{1}{2\pi} \int_{-\pi}^{\pi} e^{j \omega (m - n)} d\omega = \delta_{m n}$. Advanced derivations may invoke the Poisson summation formula to relate the DTFT to periodic spectra, confirming energy preservation under sampling assumptions, though the direct orthogonality method suffices for the core result./05:_Digital_Signal_Processing/5.06:_Discrete_-Time_Fourier_Transform_(DTFT)) ## Applications ### In Signal Processing and Engineering In signal processing, Parseval's theorem establishes the equivalence between the energy of a signal in the time domain and its representation in the frequency domain, ensuring conservation of energy during transformations such as the Fourier transform. This principle is essential for analyzing signal power distribution, where the total energy $ E = \int_{-\infty}^{\infty} |x(t)|^2 \, dt $ equals $ E = \frac{1}{2\pi} \int_{-\infty}^{\infty} |X(\omega)|^2 \, d\omega $, with $ X(\omega) $ denoting the Fourier transform. Engineers leverage this to estimate power spectral density (PSD), which quantifies how signal power is spread across frequencies; for instance, methods like the periodogram or Welch's approach rely on this equivalence to compute PSD from finite-length signals while preserving overall energy.[](https://ocw.mit.edu/courses/6-011-introduction-to-communication-control-and-signal-processing-spring-2010/8075041184d566103ce7c3f69afc5e75_MIT6_011S10_chap10.pdf)[](https://ieeexplore.ieee.org/document/9729813) For discrete-time signals processed via the discrete Fourier transform (DFT), Parseval's theorem manifests as $ \sum_{n=0}^{N-1} |x|^2 = \frac{1}{N} \sum_{k=0}^{N-1} |X|^2 $, where $ N $ is the number of samples, enabling verification of filter designs by confirming energy preservation after filtering. In digital filter engineering, this checks that linear phase filters or finite impulse response designs do not introduce unintended energy loss or gain during convolution operations approximated in the frequency domain. Similarly, in compression algorithms like JPEG, which employs the discrete cosine transform (DCT)—a real-valued variant of the DFT—Parseval's theorem guarantees that the transform preserves signal energy, allowing quantization of high-frequency coefficients to reduce data size while maintaining perceptual quality through energy redistribution.[](https://web.mit.edu/~gari/teaching/6.555/lectures/ch_DFT.pdf)[](https://arxiv.org/pdf/2102.06968) In electrical engineering contexts, such as AC circuits with periodic signals, Parseval's theorem connects the root-mean-square (RMS) value of the waveform to the RMS values of its harmonic components, expressed as the RMS of the total signal equaling the square root of the sum of squared RMS harmonics. This facilitates power calculations in systems like power distribution networks, where the theorem quantifies effective power dissipation across frequencies without direct time-domain integration. For numerical implementation, the fast Fourier transform (FFT) algorithm computes the DFT efficiently in $ O(N \log N) $ operations, allowing practical evaluation of Parseval's sums for large datasets in real-time processing.[](https://pubs.aip.org/aapt/ajp/article-pdf/33/2/99/11942934/99_1_online.pdf)[](https://web.mit.edu/~gari/teaching/6.555/lectures/ch_DFT.pdf) Practical examples illustrate these applications; in audio signal processing, Parseval's theorem verifies the energy of a digitized sound wave, such as a 1-second clip at 44.1 kHz sampling, by comparing time-domain summation $ \sum |x|^2 $ to the scaled frequency-domain sum, ensuring compression or noise reduction algorithms retain acoustic fidelity. Regarding sampling, the theorem aids in assessing aliasing effects: proper sampling above the Nyquist rate preserves energy equivalence between continuous and discrete domains, while undersampling causes aliasing that distorts frequency content and violates energy conservation unless anti-aliasing filters are applied beforehand.[](https://ese2240.seas.upenn.edu/lab2/)[](https://courses.grainger.illinois.edu/NPRE435/fa2021/notes/ch2_part2_sampling_2021.pdf) ### In Physics and Other Fields In quantum mechanics, Parseval's theorem ensures the preservation of probability norms when transforming wave functions between position and momentum spaces via the Fourier transform. For a normalized wave function $\psi(x)$ in position space, where $\int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1$, the theorem implies that the momentum-space wave function $\tilde{\psi}(p) = \frac{1}{\sqrt{2\pi \hbar}} \int_{-\infty}^{\infty} \psi(x) e^{-i p x / \hbar} dx$ satisfies $\int_{-\infty}^{\infty} |\tilde{\psi}(p)|^2 dp = 1$, confirming the unitary nature of the transform and the conservation of total probability.[](https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2016/72de1b688f3f40e1c42b26b3da29585a_MIT8_04S16_LecNotes8.pdf) This property is fundamental for maintaining the probabilistic interpretation of quantum states across conjugate variables.[](https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/Introduction_to_Nanoelectronics_%28Baldo%29/01%253A_The_Quantum_Particle/1.14%253A_Parsevals_Theorem) In optics and wave propagation, Parseval's theorem underpins energy conservation in diffraction patterns, linking the energy in the spatial domain of a wave field to that in its Fourier domain representation. For instance, in Fraunhofer diffraction, the intensity distribution in the far-field pattern corresponds to the squared modulus of the Fourier transform of the aperture function, and the theorem guarantees that the total integrated intensity equals the input energy, adjusted for scaling factors. This is evident in Fourier optics, where the relation $\int |u(x,y)|^2 dx dy = \frac{1}{\lambda^2 f^2} \int |U(\xi,\eta)|^2 d\xi d\eta$ holds for the field $u$ and its transform $U$, ensuring no loss in optical energy during propagation.[](https://qiweb.tudelft.nl/aoi/fourieranalysis/fourieranalysis.html) Such conservation is critical for validating simulations of diffractive elements and predicting power distribution in optical systems.[](https://opg.optica.org/abstract.cfm?uri=josaa-29-9-2015) In probability theory, Parseval's theorem, often invoked through its Plancherel form, relates the moments of probability distributions to integrals involving their characteristic functions, which are Fourier transforms of the probability densities. For a random variable with square-integrable density $h(y)$, the theorem states that $\int_{-\infty}^{\infty} h(y)^2 dy = \frac{1}{2\pi} \int_{-\infty}^{\infty} |\phi(t)|^2 dt$, where $\phi(t)$ is the characteristic function, allowing the L² norm of the density to be expressed in terms of the characteristic function's modulus. This connection facilitates the computation and comparison of higher-order moments, as derivatives of $\phi(t)$ yield cumulants and moments, providing bounds on distribution properties like variance through Fourier-domain integrals.[](https://do.csc.knu.ua/wp-content/uploads/2023/01/Char_funct_XIan2019.pdf) In numerical analysis, Parseval's theorem provides error bounds for approximations using orthogonal expansions, such as Fourier series, by equating the L² error norm to the sum of squared coefficients beyond the truncation point. For a function $f(x)$ approximated by its partial Fourier sum $s_N(x)$, the error satisfies $\int |f(x) - s_N(x)|^2 dx = \sum_{|n|>N} |c_n|^2$, where $c_n$ are the Fourier coefficients, enabling precise quantification of convergence rates in spectral methods. This is particularly useful in solving partial differential equations numerically, where the theorem helps estimate truncation errors without direct computation of the residual.[](https://www.jstor.org/stable/41582942) A representative example of Parseval's theorem in physical contexts is its application to the heat equation solved via Fourier series, where it verifies the conservation of total heat energy. Consider the one-dimensional heat equation $\frac{\partial u}{\partial t} = \kappa \frac{\partial^2 u}{\partial x^2}$ on $[0, L]$ with initial condition $u(x,0) = f(x)$. The solution is $u(x,t) = \sum_{n=1}^{\infty} c_n e^{-\kappa (n\pi/L)^2 t} \sin(n\pi x / L)$, and Parseval's theorem gives $\frac{1}{L} \int_0^L f(x)^2 dx = \frac{1}{2} \sum_{n=1}^{\infty} c_n^2$, relating the initial total energy $\int_0^L u(x,0)^2 dx$ to the coefficients; at later times, the decaying exponentials preserve the form, confirming energy dissipation aligns with physical laws without net loss in the integrated sense.[](https://links.uwaterloo.ca/amath353docs/set10.pdf)[](https://www.mathmods.eu/resources/downloads/finish/10-li-chen/34-heatapde20111101) ## Generalizations ### Plancherel Theorem The Plancherel theorem provides a rigorous foundation for the Fourier transform as a unitary operator on the Hilbert space $L^2(\mathbb{R})$, equipped with the inner product $\langle f, g \rangle = \int_{-\infty}^{\infty} f(x) \overline{g(x)} \, dx$. With the unitary normalization of the Fourier transform defined by

\mathcal{F}f(\xi) = \hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x \xi} , dx,

the theorem asserts that for all $f \in L^2(\mathbb{R})$, $\|\hat{f}\|_{L^2(\mathbb{R})} = \|f\|_{L^2(\mathbb{R})}$, or equivalently,

\begin{aligned} \int_{-\infty}^{\infty} |f(x)|^2 , dx &= \int_{-\infty}^{\infty} |\hat{f}(\xi)|^2 , d\xi. \end{aligned}

This identity extends to the more general form $\langle f, g \rangle = \langle \hat{f}, \hat{g} \rangle$ for $f, g \in L^2(\mathbb{R})$, establishing the Fourier transform as an isometry that preserves the $L^2$ norm and inner product. The theorem was established by Michel Plancherel in his 1910 paper, where he proved the existence of the transform for square-integrable functions and the norm preservation under this normalization, building directly on Parseval's earlier identity for Fourier series.[](https://doi.org/10.1007/BF03014877) In contrast to Parseval's theorem, which applies primarily to sufficiently regular functions (such as those in $L^1(\mathbb{R}) \cap L^2(\mathbb{R})$ or continuous periodic functions) and equates the $L^2$ norm to a sum or integral over Fourier coefficients, the Plancherel theorem guarantees the existence of the Fourier transform for *all* functions in $L^2(\mathbb{R})$ via a limiting process and establishes it as a *complete isometry* on the entire space. Parseval's identity holds pointwise for the coefficients but requires additional assumptions for convergence, whereas Plancherel ensures the transform is well-defined almost everywhere and surjective onto $L^2(\mathbb{R})$, making the Fourier basis "complete" in the $L^2$ sense without restricting to nicer subclasses of functions. This generalization is essential for treating arbitrary square-integrable signals or distributions in analysis.[](https://press.princeton.edu/books/hardcover/9780691113845/fourier-analysis) A standard proof begins by verifying the identity on the dense subspace of Schwartz functions $\mathcal{S}(\mathbb{R})$ (smooth functions with rapid decay), where the Fourier transform is an isometry by direct computation using properties like the inversion theorem and Fubini's theorem for convolutions. Since $\mathcal{S}(\mathbb{R})$ is dense in $L^2(\mathbb{R})$ and the Fourier transform extends continuously to a bounded operator on $L^2(\mathbb{R})$ (with operator norm 1), the identity holds for all $L^2$ functions by approximation: for any $f \in L^2(\mathbb{R})$, choose a sequence $\{f_n\} \subset \mathcal{S}(\mathbb{R})$ converging to $f$ in $L^2$, so $\|\hat{f}_n - \hat{f}\|_{L^2} \to 0$ and the norms converge. Surjectivity follows from the Riesz-Fischer theorem, which completes the pre-Hilbert space of images of $L^1(\mathbb{R}) \cap L^2(\mathbb{R})$ under the transform to the full $L^2(\mathbb{R})$, confirming the Fourier transform is unitary and onto. This construction avoids defining the transform pointwise for general $L^2$ functions, relying instead on $L^2$ limits.[](https://press.princeton.edu/books/hardcover/9780691113845/fourier-analysis) The Plancherel theorem underpins the spectral theorem for self-adjoint operators on $L^2(\mathbb{R})$, particularly translation-invariant ones like the differentiation operator $ \frac{d}{dx} $ or the Laplacian $-\Delta$, by showing that the Fourier transform unitarily diagonalizes them into multiplication operators in the frequency domain. For instance, the spectrum of $-\Delta$ becomes the positive reals under $\mathcal{F}$, enabling eigenvalue decompositions essential in quantum mechanics and PDEs. This unitary equivalence facilitates the analysis of operator spectra and resolvents in infinite-dimensional Hilbert spaces.[](https://press.princeton.edu/books/hardcover/9780691113845/fourier-analysis) ### Extensions to Hilbert Spaces and Beyond Parseval's identity extends naturally to general Hilbert spaces, where it characterizes the norm of any element in terms of its coefficients with respect to an orthonormal basis. Specifically, in a Hilbert space $H$ equipped with an orthonormal basis $\{\phi_n\}_{n \in I}$, for any $f \in H$, the identity states that

|f|^2 = \sum_{n \in I} |\langle f, \phi_n \rangle|^2,

with the reconstruction $f = \sum_{n \in I} \langle f, \phi_n \rangle \phi_n$ holding in the norm topology of $H$.[](https://math.ou.edu/~cremling/teaching/lecturenotes/fa-new/ln5.pdf) This formulation generalizes the classical Parseval relation from $L^2$ spaces to arbitrary separable Hilbert spaces, emphasizing the preservation of inner product structure across bases.[](https://www.math.uci.edu/~rvershyn/teaching/2010-11/602/functional-analysis.pdf) In frame theory, Parseval's identity relaxes to accommodate redundant systems in Hilbert spaces, providing robust representations beyond minimal bases. A frame $\{\psi_n\}_{n \in I}$ for $H$ satisfies $A \|f\|^2 \leq \sum_{n \in I} |\langle f, \psi_n \rangle|^2 \leq B \|f\|^2$ for some bounds $A > 0$, $B < \infty$, and all $f \in H$; a Parseval frame occurs when $A = B = 1$, yielding the exact identity $\|f\|^2 = \sum_{n \in I} |\langle f, \psi_n \rangle|^2$.[](https://faculty.sites.iastate.edu/esweber/files/inline-files/book_final.pdf) Such frames enable stable signal recovery in applications like compressed sensing, where redundancy enhances resilience to noise or loss.[](https://math.umd.edu/~rvbalan/PAPERS/MyPapers/FrameIdentity_PAMS_revised.pdf) Beyond Fourier analysis, Parseval's theorem applies to other integral transforms under suitable conditions. For wavelet transforms, orthogonal wavelet bases in $L^2(\mathbb{R})$ satisfy the identity $\|f\|^2 = \sum_{m,n \in \mathbb{Z}} |\langle f, \psi_{m,n} \rangle|^2$, where $\{\psi_{m,n}\}$ denotes the wavelet family generated by dilations and translations, preserving energy across scales and positions. Similarly, for the Laplace transform on functions with support in $[0, \infty)$, a Parseval-type relation holds: $\int_0^\infty |f(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^\infty |F(\sigma + i\omega)|^2 d\omega$ for fixed $\sigma > 0$ where the transform converges, linking time-domain energy to a strip in the complex plane. In the context of group representations, Parseval's theorem generalizes via Pontryagin duality to locally compact abelian groups $G$, where the Plancherel theorem equates the $L^2(G)$ norm to an integral over the dual group $\hat{G}$: $\|f\|_{L^2(G)}^2 = \int_{\hat{G}} |\hat{f}(\chi)|^2 d\mu(\chi)$ for a suitable Plancherel measure $\mu$, extending the unitary Fourier structure to non-Euclidean settings like circles or tori. Non-commutative extensions appear in operator algebras, where analogues of Parseval's identity relate traces in von Neumann algebras to decompositions via projections or modules. For instance, in a von Neumann algebra $M$ acting on a Hilbert space, the trace $\tau(x^* x) = \sum_i \langle x \xi_i, x \xi_i \rangle$ for a suitable basis $\{\xi_i\}$ mirrors the identity, facilitating spectral analysis in quantum mechanics.[](https://math.vanderbilt.edu/peters10/teaching/spring2013/vonNeumannAlgebras.pdf) Further generalizations to quantum groups involve Hopf-von Neumann algebras, where a Plancherel theorem equates the Haar state integral to a dual weight on representations, preserving the energy equality in non-abelian quantum settings.[](https://www.math.ru.nl/~koelink/edu/GQSF-Bizerte-2010.pdf)

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