hi,
I don't know a whole lot about transforms however, I do remember one bit of information that may be useful to you: The prefactors in the sine transform (which is of course related to the fourier transform) are actually matters of convention. The sqrt(2/pi) is not always the prefactor! The prefactor is usually chosen to be sqrt(2/pi) so that the inverse transform will also have a prefactor of sqrt(2/pi) and there will be some symmetry in the two formulas so that they look alike and are easier to remember. However, SOMETIMES people chose the transform or inverse transform (I can't recall which) to have a 2/pi prefactor, and the other to have no prefactor. I comes down to a matter of convention and it doesn't matter which one you chose, as long as you are consistent. well, all this is clear to me.
if i do iDST(DST(f(r)) on discrete data, i get the original data back (expected result). this would indicate that 1) either the sqrt(2/pi) norm shouldn't be used for discrete data (why?) 2) the sqrt(2/pi) factor is taken care of somewhere inside the DST function (how?) this is the thing i'd really love to know - is the answer 1 or answer 2 correct? in my opinion, it shouldn't matter what combination of norms i use, as soon as they are consistent - but at the moment, i have the following problem. i need to iteratively solve H(r) = C(r) + C(r)*H(r) where the star denotes convolution and H and C are functions of interest. the easiest way to solve this should be fourier transforming this, solving H(k) = C(k) + C(k)H(k) by H(k) = C(k)/(1-C(k)) and then transforming H(k) back for further processing. in practice, i calculate C, transorm it into fourier space, solve for H and transform it back, where it serves as a 'parameter' for new C. due to the nature of my functions, the 3D-FT can be replaced by 1D fourier-bessel transformation AND some constants. depending on how one distributes the normalization of FT, one arrives at, e.g., FB(f(r)) = f(k) = 4pi/k int_0^infty f(r) r sin(kr) dr iFB(f(k)) = f(r) = 1/2rpi^2 int_0^infty f(k) k sin(kr) dk this can be calculated using fourier-sine transform of a new function F(r)=f(r)*r and F(k)=f(k)*k, respectively FB(f(r)) = 4pi/k sqrt(pi/2) int_0^infty F(r) sin(kr) dr iFB(f(r)) = 1/2rpi^2 sqrt(pi/2) int_0^infty F(k) sin(kr) dk so one would expect that the story is clear - do fourier-sinus, multiply with the respective normalization constant... but the problem is, that this doesn't work. at this point i asked the question - *should* i actually use these factors (coming from continuous transforms) also in the case of discrete transforms, which are self normalized to a number of discretization points?
PS> Dear Everyone, I am trying to get my email mailing list style fixed so that I don't break mailing list etiquette. Please email me personal to check me if this email sucked. Sorry for my previous mistakes! just a comment - don't change the subject line, since it breaks the thread
best, -- Lubos _@_" http://www.lubos.vrbka.net