Super-high-efficiency approximate calculation of series sum and discrete Fourier transform

We present a super-high-efficiency approximate computing scheme for series sum and discrete Fourier transform. The summation of a series sum or a discrete Fourier transform is approximated by summing over part of the terms multiplied by corresponding weights. The calculation is valid for the function under the transform being piecewise smooth in the continuum variable. The scheme reduces significantly the requirement for computer memory storage and enhances the numerical computation efficiency without losing accuracy. By comparing with the known results of examples, we show the accuracy and the efficiency of the scheme. The efficiency can be higher than $10^6$ for the examples.