The paper promotes a new sparse approximation for fractional Fourier transform, which is based on adaptive Fourier decomposition in Hardy-Hilbert space on the upper half-plane. Under this methodology, the local polynomial Fourier transform characterization of Hardy space is established, which is an analog of the Paley-Wiener theorem. Meanwhile, a sparse fractional Fourier series for chirp \( L^2 \) function is proposed, which is based on adaptive Fourier decomposition in Hardy-Hilbert space on the unit disk. Besides the establishment of the theoretical foundation, the proposed approximation provides a sparse solution for a forced Schr\(\ddot{\textrm{o}}\)dinger equations with a harmonic oscillator.

该论文提出了一种新的分数阶傅里叶变换稀疏近似，它基于上半平面 Hardy-Hilbert 空间中的自适应傅里叶分解。在该方法下，建立了Hardy空间的局部多项式傅立叶变换表征，这是Paley-Wiener定理的类比。同时，提出了一种基于单位圆盘上Hardy-Hilbert空间自适应傅里叶分解的线性调频\(L^2\)函数的稀疏分数阶傅里叶级数。除了建立理论基础外，所提出的近似还为带有谐振子的受迫 Schr\(\ddot{\textrm{o}}\)dinger 方程提供了稀疏解。