Given value of function at some points, there is a method called spectral method could give an approximation of the derivative of at same points. The basic idea of spectral method is using the derivative of the approximation calculated by the value of as an approx of derivative of . Usually, the approximation is chosen as a polynomial. In this article, we use Fourier transform to construct this polynomial approximation.
Unbounded Discrete Grid
Firstly, we consider the Fourier spectral method on unbounded discrete grid since this is the best way to show how this method work, though it may not be very useful in practical.
Use the definitions and notions in Fourier Transform And Fourier Series, in this case , the Fourier transform (also called semi-discrete Fourier transform) is
where , the inverse Fourier transform is
We are given , and our goal is to approximate the derivative of at points in . There is a brilliant idea to have approximation of by above inverse Fourier transform formula, that is, letting
then hence is an interpolation of . Furthermore, the Fourier transform of (on ) has compact support in i.e.
Naturally, would be a good approximation of .
It’s difficult to compute from the formula above directly, so we need a smarter way to construct . Rewrite value function as
where is the Kronecker delta function: and if . Interchange the infinite sums, the Fourier transform of can be written as
and
which is coincided with Fourier transform defined before. Now let
hence is the sinc function:
Use the property of Fourier transform the inverse Fourier transform of is inverse Fourier of with a shift of , that is
Now we can conclude that the interpolation of is
by the linearity of Fourier transform. So
where
Furthermore we can also calculate to approx by
Periodic Discrete Grid
In this section, our grid is periodic and discrete: in bounded interval . Given values on this grid , we want to approximate the derivative of not only on these grid points, but also on the whole interval .
Like before, we first do Fourier transform here. In this case, this kind Fourier transform is also called discrete Fourier transform (DFT). The DFT of is
And the inverse Fourier transform (IDFT) of is
Once again, we rewrite as
and the DFT of is on . IDFT of is
where is the DFT operator and is the IDFT operator. The idea of this spectral method is extending the from to , then will also be extended to by IFT:
where the IDFT operator turn the function on unbounded grid to the function on continuous interval , extends by defining . As a consequence we have
where . At this monument, we have a interpolation of on
and the derivative of at is
Unfortunately, the derivative of is not a real number if is not a grid point., which makes the derivative martix will be a complex matrix. To fix this problem, we should use as the phase number when do DFT, instead of using . But this force ourselves to assume is even. In this case, we need a special form of IDFT. is defined to be
to make this transform have a kind of symmetric property. This IDFT can also represent by
where is the IDFT operator on and is the IDFT operator on . Now interpolation of is
and this function is called periodic sinc function, denoted by . In the end we have
as a band-limited interpolation of on . The derivative of at is
