This important paper builds a non-equispaced fast Fourier transform which is based on an exact representation of \(\exp(-ix\xi)\) as a linear superposition of the functions \(\exp(-im\xi)\). This results in a greatly simplified analysis and increased flexibility and efficiency. The algorithm outperforms than Dutt-Rokhlin’s and Beylkin’s. The application to construction problem in computerized tomography is demonstrated which shows excellent exactness and efficiency compared with previous algorithms.