A sparse approximation for fractional Fourier transform

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 \( L^2 \) $L^2$ 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 \(\ddot{\textrm{o}}\) $\ddot{\text{o}}$ dinger equations with a harmonic oscillator.