Implements the Fast Sine Transform for transformation of one-dimensional real data sets. For reference, see James S. Walker, Fast Fourier Transforms, chapter 3 (ISBN 0849371635).

There are several variants of the discrete sine transform. The present implementation corresponds to DST-I, with various normalization conventions, which are specified by the parameter DstNormalization. It should be noted that regardless to the convention, the first element of the dataset to be transformed must be zero.

DST-I is equivalent to DFT of an odd extension of the data series. More precisely, if x₀, …, x_N-1 is the data set to be sine transformed, the extended data set x₀^#, …, x_2N-1^# is defined as follows

• x₀^# = x₀ = 0,
• x_k^# = x_k if 1 ≤ k < N,
• x_N^# = 0,
• x_k^# = -x_2N-k if N + 1 ≤ k < 2N.

Then, the standard DST-I y₀, …, y_N-1 of the real data set x₀, …, x_N-1 is equal to half of i (the pure imaginary number) times the N first elements of the DFT of the extended data set x₀^#, …, x_2N-1^#
y_n = (i / 2) ∑_k=0^2N-1 x_k^# exp[-2πi nk / (2N)]     k = 0, …, N-1.

The present implementation of the discrete sine transform as a fast sine transform requires the length of the data to be a power of two. Besides, it implicitly assumes that the sampled function is odd. In particular, the first element of the data set must be 0, which is enforced in transform(UnivariateFunction, double, double, int, TransformType), after sampling.