We derive a new relation between the discrete Fourier transform of a discrete sampling set of a compactly supported function and its Fourier transform. From this relation, we obtain a new window function. We then propose a new efficient algorithm to reconstruct the original function from the discrete sampling of its Fourier transform, which can adopt the fast Fourier transform and has much better accuracy than those in the literature. Several numerical experiments are also provided, illustrating the results.471233563364