Umbral methods, function factorisation and generalisation of the Fourier transform method

Technical Summary: Umbral Methods, Function Factorisation, and Generalisation of the Fourier Transform Method

Problem Statement
The paper addresses the theoretical extension of trigonometric functions beyond the classical sine–cosine pair. While the history of mathematics contains various forms of "higher-order" trigonometric functions (e.g., elliptic functions, quasi-trigonometric functions), the authors propose a systematic method to construct trigonometric-like functions based on the factorisation of rational umbral operators. The core problem is to generalize the concept of cyclic decomposition—typically associated with roots of unity in standard exponentials—to rational umbral expressions, thereby generating $n$-fold cyclic components for any integer $n \geq 2$.

Methodology
The authors employ a two-pronged approach, transitioning from a formal algebraic framework to a rigorous analytic one:

1. Formal Umbral Calculus:

   • The study begins by defining a correspondence between rational functions and transcendental functions using an umbral operator $\hat{c}$ acting on a vacuum state $\phi_0$, where $\hat{c}^\alpha \phi_0 = 1/\Gamma(1+\alpha)$.
   • The Gaussian function $e^{-x^2}$ is treated as the umbral image of the Lorentzian rational function $1/(1+\hat{c}x^2)$.
   • By applying partial fraction decomposition to the rational operator, the authors factorise the expression using roots of unity (e.g., $\pm i$ for the quadratic case, cubic roots for the cubic case).
   • This factorisation yields a set of $n$ functions that sum to the original umbral function. These components are identified as specific cases of the two-parameter Mittag–Leffler function, $E_{\alpha, \beta}(x)$.

2. Analytic Umbral Framework:

   • The formal identities are reinterpreted using the analytic umbral framework, where the umbral action is realised via Mellin–Barnes pairings between a jump kernel and a ground state.
   • The authors demonstrate that the formal factorisation corresponds to a "spectral transmutation" and a "Mellin dilation."
   • Specifically, the cyclic components are shown to arise from the splitting of the spectral kernel $\pi/\sin(n\pi s)$, which mirrors the root-of-unity decomposition of the rational kernel.

Key Contributions and Results

• Construction of Higher-Order Trigonometric Functions:

  • Quadratic Case ($n=2$): The factorisation of the Gaussian umbral operator yields "Gaussian trigonometric functions." The cosine-like component is the Gaussian $e^{-x^2}$, and the sine-like component is its umbral companion, $e^{-x^2}\text{erfi}(x)$ (related to the Faddeeva function). These satisfy Kramers–Kronig-type relations.
  • Cubic Case ($n=3$): The method is extended to the cubic exponential $e^{-x^3}$. This yields a three-component cyclic system involving Mittag–Leffler functions $E_{1/3, 1}(-x)$, $E_{1/3, 4/3}(-x^3)$, and $E_{1/3, 5/3}(-x^3)$.
  • General Case ($n$): The authors establish a general decomposition where the rational operator $1/(1+\hat{c}x^n)$ is split into $n$ components using the $n$-th roots of unity. Each component isolates a specific cyclic sector of the expansion.

• Generalisation of the Fourier Transform:

  • The paper defines a "g-Fourier transform" ($F_n$) where the standard exponential kernel $e^{ikx}$ is replaced by a Mittag–Leffler kernel $E_{1/n, 1}(ikx)$.
  • Using the umbral correspondence, the authors derive that the g-Fourier transform of a function can be evaluated by mapping it to a standard Fourier transform with a modified spectral ground state involving Gamma functions.
  • Explicit calculations are provided for $n=2$ (Gaussian case), showing that the transform of $e^{-|x|}$ and $e^{-x^2}$ can be expressed in terms of standard functions and error functions.

• Analytic Interpretation:

  • The paper clarifies that the formal cyclic components are not merely algebraic projections but analytic contour objects. Their local expansions and sectorial asymptotics arise from different deformations of the same Mellin–Barnes integral.
  • The splitting of the rational kernel is shown to be analytically equivalent to the splitting of the Mellin kernel $\pi/\sin(n\pi s)$.

Significance and Claims
The paper claims to provide a "consistent and operationally transparent framework" for extending Fourier analysis to functions generated by non-standard exponentials. The primary significance lies in unifying the construction of higher-order trigonometric functions and generalised Fourier transforms under a single algebraic scheme based on cyclic symmetry and roots of unity.

The authors assert that:

1. Unification: The quadratic (Gaussian) and cubic cases are not isolated phenomena but specific instances of a general $n$-fold decomposition principle.
2. Operational Simplicity: The umbral method allows the evaluation of complex transforms (like the g-Fourier transform) by mapping them onto standard Fourier transforms, preserving the algebraic backbone of the analysis.
3. Analytic Rigor: The transition to the Mellin–Barnes framework provides a rigorous justification for the formal umbral manipulations, explaining the origin of the component expansions and their asymptotic behavior.

Limitations and Future Directions (as stated by the authors)
The paper maintains a modest scope regarding its conclusions:

• It does not develop a full inversion theory for the umbral Fourier transform; the construction of the anti-transform remains an open problem requiring further investigation into dual vacuums and inverse operators.
• The admissible classes of test functions and distributions for the analytic-functional setting are not fully characterised.
• While the connection to elliptic functions is speculated (via doubly periodic structures), this is proposed as a potential future extension rather than a result of the current work.
• The asymptotic and Stokes behavior of the cyclic components, though visible in the Mellin–Barnes representation, is reserved for separate treatment in future work.