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).

翻译：本文研究了基于分数阶傅里叶变换（FRFT）的傅里叶变换与拉普拉斯变换数值反演方法及闭型牛顿-柯特斯求积规则。研究表明，长度为QN的加权序列的快速FRFT可分解为两个快速FRFT的复合运算：长度为Q的加权序列的快速FRFT与长度为N的序列的快速FRFT。以方差伽马分布和广义温和稳定（GTS）分布为例，应用牛顿-柯特斯规则、复合快速FRFT及非加权快速分数阶傅里叶变换（FRFT）算法进行验证。结果表明，相较于非加权快速FRFT，复合快速FRFT在小样本量下能提供更精确的结果，且精度随权重数（Q）增加而提升。