三维数双曲谱变换 3D Hyperbolic Spectral Transform

摘要本文提出三维数空间 𝕋 上的双曲谱变换(𝒯-变换),是经典傅里叶变换在三维数代数上的自然推广。该变换引入双曲伸缩谱参数 δ₀ 与振荡频率谱参数 ω,将三维数信号分解为"振荡谱"与"伸缩谱"两个完全分离的正交通道。证明了 δ₀=0 时 𝒯-变换严格退化为经典傅里叶变换;导出了黄金衰减谱约束 |𝒯|² = C·e-δ₀/φ;验证了卷积定理、Parseval 能量守恒及逆变换重构。全部结论通过数值实验严格验证。关键词:三维数;双曲谱变换;傅里叶变换推广;黄金分割;卷积定理;能量守恒 AbstractThis paper proposes the Hyperbolic Spectral Transform (𝒯-transform) on the 3D number space 𝕋, a natural generalization of the classical Fourier transform to the 3D number algebra. The transform introduces a hyperbolic scaling parameter δ₀ and an oscillation frequency parameter ω, decomposing a 3D-number signal into fully decoupled "oscillation-spectrum" and "scaling-spectrum" channels (orthogonal due to ei=0). We prove that at δ₀=0 the 𝒯-transform reduces exactly to the classical Fourier transform, derive the golden-decay spectral constraint |𝒯|²=C·e-δ₀/φ, and verify the convolution theorem, Parseval energy conservation, and inverse-transform reconstruction numerically.