Parseval–Rayleigh Identities Overview

• Parseval–Rayleigh identities are families of equalities and inequalities that express integrals, norms, or traces in one domain as coefficient sums in a dual representation.
• They are applied in contexts ranging from classical Fourier analysis to Hilbert-space frames and operator theory, converting complex integrals into manageable sums.
• Recent developments extend these identities into weak forms and operator-valued frameworks, enhancing applications in optimization, spectral analysis, and algebraic geometry.

Parseval–Rayleigh identities are families of equalities and closely related inequalities that express a quantity computed in one representation—typically an integral, an inner product, a norm, or a trace-like functional—in terms of coefficients or samples in another representation. In classical Fourier analysis they equate an $L^2$ quantity with a sum over Fourier coefficients; in modern work the same organizing principle appears in mixed Fourier-series/Fourier-transform formulas, generalized integral transforms, frame theory, operator theory, and positive-characteristic commutative algebra. The recent literature shows that the label “Parseval–Rayleigh” no longer refers only to the standard $L^2$ Plancherel setting, but to a wider pattern of reconstruction, duality, and energy decomposition across analytic and algebraic contexts (Ghorbanpour et al., 2017, Cai et al., 2020, Caine et al., 20 May 2025, Pochekai, 30 Apr 2026).

1. Classical Fourier identities and the Rayleigh theme

In the standard Fourier-series setting, Parseval’s identity states that for a Riemann-integrable function $f$ on $[0,1]$ with Fourier coefficients

$$c_n(f)=\int_0^1 f(t)e^{-2\pi i n t}\,dt,\qquad n\in\mathbb Z,$$

one has

$$\int_0^1 |f(x)|^2\,dx=\sum_{n=-\infty}^{\infty}|c_n(f)|^2.$$

This is the formulation used to derive values of the Riemann zeta function at even integers from Bernoulli polynomials (Ghorbanpour et al., 2017).

The same paper proves that for $k\ge 1$, the nonzero Fourier coefficients of the Bernoulli polynomial $B_k(t)$ are

$c_n(B_k)=-\frac{k!}{(2\pi i n)^k}\qquad (n\ne 0),$

and combines this with

$$\int_0^1 |B_k(t)|^2\,dt=\frac{(-1)^{k-1}(k!)^2B_{2k}}{(2k)!}$$

to obtain

$L^2$0

Here the Parseval mechanism is exact: the squared $L^2$1-norm of a function is identified with the squared moduli of its Fourier coefficients, and the coefficient decay encodes $L^2$2 (Ghorbanpour et al., 2017).

A broader measure-theoretic restatement replaces intervals and trigonometric systems by bounded measurable sets $L^2$3 and arbitrary mutually orthogonal collections $L^2$4. For bounded, positive, measurable $L^2$5 on $L^2$6, if

$L^2$7

and $L^2$8, then

$L^2$9

For general bounded measurable $f$0, the same principle is applied on sign-definite pieces $f$1, yielding

$f$2

This formulation shifts emphasis from Fourier series to orthogonality and measure additivity (Siktar, 2019).

The “Rayleigh” aspect appears whenever such identities are used to compare spectral or coefficient data with energies, traces, or partial sums. In the classical analytic setting, this is most familiar in the equivalence between an $f$3-norm and a coefficient norm; in later sections the same structural role is played by frame coefficients, transform kernels, and Frobenius-twisted monomial pairings.

2. Mixed Fourier formulas and weak Parseval identities

A significant extension of the classical theory concerns products of non-periodic and periodic functions. The mixed Parseval–Plancherel formula states that if $f$4, $f$5 is $f$6-periodic, and

$f$7

then

$f$8

where $f$9 is the Fourier transform of $[0,1]$0 and $[0,1]$1 are the exponential Fourier coefficients of $[0,1]$2. For a $[0,1]$3-periodic function,

$[0,1]$4

This formula mixes Fourier transform samples of $[0,1]$5 with Fourier-series coefficients of $[0,1]$6, and the paper emphasizes that it allows the evaluation of oscillatory integrals (Kouba, 2013).

A later development replaces the classical mixed Parseval condition by a weaker and often more practical hypothesis. Let $[0,1]$7 be a periodic, integrable function of period $[0,1]$8, and let $[0,1]$9 have compact support, with $$c_n(f)=\int_0^1 f(t)e^{-2\pi i n t}\,dt,\qquad n\in\mathbb Z,$$0 of bounded variation near all sampling points. Then

$$c_n(f)=\int_0^1 f(t)e^{-2\pi i n t}\,dt,\qquad n\in\mathbb Z,$$1

where $$c_n(f)=\int_0^1 f(t)e^{-2\pi i n t}\,dt,\qquad n\in\mathbb Z,$$2 and

$$c_n(f)=\int_0^1 f(t)e^{-2\pi i n t}\,dt,\qquad n\in\mathbb Z,$$3

The stated advantage is that the identity requires compact support and local bounded variation for $$c_n(f)=\int_0^1 f(t)e^{-2\pi i n t}\,dt,\qquad n\in\mathbb Z,$$4, rather than global bounded variation or square-integrability, and it produces a finite sum rather than an often divergent infinite sum (Cai et al., 2020).

A central special case is obtained by taking $$c_n(f)=\int_0^1 f(t)e^{-2\pi i n t}\,dt,\qquad n\in\mathbb Z,$$5, the $$c_n(f)=\int_0^1 f(t)e^{-2\pi i n t}\,dt,\qquad n\in\mathbb Z,$$6-fold convolution of the rectangle function $$c_n(f)=\int_0^1 f(t)e^{-2\pi i n t}\,dt,\qquad n\in\mathbb Z,$$7, whose Fourier transform is $$c_n(f)=\int_0^1 f(t)e^{-2\pi i n t}\,dt,\qquad n\in\mathbb Z,$$8. One then has

$$c_n(f)=\int_0^1 f(t)e^{-2\pi i n t}\,dt,\qquad n\in\mathbb Z,$$9

For $$\int_0^1 |f(x)|^2\,dx=\sum_{n=-\infty}^{\infty}|c_n(f)|^2.$$0, the sum can be restricted to $$\int_0^1 |f(x)|^2\,dx=\sum_{n=-\infty}^{\infty}|c_n(f)|^2.$$1, and when the support is sufficiently small only the $$\int_0^1 |f(x)|^2\,dx=\sum_{n=-\infty}^{\infty}|c_n(f)|^2.$$2 term remains: $$\int_0^1 |f(x)|^2\,dx=\sum_{n=-\infty}^{\infty}|c_n(f)|^2.$$3 The paper presents this as a Fourier-analytic foundation for Lobachevsky-type integrals (Cai et al., 2020).

Illustrative formulas include

$$\int_0^1 |f(x)|^2\,dx=\sum_{n=-\infty}^{\infty}|c_n(f)|^2.$$4

valid for all $$\int_0^1 |f(x)|^2\,dx=\sum_{n=-\infty}^{\infty}|c_n(f)|^2.$$5, and the identities

$$\int_0^1 |f(x)|^2\,dx=\sum_{n=-\infty}^{\infty}|c_n(f)|^2.$$6

$$\int_0^1 |f(x)|^2\,dx=\sum_{n=-\infty}^{\infty}|c_n(f)|^2.$$7

The $$\int_0^1 |f(x)|^2\,dx=\sum_{n=-\infty}^{\infty}|c_n(f)|^2.$$8 identity is said to coincide with Jolany’s result, while the $$\int_0^1 |f(x)|^2\,dx=\sum_{n=-\infty}^{\infty}|c_n(f)|^2.$$9 case is described as new to that paper (Cai et al., 2020).

These results clarify a common misconception: Parseval-type identities are not restricted to globally square-integrable pairs. The cited weak form works precisely where the classical mixed Parseval formula

$k\ge 1$0

may be unavailable because the relevant functions fail bounded-variation or square-integrability assumptions (Cai et al., 2020).

3. Integral transforms, special functions, and explicit evaluations

The literature also uses “Parseval–Goldstein type” to describe identities that connect distinct integral transforms rather than Fourier objects alone. One such framework introduces the generalized Laplace-type transform

$k\ge 1$1

and the generalized Stieltjes-type transform

$k\ge 1$2

If $k\ge 1$3, then applying the generalized Laplace transform twice yields the generalized Stieltjes transform: $k\ge 1$4 This is stated as Lemma 1 in the transform-theoretic development (Albayrak, 2023).

The main Parseval–Goldstein theorem in that setting is

$k\ge 1$5

together with the symmetric identity

$k\ge 1$6

For $k\ge 1$7, the paper states that this recovers the classical Parseval–Goldstein theorem for Laplace and Stieltjes transforms (Albayrak, 2023).

The same paper develops mixed identities involving generalized Fourier sine and cosine transforms and a theorem involving a hypergeometric kernel $k\ge 1$8. It explicitly presents these results as tools for evaluating improper integrals of power-law, exponential, trigonometric, Bessel, and hypergeometric type (Albayrak, 2023).

Special functions enter Parseval–Rayleigh theory in other, more classical, ways as well. One example is the Rayleigh–Sneddon identity for the positive zeros $k\ge 1$9 of the Bessel function $B_k(t)$0: $B_k(t)$1 A Laplace-transform derivation introduces

$B_k(t)$2

whose inverse transform is

$B_k(t)$3

Taking $B_k(t)$4 recovers the Rayleigh–Sneddon sum (Giusti et al., 2016).

The weak Parseval framework for periodic functions also yields Bessel-function formulas: $B_k(t)$5 and

$B_k(t)$6

These are presented as cases where the classical Parseval formula is inapplicable, but the extended identity remains effective (Cai et al., 2020).

A plausible implication is that the modern theory treats “Parseval–Rayleigh identity” less as a single theorem and more as a reusable design pattern: identify a transform pair or coefficient system, then prove an exact duality converting difficult integrals or sums into tractable finite or rapidly convergent expressions.

4. Parseval frames and operator-theoretic generalizations

In Hilbert-space frame theory, a Parseval frame $B_k(t)$7 for $B_k(t)$8 is a spanning set such that for all $B_k(t)$9,

$c_n(B_k)=-\frac{k!}{(2\pi i n)^k}\qquad (n\ne 0),$0

Equivalently, if $c_n(B_k)=-\frac{k!}{(2\pi i n)^k}\qquad (n\ne 0),$1 has columns $c_n(B_k)=-\frac{k!}{(2\pi i n)^k}\qquad (n\ne 0),$2, then

$c_n(B_k)=-\frac{k!}{(2\pi i n)^k}\qquad (n\ne 0),$3

This generalizes the orthonormal basis property to redundant systems (Caine et al., 20 May 2025).

A recent optimization-theoretic treatment introduces the total frame energy

$c_n(B_k)=-\frac{k!}{(2\pi i n)^k}\qquad (n\ne 0),$4

where $c_n(B_k)=-\frac{k!}{(2\pi i n)^k}\qquad (n\ne 0),$5 are prescribed vector norms. The first term measures deviation from the Parseval identity, and the second measures deviation from the prescribed norms. The global minima are precisely the matrices $c_n(B_k)=-\frac{k!}{(2\pi i n)^k}\qquad (n\ne 0),$6 satisfying $c_n(B_k)=-\frac{k!}{(2\pi i n)^k}\qquad (n\ne 0),$7 and $c_n(B_k)=-\frac{k!}{(2\pi i n)^k}\qquad (n\ne 0),$8 for all $c_n(B_k)=-\frac{k!}{(2\pi i n)^k}\qquad (n\ne 0),$9. The paper states that this nonconvex function has no spurious local minimizers, extends the Benedetto–Fickus theorem to a non-compact setting, and implies that gradient descent converges to an equal norm Parseval frame from a dense open set of initial data (Caine et al., 20 May 2025).

The same work uses this optimization result to study the topology of frame spaces $$\int_0^1 |B_k(t)|^2\,dt=\frac{(-1)^{k-1}(k!)^2B_{2k}}{(2k)!}$$0. It gives the admissibility criterion

$$\int_0^1 |B_k(t)|^2\,dt=\frac{(-1)^{k-1}(k!)^2B_{2k}}{(2k)!}$$1

and proves connectivity consequences under inequalities involving a constant $$\int_0^1 |B_k(t)|^2\,dt=\frac{(-1)^{k-1}(k!)^2B_{2k}}{(2k)!}$$2, including path-connectedness for certain spaces of real Parseval frames (Caine et al., 20 May 2025).

Operator-valued and partial-sum versions of Parseval identities appear for Hilbert–Schmidt frames. If $$\int_0^1 |B_k(t)|^2\,dt=\frac{(-1)^{k-1}(k!)^2B_{2k}}{(2k)!}$$3 is a Parseval HS-frame, then for all $$\int_0^1 |B_k(t)|^2\,dt=\frac{(-1)^{k-1}(k!)^2B_{2k}}{(2k)!}$$4 and $$\int_0^1 |B_k(t)|^2\,dt=\frac{(-1)^{k-1}(k!)^2B_{2k}}{(2k)!}$$5,

$$\int_0^1 |B_k(t)|^2\,dt=\frac{(-1)^{k-1}(k!)^2B_{2k}}{(2k)!}$$6

An associated inequality is

$$\int_0^1 |B_k(t)|^2\,dt=\frac{(-1)^{k-1}(k!)^2B_{2k}}{(2k)!}$$7

These statements generalize earlier vector-frame identities to the Hilbert–Schmidt setting (Poria, 2016).

Parseval frames also generate explicit Hamiltonians. For a Parseval frame $$\int_0^1 |B_k(t)|^2\,dt=\frac{(-1)^{k-1}(k!)^2B_{2k}}{(2k)!}$$8 and real numbers $$\int_0^1 |B_k(t)|^2\,dt=\frac{(-1)^{k-1}(k!)^2B_{2k}}{(2k)!}$$9, one defines

$L^2$00

with domain restrictions required in the unbounded case. In this setting the generalized Parseval–Rayleigh identity takes the form

$L^2$01

The papers on bounded and unbounded Hamiltonians emphasize that the coefficients $L^2$02 in a frame expansion need not coincide with the actual spectrum of the operator, because frames are generally not bases (Bagarello et al., 2020, Bagarello et al., 2023).

That distinction is formalized by the notion of $L^2$03-connection: a pair $L^2$04 is $L^2$05-connected to an orthonormal basis $L^2$06 if

$L^2$07

The frame-based expansion coefficients $L^2$08 and the diagonal spectral data $L^2$09 may differ. The finite-dimensional theory also gives Rayleigh-type inequalities such as

$L^2$10

and trace relations between the true eigenvalues and the frame coefficients (Bagarello et al., 2020).

A separate construction produces Parseval frames of piecewise constant functions in $L^2$11. If

$L^2$12

with the operators satisfying

$L^2$13

then the family

$L^2$14

is a Parseval frame, and hence

$L^2$15

The paper also shows that this frame can be dilated to an orthonormal basis in a larger Hilbert space (Dutkay et al., 2018).

These developments dispel another common misconception: in modern usage, Parseval–Rayleigh identities are not confined to orthonormal expansions. Redundancy, operator-valued coefficients, and non-basis decompositions are integral to the contemporary theory.

5. Positive-characteristic algebra, residue maps, and volume maps

A distinct but increasingly active strand of research transports Parseval–Rayleigh identities into graded Artinian Gorenstein and Cohen–Macaulay algebra. For a homogeneous complete intersection

$L^2$16

over a field of positive characteristic $L^2$17, the quotient is Artinian Gorenstein with socle degree

$L^2$18

The residue map

$L^2$19

is normalized by choosing a matrix $L^2$20 with

$L^2$21

and requiring $L^2$22 (Adiprasito et al., 7 Nov 2025).

If $L^2$23 denotes the set of monic degree-$L^2$24 monomials in $L^2$25, then for any $L^2$26,

$L^2$27

This is presented as the main Parseval–Rayleigh identity for homogeneous complete intersections, valid in any positive characteristic (Adiprasito et al., 7 Nov 2025).

A more general theorem for graded Artinian Gorenstein algebras over a field of characteristic $L^2$28 begins with $L^2$29, an Artinian Gorenstein quotient $L^2$30 of socle degree $L^2$31, a volume functional $L^2$32, and the Frobenius-power quotient $L^2$33 of socle degree

$L^2$34

There exists a unique element

$L^2$35

with $L^2$36, and for homogeneous $L^2$37,

$L^2$38

In the complete intersection case,

$L^2$39

recovering the more explicit formula above (Pochekai, 30 Apr 2026).

The most abstract formulation is given for Artinian quotients $L^2$40 of Cohen–Macaulay algebras in characteristic $L^2$41. Writing

$L^2$42

for the volume map and $L^2$43 for the Frobenius trace on the canonical module, the central identity is

$L^2$44

The paper explicitly describes this as the unifying conceptual core behind concrete Parseval–Rayleigh identities (Adiprasito et al., 4 May 2026).

In semigroup algebras of lattice polytopes, the same phenomenon is expressed as identities for the fundamental class. For an interior lattice point $L^2$45 at height $L^2$46,

$L^2$47

with a characteristic-$L^2$48 version

$L^2$49

The paper proves that these Parseval–Rayleigh identities are equivalent to a system of differential equations for the volume map, and uses them to establish strong Lefschetz properties and unimodality statements for $L^2$50-polynomials (Adiprasito et al., 17 Sep 2025).

This algebraic literature shows that the phrase “Parseval–Rayleigh identity” can denote an exact reconstruction formula for a residue or volume functional rather than an $L^2$51-norm equality. The common structure is still recognizable: a distinguished functional is recovered from basis values through a bilinear or Frobenius-twisted pairing.

6. Applications, significance, and scope of the concept

Across the cited literature, Parseval–Rayleigh identities serve several recurring functions. In harmonic analysis they evaluate oscillatory, Lobachevsky-type, sinc, and Bessel integrals, often converting them into finite sums or rapidly convergent series (Kouba, 2013, Cai et al., 2020, Giusti et al., 2016). In transform theory they connect generalized Laplace, Stieltjes, and Fourier-type transforms and provide explicit evaluations of improper integrals involving powers, exponentials, Bessel functions, and hypergeometric kernels (Albayrak, 2023). In number theory they recover Euler’s formula for $L^2$52 by expressing the $L^2$53-norm of Bernoulli polynomials through Fourier coefficients (Ghorbanpour et al., 2017).

In frame theory, the Parseval identity becomes a structural condition for redundant expansions, optimization landscapes, and topological classification of frame spaces (Caine et al., 20 May 2025). In operator theory, it underlies Hamiltonian constructions based on Parseval frames and clarifies the gap between frame-expansion coefficients and the actual spectrum (Bagarello et al., 2020, Bagarello et al., 2023). In operator-valued frame theory it yields exact identities and inequalities for partial frame sums (Poria, 2016). In constructive analysis, it supports explicit Parseval frames of piecewise constant functions together with dilation to orthonormal bases (Dutkay et al., 2018).

In positive-characteristic commutative algebra and algebraic combinatorics, Parseval–Rayleigh identities control residue maps and volume maps, furnish differential descriptions of fundamental classes, and imply anisotropy, Hard Lefschetz, strong Lefschetz, and related consequences for complete intersections, simplicial spheres, semigroup algebras, and other Gorenstein settings (Adiprasito et al., 7 Nov 2025, Pochekai, 30 Apr 2026, Adiprasito et al., 4 May 2026, Adiprasito et al., 17 Sep 2025). The papers explicitly connect these identities to proofs of the Strong Lefschetz Property in characteristic $L^2$54, the $L^2$55-theorem for simplicial spheres, and the Ohsugi–Hibi conjecture (Adiprasito et al., 7 Nov 2025, Adiprasito et al., 4 May 2026, Adiprasito et al., 17 Sep 2025).

A concise way to summarize the modern landscape is that Parseval–Rayleigh identities now inhabit at least four mathematically distinct regimes: Fourier and transform analysis, Hilbert-space frame theory, spectral/operator constructions, and Frobenius-sensitive algebraic geometry and combinatorics. This suggests that the unifying content of the term is not a particular formula but a principle: an exact passage from an object in a “physical,” “spatial,” or “algebraic” domain to coefficients, pairings, or traces in a dual representation.