A compressive sensing (CS) reconstruction method for polynomial phase signals is proposed in this paper. It relies on the Polynomial Fourier transform, which is used to establish a relationship between the observation and sparsity domain. Polynomial phase signals are not sparse in commonly used domains such as Fourier or wavelet domain. Therefore, for polynomial phase signals standard CS algorithms applied in these transformation domains cannot provide satisfactory results. In that sense, the Polynomial Fourier transform is used to ensure sparsity. The proposed approach is generalized using time-frequency representations obtained by the Local Polynomial Fourier transform (LPFT). In particular, the first-order LPFT can produce linear time-frequency representation for chirps. It provides revealing signal local behavior, which leads to sparse representation. The theory is illustrated on examples.