From Trigonometric to Hyperelliptic Functions: A Unified Framework for Fourier Analysis

Fourier analysis, based on trigonometric functions, Decomposes functions into superpositions of simple harmonic oscillations; this is the lowest-order form of the entire harmonic analysis hierarchy. By introducing more complicated periodic structures, differential operators, and symmetries, one naturally generalizes to hyperbolic functions, elliptic functions, hyperelliptic functions,and even higher-order transcendental functions. This paper systematically develops this hierarchy and establishes a unified mathematical framework: on a torus (real or complex) of arbitrary genus g, every square-integrable function admits a Fourier-type expansion as a multiple Fourier series Pn∈Zg cne 2πin·x (discrete spectrum); on the corresponding non-compact spaces (such as the real line, the multiplicative group, the upper half-plane, Euclidean space,etc.) we obtain invertible integral transforms (continuous spectrum), including the Fourier transform, Mellin transform, theta-kernel transform, higherdimensional Fourier transform, and spectral decomposition of automorphic forms. The theta function, as the paradigm of multiple Fourier series, pervades all levels. We provide rigorous statements and proofs, including complete derivations of the completeness of Lamé series, the invertibility of the theta-kernel transform, the multiple Fourier expansion of hyperelliptic functions, and the invertibility of the higher-dimensional theta transform. We also discuss the orthogonality and convergence conditions of basis functions at each level, and relate the Mellin transform to wavelet scale analysis.