Fourier Transform for Operators

• Fourier Transform for Operators is a framework that applies Fourier analysis to operators, transforming differential and algebraic structures for diverse applications.
• It encapsulates multiple constructions, including operational calculus for PDEs, unitary re-realizations in quantum systems, and non-Abelian transforms over complex domains.
• The theory underpins applications ranging from spectral analysis in non-Euclidean settings to quantum signal processing and operator-valued time-frequency synthesis.

A Fourier transform for operators denotes a family of constructions in which Fourier analysis acts on operators rather than only on functions. In the most classical sense, it is an operational calculus: if $L$ acts on $f$, one seeks an operator $\widetilde L$ such that $\mathcal F(Lf)=\widetilde L(\mathcal Ff)$. In a second sense, it is a unitary similarity or representation change sending one observable to another, as in continuous position–momentum duality or discrete number–phase duality. In a third sense, the input itself is an operator, and the output is a phase-space function or another operator, as in the Fourier–Wigner transform, operator-valued short-time Fourier transforms, and Hilbert–Schmidt operator Fourier theory (Neretin, 2020, Perez-Leija et al., 2015, Luef et al., 2024).

1. Classical meaning and conceptual variants

On $\mathbb{R}$, the model correspondence is

$$\mathcal F\!\left(\frac{d}{dx}f\right)(\xi)=-i\xi\,\mathcal Ff(\xi),\qquad \mathcal F(xf)(\xi)=i\frac{d}{d\xi}\mathcal Ff(\xi),$$

so differentiation becomes multiplication and multiplication becomes differentiation. This is the prototype of Fourier transform for operators: the transform does not merely move functions between realizations; it transports algebraic and differential structure (Neretin, 2020).

That prototype has three non-equivalent extensions. First, one may transport operators under non-Euclidean Fourier transforms, where the images are no longer purely differential or multiplicative. Second, one may apply a Fourier-type unitary transform to one operator and obtain a conjugate observable, as in finite-dimensional number–phase constructions. Third, one may define a Fourier transform whose argument is itself an operator, producing a function on phase space, a reproducing-kernel-valued object, or another operator-space element. The literature surveyed here uses all three meanings, and a persistent source of confusion is to treat them as interchangeable. They are related, but they live in different categories: function transforms with transported operators, operator-valued transforms, and unitary re-realizations of operator algebras (Perez-Leija et al., 2015, Luef et al., 2024, Dörfler et al., 2022).

2. Operational calculus from Euclidean to non-Abelian settings

In the Euclidean case, the operational calculus is abelian and exact: the Fourier variable is a character parameter, and differential operators become multiplication operators or first-order differential operators in the dual variable. This underlies the standard spectral treatment of constant-coefficient PDEs and is the benchmark against which non-Euclidean generalizations are measured (Neretin, 2020).

On the Lobachevsky plane $\mathbb H$, Neretin’s Fourier transform $\mathcal J$ intertwines the geometric $SL(2,\mathbb R)$-action on $L^2(\mathbb H,du)$ with principal series representations $f$0 on functions of $f$1. The transformed operators are no longer purely differential: basic vector fields on $f$2 become differential–difference operators in $f$3, with shift operators

$f$4

A representative identity is the transformed $f$5, whose image involves $f$6 and $f$7. The crucial structural feature is that shifts act in the spectral parameter $f$8, i.e. in a direction transversal to the Plancherel contour $f$9 (Neretin, 2020).

For $\widetilde L$0, the same phenomenon becomes fully explicit at the level of polynomial differential operators. Fourier images of multiplication by coordinates and partial derivatives are finite sums

$\widetilde L$1

where $\widetilde L$2 shift the principal-series parameters $\widetilde L$3 by $\widetilde L$4, and the coefficients are rational in $\widetilde L$5 with poles along $\widetilde L$6. Here the non-abelian Fourier transform converts local differential operators on the group into differential–difference operators on representation parameters, reflecting both Mellin-type exponent shifts and meromorphic dependence of intertwining operators (Neretin, 2018).

A plausible implication is that “Fourier transform for operators” becomes progressively less about diagonalization and more about controlled algebra transport as symmetry moves from abelian to semisimple.

3. Functional calculus and operator-valued synthesis

For constant-coefficient differential operators, Fourier analysis yields a direct functional calculus. If $\widetilde L$7 is the directional derivative, then

$\widetilde L$8

and therefore

$\widetilde L$9

The operator $\mathcal F(Lf)=\widetilde L(\mathcal Ff)$0 is thus represented by the symbol $\mathcal F(Lf)=\widetilde L(\mathcal Ff)$1, and in physical space becomes a convolution kernel obtained by inverse Fourier transform. The exponential case reproduces the translation operator $\mathcal F(Lf)=\widetilde L(\mathcal Ff)$2, while inverse-derivative and trigonometric operator functions yield explicit singular or oscillatory kernels (Stenlund, 2021).

In quantum computation, the same principle appears in a spectral-synthesis form. Given a Hermitian operator $\mathcal F(Lf)=\widetilde L(\mathcal Ff)$3 and an oracle for $\mathcal F(Lf)=\widetilde L(\mathcal Ff)$4, Fourier-based quantum signal processing approximates $\mathcal F(Lf)=\widetilde L(\mathcal Ff)$5 by a truncated Fourier series

$\mathcal F(Lf)=\widetilde L(\mathcal Ff)$6

and realizes this approximation through a single-qubit QSP construction whose matrix element reproduces the target series. The method uses one qubit ancilla regardless the degree of the approximating series, provides a classical algorithm for computing the phase parameters from the Fourier coefficients, and is compatible with Trotterised Hamiltonian simulations schemes and hybrid digital-analog approaches (Silva et al., 2022).

For fractional-order operators, the Fourier multiplier viewpoint remains primary on $\mathcal F(Lf)=\widetilde L(\mathcal Ff)$7: $\mathcal F(Lf)=\widetilde L(\mathcal Ff)$8. On bounded domains, however, the transform no longer diagonalizes the full boundary problem. Instead, Fourier factorization in the normal variable leads to $\mathcal F(Lf)=\widetilde L(\mathcal Ff)$9-transmission spaces, the boundary factor $\mathbb{R}$0, and the local Dirichlet trace $\mathbb{R}$1. This marks a shift from whole-space multiplier calculus to boundary-adapted pseudodifferential calculus (Grubb, 2022).

4. Fourier transforms whose inputs are operators

The Fourier–Wigner transform is an operator-valued analogue of the classical Fourier transform on phase space. For trace-class $\mathbb{R}$2 on $\mathbb{R}$3,

$\mathbb{R}$4

where $\mathbb{R}$5 is the Schrödinger representation. For rank-one $\mathbb{R}$6, this reduces to the cross-ambiguity function $\mathbb{R}$7. The transform is one fiber of the group Fourier transform of the Heisenberg group and satisfies an operator Hausdorff–Young inequality,

$\mathbb{R}$8

Its restriction theory is equivalent to restriction for the symplectic Fourier transform $\mathbb{R}$9 on $$\mathcal F\!\left(\frac{d}{dx}f\right)(\xi)=-i\xi\,\mathcal Ff(\xi),\qquad \mathcal F(xf)(\xi)=i\frac{d}{d\xi}\mathcal Ff(\xi),$$0, and its dual extension operator

$$\mathcal F\!\left(\frac{d}{dx}f\right)(\xi)=-i\xi\,\mathcal Ff(\xi),\qquad \mathcal F(xf)(\xi)=i\frac{d}{d\xi}\mathcal Ff(\xi),$$1

yields Schatten estimates from classical restriction theorems. In this framework, the Weyl transform of a smooth measure on a compact hypersurface with non-vanishing Gaussian curvature lies in $$\mathcal F\!\left(\frac{d}{dx}f\right)(\xi)=-i\xi\,\mathcal Ff(\xi),\qquad \mathcal F(xf)(\xi)=i\frac{d}{d\xi}\mathcal Ff(\xi),$$2 iff $$\mathcal F\!\left(\frac{d}{dx}f\right)(\xi)=-i\xi\,\mathcal Ff(\xi),\qquad \mathcal F(xf)(\xi)=i\frac{d}{d\xi}\mathcal Ff(\xi),$$3, resolving the Mishra–Vemuri conjecture (Luef et al., 2024).

Time–frequency analysis admits a parallel operator lift. For Hilbert–Schmidt operators $$\mathcal F\!\left(\frac{d}{dx}f\right)(\xi)=-i\xi\,\mathcal Ff(\xi),\qquad \mathcal F(xf)(\xi)=i\frac{d}{d\xi}\mathcal Ff(\xi),$$4,

$$\mathcal F\!\left(\frac{d}{dx}f\right)(\xi)=-i\xi\,\mathcal Ff(\xi),\qquad \mathcal F(xf)(\xi)=i\frac{d}{d\xi}\mathcal Ff(\xi),$$5

defines an operator-valued short-time Fourier transform. It satisfies an operator Moyal identity, yields a vector-valued reproducing kernel Hilbert space $$\mathcal F\!\left(\frac{d}{dx}f\right)(\xi)=-i\xi\,\mathcal Ff(\xi),\qquad \mathcal F(xf)(\xi)=i\frac{d}{d\xi}\mathcal Ff(\xi),$$6, and leads to coorbit spaces of operators $$\mathcal F\!\left(\frac{d}{dx}f\right)(\xi)=-i\xi\,\mathcal Ff(\xi),\qquad \mathcal F(xf)(\xi)=i\frac{d}{d\xi}\mathcal Ff(\xi),$$7 defined by mixed-norm conditions on $$\mathcal F\!\left(\frac{d}{dx}f\right)(\xi)=-i\xi\,\mathcal Ff(\xi),\qquad \mathcal F(xf)(\xi)=i\frac{d}{d\xi}\mathcal Ff(\xi),$$8. The admissible windows are exactly those operators $$\mathcal F\!\left(\frac{d}{dx}f\right)(\xi)=-i\xi\,\mathcal Ff(\xi),\qquad \mathcal F(xf)(\xi)=i\frac{d}{d\xi}\mathcal Ff(\xi),$$9 for which $\mathbb H$0, and these windows fully classify the operators generating equivalent norms on classical modulation spaces (Dörfler et al., 2022).

A distinct but related construction works directly on the Hilbert space $\mathbb H$1 of Hilbert–Schmidt operators. Quantum Hermite functions $\mathbb H$2 form an orthonormal basis of $\mathbb H$3, and the induced Fourier transform on $\mathbb H$4 diagonalizes on that basis: $\mathbb H$5 This supplies Plancherel theory, a Hausdorff–Young type estimate for Schatten classes $\mathbb H$6, and a Hardy-type theorem whose only simultaneous Gaussian-decay fixed point is a multiple of $\mathbb H$7 (Garg et al., 13 Feb 2026).

5. Unitary re-realizations: Bargmann, number–phase, and finite dimensions

One of the clearest forms of Fourier transform for operators is unitary re-realization. Under the Bargmann transform $\mathbb H$8, the fractional Fourier transform becomes the composition operator

$\mathbb H$9

Thus a non-local oscillatory integral operator on $\mathcal J$0 becomes rotation in the complex plane on Fock space. The same framework transports the Hilbert transform to an integral operator on $\mathcal J$1 with kernel involving the entire function $\mathcal J$2, and transports wavelet transforms to kernel operators of reproducing-kernel type (Dong et al., 2016).

In finite-dimensional quantum optics, the Pegg–Barnett phase operator is obtained by discrete Fourier transform of the truncated number operator

$\mathcal J$3

With $\mathcal J$4 the $\mathcal J$5-dimensional DFT and $\mathcal J$6,

$\mathcal J$7

This is the precise discrete analogue of the position–momentum relation. Moreover, the cyclic shift operator $\mathcal J$8 in circular waveguide arrays satisfies

$\mathcal J$9

so the London–Susskind–Glogower structure appears directly in the array Hamiltonian $SL(2,\mathbb R)$0 (Perez-Leija et al., 2015).

A common misconception is that number and phase are connected by the same continuous Fourier transform as position and momentum. In the Pegg–Barnett construction, the relation is discrete and finite-dimensional; the limit $SL(2,\mathbb R)$1 is taken only after physically relevant quantities are computed (Perez-Leija et al., 2015).

6. Specialized operator classes, non-Abelian transforms, and rigidity

Triebel’s Fourier operators take the form

$SL(2,\mathbb R)$2

with $SL(2,\mathbb R)$3 in a Hörmander-type symbol class. On Besov spaces $SL(2,\mathbb R)$4, these operators are compact under explicit conditions on $SL(2,\mathbb R)$5, and their eigenvalues satisfy polynomial decay with possible logarithmic corrections via entropy estimates for $SL(2,\mathbb R)$6 and Carl’s inequality. Here “Fourier transform for operators” means building an operator class by composing $SL(2,\mathbb R)$7 with pseudodifferential structure and then analyzing its spectrum (Triebel, 2022).

On Fock space, Toeplitz operators admit a Fourier decomposition at the symbol level: $SL(2,\mathbb R)$8 where each $SL(2,\mathbb R)$9 has compact support in the frequency domain. This decomposition yields boundedness criteria in terms of Carleson measures attached to derivatives of the heat transform $L^2(\mathbb H,du)$0, recovers compactness from vanishing Berezin transform $L^2(\mathbb H,du)$1 at infinity, and gives Schatten estimates for products of Toeplitz operators without invoking full pseudodifferential calculus (Wu et al., 2021).

Over finite fields, non-Abelian Fourier transforms $L^2(\mathbb H,du)$2 are constructed on paraspherical spaces $L^2(\mathbb H,du)$3 using a Slipper pairing $L^2(\mathbb H,du)$4 and kernels coming from $L^2(\mathbb H,du)$5-sheaves on Wang monoids. These transforms are normalized intertwining operators for parabolic induction. For opposite $L^2(\mathbb H,du)$6 parabolics of $L^2(\mathbb H,du)$7, the construction reduces to the classical linear Fourier transform on $L^2(\mathbb H,du)$8; for opposite Borels in $L^2(\mathbb H,du)$9 and opposite Siegel parabolics in $f$00, it becomes a Kloosterman Fourier transform on a quadric cone, with inversion on a natural subspace of special functions (Slipper, 2024).

At the level of abstract Fourier operators on $f$01, Henstock–Kurzweil integration gives a concrete integral realization of $f$02 on a dense subspace $f$03, together with differentiability formulas under the weaker hypothesis $f$04. This provides a generalized-integral model for an otherwise abstract operator extension (Arredondo et al., 2020).

Finally, rigidity results show that the product–convolution identities characterizing the Fourier transform are themselves stable only in a trivial sense: if a bijection $f$05 on a sufficiently large smooth function space satisfies

$f$06

under the hypotheses stated in the paper, then the corresponding identity holds exactly. On Schwartz space, previously known theorems then force $f$07 to be, up to diffeomorphism and possibly complex conjugation, a Fourier transform of the usual type (König et al., 2024).

In aggregate, these developments show that “Fourier transform for operators” is not a single theorem but a family of structurally related doctrines: transport of operator algebras under transforms, construction of transforms whose arguments are operators, and unitary equivalences that expose hidden canonical forms. Across Euclidean, hyperbolic, phase-space, quantum, pseudodifferential, and finite-field settings, the recurring principle is that Fourier analysis acts as a mechanism for reorganizing operator structure, often replacing locality by spectral simplicity, or replacing noncommutative geometry by differential–difference calculus (Neretin, 2020, Luef et al., 2024, Slipper, 2024).