Real fast Fourier transform on quasi-equidistant sample points

Trigonometric polynomial interpolation of periodic functions with period 2π on equidistant points in the interval [0, 2π) is a well-known and effective approximation tool. A standard numerical procedure for implementing this method is based on doubling the number of interpolation points at each step, so that the ordinary Fast Fourier Transform (FFT) technique is applicable. In this paper, a set is called a quasi-equidistant point set when it is the union of equidistant point sets with the same size but with mutually different phases. A fast algorithm is proposed for trigonometric polynomial interpolation on quasi-equidistant sample points for real periodic functions. The proposed algorithm is a generalization of the real FFT, but still requires n log2 n+O(n) real arithmetic operations, where n is the number of interpolation points. With the quasi-equidistant point set and the algorithm for the interpolation on them, it is possible to construct an efficient scheme for automatic function approximation in which the rate at which the number of sample points increases is less than 2 and can be arbitrarily close to 1.