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 On Wed, Apr 22, 2015 at 7:09 AM, Warren D Smith <warren.wds@gmail.com> wrote:
On 4/22/15, Warren D Smith <warren.wds@gmail.com> wrote:
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.
--or complex coefficient.
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