For approximating a function of D variables, the typical fourier series is SUM_A B_A * exp(i*X*A) where X=(x1,x2,...,xD) is the coordinate-vector, and A=(a1,a2,...,aD) is a vector of integers, and B_A is a real coefficient. Another kind of fourier series would be SUM_{A,T} C_{A,T} * sin(i*x1*a1)*cos(i*x2*a2)*cos(i*x3*a3)*...*sin(i*xD*aD) where each function can be either sin or cos, as specified by a D-bit word T. Now you might say "the two are the same" which'd be true in some sense... but... Each term in the first series is a "plane wave." But each term in the 2nd series is a function that is not a plane wave, but rather something "pebbly looking." Also, the 2nd series has more coefficients, a factor 2^D or 2^(D-1) more, Which it seems to me makes a difference. I mean, if you view series #2 as arising from expanding series #1, the "extra" coefficients do not supply any extra information. But if you truncate both series at some given set of terms, then series #2 can provide a bitter fit than series #1 can, by taking advantage of the underdetermination of the coefficients to provide a better least-square fit. So when you view it that way, you see series #2 is actually a strictly wider class of series, capable of more feats, than series #1. For some reason, everybody(?) has focused on series #1, but I would think series #2 is the better choice, a goodly fraction of the time, in applications. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)