Enhanced the Fast Fractional Fourier Transform (FRFT) scheme using the closed Newton-Cotes rules

The paper considers the fractional Fourier transform (FRFT)--based numerical inversion of Fourier and Laplace transforms and the closed Newton Cotes quadrature rules. It is shown that the fast FRFT of a QN-long weighted sequence is the composite of two fast FRFTs: the fast FRFT of a Q-long weighted sequence and the fast FRFT of an N-long sequence. The Newton-Cotes rules, the composite fast FRFT, and non-weighted fast Fractional Fourier transform (FRFT) algorithms are applied to the Variance Gamma distribution and the Generalized Tempered Stable (GTS) distribution for illustrations. Compared to the non-weighted fast FRFT, the composite fast FRFT provides more accurate results with a small sample size, and the accuracy increases with the number of weights (Q).