From: Neil Bickford <techie314@gmail.com> Unfortunately (if I've interpreted the above correctly), it looks like expanding the D-bit version over individual bits of T just results in a product of a whole bunch of phase-shifted sine waves, scaled by some amount*.
This can't represent arbitrary functions (or even as many as any individual Fourier transform can), because it has zeroes all over the place.**
*Since C_{A,0...}*Sin[i*x1*a1]*(rest of the sum...) + C_{A,1...}*Cos[i*x1*a1]*(rest of the sum...) = Sqrt[c1^2+c2^2]*Sin[i*x1*a1 + ArcTan[c2/c1]]*(rest of the sum...).
** Unless the series expresses a constant, in which case it's, um, well, not extraordinarily interesting.
--Neil Bickford
--WDS responds: The fact that sin(x) and cos(x) have many zeros is no obstacle to SUM_k a[k] * sin(x*k) + b[k] * cos(x*k) representing any 2*pi-periodic nice-enough function. This sum is, in fact, a standard 1-dimensional fourier series, which DOES represent any 2*pi-periodic nice-enough function. Therefore I fail to understand your "This can't represent arbitrary functions (or even as many as any individual Fourier transform can), because it has zeroes all over the place... unless a constant." If you expand exp(i*x) = cos(x)+i*sin(x) and exp(i*x+i*y) = [cos(x)+i*sin(x)]*[cos(y)+i*sin(y)] = cos(x)*cos(y)+i*sin(x)*cos(y)+...-sin(x)*sin(y) etc, then the usual fourier series in the form USUAL FOURIER = SUM_A C_A * exp(i*A.X) summed over integer vectors A -- which will represent any nice-enough complex-valued function of D variables that is periodic in a D-dimensional box with sidelengths 2*pi -- is converted via the expansion process to a form like my second kind of fourier series. In fact, it is the same as my second kind of fourier series, EXCEPT that the coefficients of the new series are determined (via the expansion process) from a much small number of coeffs of the old series, so the "extra information" in all those extra coefficients is illusory. But if we now permit the extra coeffs to be altered independently so that that extra information really is there, then we get a wider class of series -- which was my point. And it still seems to me to be a good point, I'm not seeing anything wrong about it. This "new" kind of fourier series seems likely to me to be more useful than the old plane-wave kind for many applications. Which is more useful? Probably depends on the application.