A New Harmonic Analysis Framework: Lissajous Figure Decomposition via Theta Functions

We present a novel harmonic decomposition method based on Lissajous figures and Jacobi theta functions. Unlike the classical Fourier series which uses a single frequency basis {einωt}, our approach employs a two-frequency basis {ϑk(ω1t + iω2t|τ)}4k=1, naturally suited for signals with two independent frequency components. We develop a rigorous mathematical theory, prove completeness and orthogonality of the basis, establish exponential convergence rates, and demonstrate numerical performance with 99.88% reconstruction accuracy forshort-duration signals. In the second part, we apply the method to spectral complexity analysis, introducing an entropy-based criterion that distinguishes the determinant (P-complete) from the permanent (#P-complete).