Generalized Legendre Transforms (GLFTs)

• Generalized Legendre Transforms are a class of transformations that extend classical convex conjugation by incorporating affine deformations and parameter-dependent dualities.
• They exhibit order-reversing involutivity and affine covariance, ensuring stability under domain and range reparameterizations with explicit dual pair formulas.
• GLFTs play a critical role in physics and mathematics by reformulating generating functionals, statistical ensembles, and geometric structures in integrable and quantum systems.

A Generalized Legendre Transform (GLFT) is a broad class of transformations that extend the classical Legendre–Fenchel transform beyond its canonical convex conjugation paradigm. GLFTs encompass affine-deformed conjugates, parameter-dependent symmetries for integrable systems, functional transforms for fields with multiple hierarchies of scales, and applications ranging from physics (statistical mechanics, quantum field theory) to pure mathematics (convex analysis, geometry of Frobenius manifolds, special function theory). The defining feature of GLFTs is their role in generating dual or alternate descriptions of systems—analytic, algebraic, or geometric—by systematically exchanging "primal" and "dual" parameters, often via non-trivial deformation or in the presence of additional structural (e.g., affine, symplectic, or combinatorial) freedoms.

1. Foundational Concepts and Formal Definitions

The classical Legendre transform acts on a convex function $F: \mathbb{R}^n \to \mathbb{R}\cup\{\infty\}$ by

$$(LF)(\eta) = \sup_{\theta \in \mathbb{R}^n} \{ \langle\theta, \eta\rangle - F(\theta) \}$$

yielding the convex conjugate. The transform is an involutive, order-reversing operation that features prominently in convex analysis, thermodynamics, and optimization.

A GLFT, in its most general algebraic form, is a one-parameter family of invertible, order-reversing transforms on the space of lower semicontinuous convex functions. According to the Artstein-Avidan–Milman theorem, any such GLFT is necessarily an affine deformation of the Legendre transform (Nielsen, 28 Jul 2025). Explicitly, for parameters $\lambda > 0$, $E \in \mathrm{GL}(\mathbb{R}^n)$, $f, g \in \mathbb{R}^n$, and $h \in \mathbb{R}$,

$$(\mathcal{T}F)(\eta) = \lambda\,(LF)(E\eta + f) + \langle\eta, g\rangle + h$$

where $LF$ denotes the ordinary Legendre–Fenchel transform. The GLFT class is closed under composition and naturally incorporates transformations relevant for applications, including scaling, reparametrization, and translation of both the domain and range.

In statistical mechanics and field theory, the GLFT is understood as the process of exchanging variables (such as energy for temperature) by a more general "link" function or cost. For instance, replacing the linear cost with $$C(\xi, \eta) = (1/\gamma)\log(1 + \gamma\,\xi^k\eta_k)$$ produces a deformed conjugacy suitable for generalized entropy frameworks (Morales et al., 2022).

2. Structural and Algebraic Properties

GLFTs exhibit several structural properties:

• Order-Reversing Involutivity: Applying a GLFT twice (up to coordinate change and possible sign/action) recovers the original function, owing to the underlying geometric duality.
• Affine Covariance: The class is stable under affine changes of variables; affine deformations commute with the transform up to parameter reshuffling (Nielsen, 28 Jul 2025, Iusem et al., 2017).
• Explicit Dual Pair Formulas: For $$F_P(\theta) = \lambda F(A\theta + b) + \langle\theta, c\rangle + d$$, one has

$$\mathcal{L}_P(F)(\eta) = (LF)_{P}(\eta) = L(F_{P^\circ})(\eta)$$

with $P^\circ$ an explicit involutive parameter map.

• Generalized Envelopes and Convolutions: When generalizing the duality for weight functions, GLFTs may act via infimal/supremal convolutions:

$$[\sigma \check{\ast} \tau](t) = \inf_{s > 0}\{\sigma(s) + \tau(t/s)\}, \quad [\sigma \hat{\ast} \tau](t) = \sup_{s > 0}\{\sigma(s) - \tau(s/t)\}$$

(Schindl, 12 May 2025).

In integrable systems and Frobenius manifold theory, GLFTs manifest as coordinate or metric changes generated by invertible vector fields or "Legendre fields," preserving associative products and relating entire hierarchies of Hamiltonian flows (Liu et al., 23 Nov 2024).

3. Geometric and Symplectic Interpretations

In Hessian geometry and information geometry, GLFTs correspond to coordinate transformations on manifolds with a strictly convex potential. This leads to the concept of dual affine connections $(\nabla, \nabla^*)$:

• The primal connection $\nabla$ is flat in "primal" coordinates $\theta$.
• The dual connection $\nabla^*$ is flat in "dual" coordinates $n = \partial\varphi/\partial\theta$.

The Legendre transform establishes a mapping between $(\theta, \varphi(\theta))$ and $(n, \psi(n))$, where $$\psi(n) = \langle\theta, n\rangle - \varphi(\theta)$$ (Gauvin, 6 Mar 2025). These dual coordinates underpin the geometry of dually flat statistical manifolds, yielding Bregman divergences and their generalizations. When the dual affine connections are deformed (e.g., by a Rényi or $\alpha$-divergence), the geometry ceases to be dually flat and exhibits curvature both in its metric and symplectic structure (Morales et al., 2022).

Deforming the link function in the GLFT (e.g., to a logarithmic function) gives rise to new symplectic forms, modified metrics, and Kähler structures. The divergence function produced by the GLFT may serve as a generating potential for both Riemannian and symplectic geometry, capturing the interplay between dual variables even in non-canonical (curved) contexts.

4. Applications in Physics and Applied Mathematics

GLFTs play a central role in multiple domains:

• Statistical Mechanics: In thermodynamics, GLFTs generalize the transition between different ensembles by systematically exchanging variables and reformulating the entropy, free energy, and partition function in terms of conjugate pairs. The symmetry and invertibility of multivariate GLFTs underpin the equivalence between microcanonical, canonical, and grand canonical ensembles (Wu et al., 3 Apr 2024).
• Quantum Field Theory: The passage from generating functionals to effective actions is implemented via an algebraic (combinatorial) generalization of the Legendre transform. Here, GLFTs generate correspondences between tree, connected, and full graph expansion coefficients, and their quasi-involutive character is rooted in homological (Euler characteristic) relations between vertices and edges of Feynman diagrams (Jackson et al., 2016).
• Frobenius Manifolds and Integrable Systems: Applying GLFTs (as Legendre or reciprocal transformations) relates prepotentials and principal hierarchies, generating symmetries between solutions of the WDVV equations; these connect rational and trigonometric V-systems and manifest as "rotations" of metrics and flows in hierarchies (Liu et al., 23 Nov 2024, Feigin et al., 29 Jul 2024, Strachan et al., 2016).
• Convex Analysis and Weight Functions: Generalized conjugates for weight functions encode the pointwise product and quotient of underlying sequences, with precise control of growth indices and regularity properties crucial for weighted ultradifferentiable functional spaces (Schindl, 12 May 2025, Schindl, 23 May 2025).

5. Construction, Authentication, and Analytical Methodologies

Methodologies for constructing and verifying GLFT pairs rely on:

Technique	Principle	Applicability
Supremum/Infimum Convolution	$g(m) = \sup_x \{mx - f(x)\}$; generalized via kernels	Convex, non-smooth
Inversion/Formal Duality	$g(m) = m(f')^{-1}(m) - f((f')^{-1}(m))$	Differentiable, invertible $f'$
Integral Representation	$g(m) = \int (f')^{-1}(m) dm$	Analytical approximation
Series Expansion	Taylor or Lagrange–Bürmann inversion at a (dual) basepoint	Approximate, special function

These approaches generalize naturally when the dual coordinate, kernel, or integration is deformed or involves more general operations, such as subgradients in non-smooth cases or functionals on Banach/Hilbert spaces with additional affine structure (Kolt et al., 2022, Iusem et al., 2017, Nielsen, 28 Jul 2025). In the context of weight functions, explicit algebraic or supremal convolution formulas determine how mapping properties and indices are transformed (Schindl, 12 May 2025).

6. Implications for Operator Theory and Function Spaces

GLFTs are exploited to analyze the mapping and continuity properties of operators between function spaces defined by weight matrices (as in Gelfand–Shilov or Braun–Meise–Taylor classes). The action of a GLFT on the weight function translates into pointwise matrix operations (product or quotient), and the cumulative effect on growth indices determines sharp inclusion relations and regularity criteria for the action of linear and nonlinear operators such as composition or resolvent operators (Schindl, 23 May 2025).

7. Information Geometry and Invariant Structure

In information geometry, GLFTs correspond to the canonical ambiguity in affine coordinate selection on dually flat (Hessian) manifolds. The invariance under such affine transformations underscores the geometric robustness of divergences, metric structures, and duality relations derived from convex conjugate potentials. GLFTs preserve fundamental geometric quantities (such as Bregman divergence, Fisher metric) up to possible metric scaling and affine coordinate changes, which coincide with the structure of exponential families and their dual geometries (Nielsen, 28 Jul 2025).

Summary Table: Generalized Legendre Transforms—Key Structures and Applications

Area	GLFT Manifestation	Reference
Convex Analysis	Affine-deformed conjugates	(Iusem et al., 2017, Nielsen, 28 Jul 2025)
Weight Functions	Infimal/supremal convolution, envelope	(Schindl, 12 May 2025, Schindl, 23 May 2025)
Frobenius Manifolds	Metric/coordinate rotation via Legendre fields	(Liu et al., 23 Nov 2024, Strachan et al., 2016, Feigin et al., 29 Jul 2024)
Thermodynamics	Ensemble transformation, entropy reparam.	(Wu et al., 3 Apr 2024, Gauvin, 6 Mar 2025)
QFT/Combinatorics	Algebraic duality of generating series	(Jackson et al., 2016)
Information Geometry	Dually flat manifolds, Bregman/α-divergences	(Morales et al., 2022, Nielsen, 28 Jul 2025)

Concluding Remarks

The theory of generalized Legendre transforms furnishes a unifying language to express duality, symmetry, and transformation properties across a wide range of mathematical, physical, and geometric contexts. Its current reach includes but is not limited to convex analysis, spectral theory, integrable systems, quantum field theory, and information geometry. The algebraic, geometric, and analytic structures underlying GLFTs not only generalize the O.G. Legendre–Fenchel framework but also generate new symmetries, invariants, and computational tools for dealing with multi-parameter, non-flat, or otherwise deformed systems.