# Discretising the Fourier Integral gives a high condition number

I have the following integral equation $$\int_0^1 e^{-2\pi i s t} f(t)\, \text{d} t = g(s), \hspace{3em} -1/2 \leq s \leq 1/2$$ where $$f$$ is to be found, and $$g$$ is known. I believe this problem is equivalent to finding the inverse Fourier transform of $$g$$.

This can be viewed as a Fredholm equation of the first kind. The function $$g$$ is known on $$n=129$$ points, which are equally spaced and symmetric about $$0$$. I have used the midpoint quadrature method for $$t$$ to discretise this problem and obtain a linear system $$A x = b,$$ where $$x$$ corresponds to $$f$$ evaluated at the quadrature points. The problem comes from the fact that the discretisation matrix $$A$$ in this problem has a huge condition number ($$\sim 10^{16}$$). My understanding of the Fourier transform is that this problem should be well-posed, or at least stable, and I'm struggling to figure out why the condition number am obtaining seems to be so high.

• If the condition number is 10^16, you will be fine solving this in quad precision (and will run into trouble in double precision). Also, if $f$ is not periodic, then perhaps use Gaussian quadrature nodes/weights instead of the midpoint rule. The other answer suggests that the problem itself is ill-conditioned so I don't know if that's a panacea, but learning to use multiple precisions will be very useful. Apr 27, 2019 at 0:19