A SUMMABILITY METHOD FOR THE ARITHMETIC FOURIER-TRANSFORM

The Arithmetic Fourier Transform (AFT) is an algorithm for the computation of Fourier coefficients, which is suitable for parallel Processing and in which there are no multiplications by complex exponentials. This is accomplished by the use of the Mobius function and Mobius inversion. However, the algorithm does require the evaluation of the function at an array Of irregularly spaced points. In the case that the function has been sampled at regularly spaced points, interpolation is used at the intermediate points of the array. Generally the AFT is most effective when used to calculate the Fourier cosine coefficients of an even function. In this paper a summability method is used to derive a modification of the AFT algorithm. The proof of the modification is quite independent of the AFT itself and involves a summation by primes. One advantage of the new algorithm is that with a suitable sampling scheme low Order Fourier coefficients may be calculated without interpolation.