For periodic not-necessarily smooth $f$ and a range of $m$, say $0\ldots 31$, I want to compute $\int_{-\pi}^\pi f(t) \cos (mt) dt$ (and maybe the same integral with $\sin$ instead of $\cos$) to machine precision (or close-ish to it).

The obvious thing to do is to evaluate $f$ at an equispaced grid and take a DFT, but this is only giving me linear convergence, and getting enough precision will take a very fine grid.

The other obvious thing (which I haven't yet tried) would be to pre-compute $w_i \cos(m t_i)$ for $w_i$ $t_i$ the Legendre weights and points (scaled to $(-\pi,\pi)$), then at evaluation time compute $f(t_i)$ and find all the integrals with a matvec. (All else being equal, I'd prefer not to have to have a bunch of pre-computed data to keep track of, but it's better than re-computing it every time if that's the only other option)

My only other idea (also not yet tried) was to figure out a closed-form solution for $\int_{-\pi}^\pi T_n(t/\pi)\cos(mt)$ for $T_n$ the Chebyshev basis, then at evaluation time do a fast Chebyshev transform to get $f \approx \sum_i c_i T_i $ and use that to compute the integrals.

None of these seem particularly satisfying.

So my question is: is there some neat technique I'm missing that will let me compute banks of these coefficients with spectral (or at least quadratic) accuracy and with good complexity?

  • $\begingroup$ There are several numerical integration techniques and probably several for computing Fourier coefficients. What references have you looked at? $\endgroup$
    – Somos
    Mar 28, 2019 at 2:34
  • 1
    $\begingroup$ There's literature out there but they tend to assume smooth $f$. Numerical Recipes (3rd edition) 13.9 has discussion for smooth (of unclear order) non-periodic, TOMS649 has one for p-order smooth. Lyness has ams.org/journals/mcom/1971-25-113/S0025-5718-1971-0293846-4 but the points of non-smoothness are needed. I did find this from Patterson : link.springer.com/article/10.1007%2FBF01399083 that looks like the third idea above, which is promising. $\endgroup$
    – JCK
    Mar 28, 2019 at 18:44
  • $\begingroup$ Perhaps an obvious question, but why do you want these integrals evaluated? What will you do with their values? $\endgroup$
    – Somos
    Mar 28, 2019 at 18:58
  • $\begingroup$ I'm interested in decomposing functions on the unit disk in terms of Zernike polynomials, and am following Boyd&Yu sciencedirect.com/science/article/pii/… , in particular Eq (141) in which $p_n(r)$ for Zernike polynomials is a fourier integral. $\endgroup$
    – JCK
    Mar 28, 2019 at 21:34


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.