Quadratic Transform Overview

• Quadratic transforms are nonlinear operators using quadratic phases to generalize classical Fourier and related transforms.
• They underpin advanced methods in harmonic analysis, signal processing, and fractional programming with robust inversion and convolution properties.
• Their structured algebraic and analytic features enable efficient computation and innovative applications in quantum information and optimization.

A quadratic transform is a general term encompassing a family of nonlinear transformations, integral or discrete operators, and optimization techniques in which quadratic phases, quadratic forms, or quadratic surrogates play a central role. Quadratic transforms are pervasive across harmonic analysis, signal processing, mathematical optimization, time–frequency analysis, operator theory, and algebraic geometry. This article surveys main classes of quadratic transforms, their algebraic structures, analytic properties, and applications in computational mathematics and information sciences.

1. Quadratic Transforms in Integral Operator Theory

Quadratic phase transforms generalize the classical Fourier, fractional Fourier, and linear canonical transforms by introducing quadratic terms in the phase function of the integral kernel. The general form for the Quadratic Phase Fourier Transform (QPFT) on $L^1 \cap L^2(\mathbb{R})$ is

$$Q_\Lambda[f](u) = \int_{-\infty}^\infty f(x) K_\Lambda(x, u)\, dx,$$

with kernel

$$K_\Lambda(x, u) = \frac{1}{\sqrt{2\pi\,|B|}}\exp\left[-i\left( A x^2 + B x u + C u^2 + D x + E u \right)\right], \quad \Lambda = (A, B, C, D, E) \in \mathbb{R}^5,\, B \neq 0.$$

Specializations of the parameters $A,B,C,D,E$ recover the fractional Fourier transform (FrFT), linear canonical transform (LCT), special affine Fourier transform, and the classical Fourier transform as limiting cases (Gupta et al., 2024, Varghese et al., 5 May 2025, Varghese et al., 5 Jul 2025).

Key analytic properties:

• Inversion formula: The QPFT is invertible, with the explicit inverse obtained by complex conjugation of kernel phases and sign reversal of parameters.
• Energy conservation: Parseval/Plancherel identities hold, ensuring isometry on $L^2$.
• Convolution theorem: There is a chirp-convolution structure:

$$(f \star_\Lambda g)(x) = \int f(t) g(x-t) e^{2iA t(x-t)} dt,$$

such that

$$Q_\Lambda[f \star_\Lambda g](u) = Q_\Lambda[f](u) Q_\Lambda[g](u) \times \text{(phase factor)}$$

• Uncertainty principles: Quadratic extensions of Heisenberg, logarithmic, local (Donoho–Stark), and Rényi uncertainty principles are established (Gupta et al., 2024, Dar et al., 2022).

These transforms have multidimensional analogues, with the multidimensional QPFT using a symplectic block matrix parameterization and admitting associated multidimensional convolution theorems (Varghese et al., 5 May 2025).

2. Quadratic Transforms in Optimization: Quadratic Transform for Fractional Programming

A distinct and influential usage of the quadratic transform arises in fractional programming (FP), where optimization objectives involve ratios, such as SINR or generalized Rayleigh quotients. The quadratic transform in this context is an algorithmic device to decouple nonconvex constraints by introducing auxiliary variables and surrogate objectives: $$\sum_{i=1}^n \frac{A_i(x)}{B_i(x)} \;\Longrightarrow\; \max_{x,y}\; \sum_{i=1}^n [\,2y_i\sqrt{A_i(x)} - y_i^2 B_i(x)\,]$$ with $y_i^{\star} = \sqrt{A_i(x)}/B_i(x)$ at the stationary point (Shen et al., 13 Mar 2025, Shen et al., 2023).

The quadratic transform enjoys these features:

• Majorization-Minimization structure: Alternating updates in auxiliary and main variables increase the original objective monotonically, converging to stationary points.
• Closed-form updates: Inner minimizations/maximizations in auxiliary variables have explicit solutions.
• Matrix generalizations: The method extends to matrix ratios and MIMO problems, yielding surrogates of the form $$\operatorname{Tr}(2 Y_i^H \sqrt{A_i(x)} - Y_i^H B_i(x) Y_i)$$.
• Accelerated variants: Gradient-projection and Nesterov-accelerated versions achieve faster $O(1/k^2)$ convergence rates under Lipschitz gradient assumptions (Shen et al., 2023).

Applications include:

• Multi-user power control and beamforming (sum-log-ratio objectives)
• CRB minimization in radar estimation
• Normalized cut objectives in spectral clustering
• SVM margin maximization (via fractional max–min programming)
• Massive MIMO optimization avoiding large matrix inversions (Shen et al., 13 Mar 2025, Shen et al., 2023).

Comparison to classical FP methods (e.g., Dinkelbach) shows that quadratic transforms efficiently handle sum-of-ratios and matrix-valued ratios inaccessible to classical Dinkelbach techniques (Shen et al., 13 Mar 2025).

3. Discrete Quadratic and Quantum Quadratic Transforms

3.1 Quadratic Discrete Fourier Transform

The quadratic discrete Fourier transform (QDFT) extends the standard DFT by including quadratic phases: $$(\mathbf{F}_{r,a})_{n\alpha} = \frac{1}{\sqrt{d}}\, q^{ -\tfrac{a}{2} n^2 + (\alpha - \frac{d-1}{2} r) n - \tfrac{(d-1)^2}{4} r }$$ where $q = \exp(2\pi i / d)$, $a \in \mathbb{Z}/d\mathbb{Z}$, $r \in \mathbb{R}$. For $r=a=0$, this reduces to the standard DFT. In quantum information, the QDFT enables explicit constructions of complete sets of mutually unbiased bases in prime dimensions (Kibler, 2010).

3.2 Quadratic Quantum Fourier Transform

The quantum quadratic Fourier transform (QQFT) is a number-conserving unitary on Fock spaces, constructed as

$$\hat{V} = \exp \biggl( \frac{2\pi i}{L} \sum_{m,n=0}^{L-1} A_{m,n}\, \hat{\psi}_m^\dag \hat{\psi}_n \biggr),$$

where $\hat{\psi}_n^\dag$ are creation operators. QQFT is QFT on the single-particle sector but imparts genuinely quadratic many-body phases. It enables programmable Hamiltonian engineering in synthetic quantum materials, allowing local implementation (depth $O(N \log N)$ in system size) of nonlocal or symmetry-rich models, including Poincaré-invariant crystals and flat Chern bands. The protocol is robust to experimental noise tolerances within current capabilities (Wang et al., 2022).

4. Quadratic Transforms in Harmonic and Time–Frequency Analysis

Extensions of the QPFT and their variants are fundamental to fine-scale signal representation, time–frequency localization, and adaptive filtering:

• Windowed QPFT: Combines nonstationary localization with quadratic phase sensitivity. The WQPFT admits a reproducing kernel Hilbert space structure, explicit inversion, and a suite of convolution theorems, both in the time- and frequency-domain (Varghese et al., 5 Jul 2025).
• Quadratic-phase wave packet transform: Achieves optimal localization for quadratic-phase signals by blending the QPFT and classical wave packet machinery (Bhat et al., 2022).
• Multiresolution analysis: Wavelet systems and Stockwell transforms constructed over the QPFT domain support adaptive decompositions tailored to quadratic-phase phenomena (Gupta et al., 2024).
• Uncertainty principles: All standard (Heisenberg, logarithmic, Rényi, Donoho–Stark) uncertainty principles adapt with explicit dependence on the quadratic phase parameter(s) (Gupta et al., 2024, Dar et al., 2022, Gupta et al., 2022, Bhat et al., 2022).
• Non-separable kernels: The multi-parameter, multi-matrix QPFT framework enables the construction of Wigner-type and STFT-type time-frequency distributions that retain energy conservation, cross-term suppression, and explicit analytic relations to standard transforms (Chauhan et al., 8 Sep 2025).
• Extensions to Dunkl and Bessel analysis: Quadratic-phase Dunkl transforms generalize QPFT to reflection-invariant settings, maintaining unitary inversion, robust convolution theory, and Heisenberg-type uncertainty (Saoudi, 26 Dec 2025).

Quaternionic and hypercomplex formulations (QQPFT, DQQPFT) further expand the applicability of quadratic transforms to multichannel, color, and multidimensional signals, preserving invertibility, modulation, and convolution structure (Dar et al., 2022, Dar, 2024, Gupta et al., 2022).

5. Algebraic and Functional Transformations

Quadratic transformations occur as change-of-variable maps and interpolation kernels in algebra and analysis:

• Affine and polynomial maps: Classification of invertible quadratic maps on $\mathbb{R}^2$ shows that, up to affine equivalence, only the map $(x, y) \mapsto (x+y^2, y)$ is invertible, with applications in Diophantine parameterizations such as the perfect cuboid problem (Sharipov, 2015).
• Elliptic and hypergeometric kernels: Quadratic transformations among multivariable integrals (Selberg–type, elliptic hypergeometric) are mediated by advanced interpolation kernels, which are also kernels of integral transforms in double affine Hecke algebra theory. Their braid and difference equations recover and generalize classical quadratic and quartic transformation identities for special functions (Rains, 2014).
• Generalized hypergeometric identities: Quadratic transformations for hypergeometric series, as exemplified by generalizations of Exton's quadratic transformation, are proved via summation theorems or manipulation of Bailey chains (Kim et al., 2014).

6. Extensions, Offshoots, and Applications

Quadratic transforms have continued to motivate diverse mathematical and engineering innovations:

• Matrix biorthogonal polynomials: Imposing the quadratic map $z \to z^2$ on matrix-valued polynomial systems enforces block structure on Gram matrices and orthogonality, producing new forms of Christoffel and kernel relations (Behera, 2021).
• Signal processing and adaptive filtering: QPFT and quadratic transform-based approaches directly address the properties and optimization of radar waveforms, time-varying filtering, MIMO channel estimation, and graph-spectral segmentation (Shen et al., 13 Mar 2025, Shen et al., 2023, Dar, 2024).
• Numerical and computational aspects: Fast algorithmic implementations are available via decompositions into chirp multiplications and FFTs, particularly for quaternionic DQQPFT, resulting in $O(N^2 \log N)$ complexity (Dar, 2024).

7. Summary Table of Major Quadratic Transform Classes

Transform Class	Domain / Setting	Key property / Application	Reference
Quadratic Phase FT (QPFT)	$L^2(\mathbb{R})$	Generalizes FT, FrFT, LCT, convolution	(Gupta et al., 2024, Varghese et al., 5 May 2025)
Quadratic Transform (FP)	Optimization	Decouples multi-ratio nonconvex programs	(Shen et al., 13 Mar 2025, Shen et al., 2023)
Discrete QDFT/QQFT	$\mathbb{C}^d$, ℋ	Mutually unbiased bases, quantum control	(Kibler, 2010, Wang et al., 2022)
Multivariate Kernel	Elliptic Hypergeometric	Symmetry, quadratic transformation	(Rains, 2014)
QQPFT/DQQPFT	Quaternionic/Hyp. signals	Multichannel time-frequency analysis	(Dar et al., 2022, Dar, 2024)
Quadratic Dunkl	Reflection-invariant	Harmonic analysis, uncertainty principle	(Saoudi, 26 Dec 2025)

The quadratic transform concept thus provides a unifying algebraic and analytic formalism underpinning fundamental advances across optimization, harmonic analysis, quantum information, and special function theory.