Chrestenson Transform and Its Relations with Fourier Transform

In this paper, through series basic mathematical expression we show the relations between Chrestenson transform and Fourier´s transform in discrete form. We show that under certain conditions Discrete Chrestenson Transform is equivalent to Discrete Fourier Transform(DFT), and so is Fast Discrete Chrestenson Transform equivalent to FFT under certain conditions. We also show that coefficient distribution of signal´s Chrestenson transform only depends upon p in Chrestenson function instead of data samples p^m, where m is any integer larger than 1 and p is any integer larger than 2. We first derive series of discrete Chrestenson Transform expressions generally, and then consider various cases for sample numbers and frequency samples number. Lastly, we also show that samples of p^m do not affect coefficient distribution of signal´s Chrestenson transform but can better reduce the correlations between all coefficients statistically for p-adic process.