First I want to correct Wikipedia. They say Gauss had something 2 years
before Fourier published his work. That's BS !
In those times, nothing could exist if not publihed and wrt Fourier,
he was banned by J. Lagrange from publishing until Lagrange died.
So, Fourier had to wait 12 years before publishng ... then 10 years
before Gauss plagia Fourier.
Through the years, Wikipedia had decreased in interest [near useless].

"fft" is not possible from Smath as is.
Smath takes Sigma of vector but not Sigma of function.
That excludes extracting coefficients of a vector. Coefficients it
spits are not correct and the reconstruction not correct either,
even if given the correct coefficients. At least, I found no way.

Would be great some of you collab spare time on "Smath fft".

There is not alternative than getting the FFTW code and
implement Strictly Mathcad 8 Pro ... 11 style, specifically
for this applicatioon indexing from '0'.
I hate Smath indexing from '1' . ONE HUNDRED PERCENT [100 %]
of numerial maths range 0 .. 1

Jean

Fourier Sampling.sm (43 KiB) downloaded 79 time(s).
Fourier Budapest.sm (118 KiB) downloaded 69 time(s).

Hello Jean,

I've fixed your beautiful Budapest algorithm example; I think all is just about confusion on indexing. My suggestion to keep all the simplest as possible is to leave all the indexing issues in the indices (use index+1 in the matrix/vector when you are in a one-based environment and you have a zero-based code and use index-1 when you are in the opposite case).

Fourier Budapest.sm (97 KiB) downloaded 79 time(s).