In regression with a high-dimensional predictor vector, it is important to estimate the central and central mean subspaces that preserve sufficient information about the response and the mean response. Using the Fourier transform, we have derived the candidate matrices whose column spaces recover the central and central mean subspaces exhaustively. Under the normality assumption of the predictors, explicit estimates of the central and central mean subspaces are derived. Bootstrap procedures are used for determining dimensionality and choosing tuning parameters. Simulation results and an application to a real data are reported. Our methods demonstrate competitive performance compared with SIR, SAVE, and other existing methods. The approach proposed in the article provides a novel view on sufficient dimension reduction and may lead to more powerful tools in the future.