Improve Fourier transform capabilities

[Anonyame]

我试了试SymPy，发现好多函数的傅里叶变换都算不出来。常数函数与符号函数的傅里叶变换也算不出来（捂脸）。如果SymPy可以在这些信号处理的问题上做出一些改进，我将感激不尽！感谢你们！

I've tried SymPy, but I found that the Fourier transforms of many functions could not be calculated. Simply, the Fourier transforms of constant function, as well as sign function could not be computed either (facepalm). I would appreciate it if SymPy can make some improvements in signal processing problems! Thx!

[Anonyame]

非常感谢你们开发的这个工具箱！

Thank you so much for developing this useful toolkit!

[oscarbenjamin]

Specifically the issues are:

In [7]: fourier_transform(1, x, k) # should be DiracDelta(x)
Out[7]: FourierTransform(1, x, k)

In [8]: inverse_fourier_transform(DiracDelta(x), x, k)
Out[8]: 1

In [12]: fourier_transform(sign(x), x, k) # should be 2/(x*I)
Out[12]: FourierTransform(sign(x), x, k)

In [11]: inverse_fourier_transform(2/(x*I), x, k) # should be sign(x)
Out[11]: 
                            ⎛1      ⎞
-2⋅ⅈ⋅InverseFourierTransform⎜─, x, k⎟
                            ⎝x      ⎠

[Anonyame]

Specifically the issues are:

In [7]: fourier_transform(1, x, k) # should be DiracDelta(x)
Out[7]: FourierTransform(1, x, k)

In [8]: inverse_fourier_transform(DiracDelta(x), x, k)
Out[8]: 1

In [12]: fourier_transform(sign(x), x, k) # should be 2/(x*I)
Out[12]: FourierTransform(sign(x), x, k)

In [11]: inverse_fourier_transform(2/(x*I), x, k) # should be sign(x)
Out[11]: 
                            ⎛1      ⎞
-2⋅ⅈ⋅InverseFourierTransform⎜─, x, k⎟
                            ⎝x      ⎠

Indeed. Is there a way to avoid this result, please?

[oscargus]

If someone is interested in doing this, take a look at the Laplace transform where @hanspi42 made a great job extending the capabilities enormously. A similar approach with "tables" of precomputed results can be used here.

[sylee957]

There could also be some algebraic approach for fourier transformation
https://arxiv.org/pdf/2009.12198.pdf
If someone is able to comprehend that paper, it can give some insight for completing the difficult problem solving in this domain.
But it wouldn't deprecate the need to implement the usual Heuristic/Pattern-Matching/Term-Rewriting approach to this problem.