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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?

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  • $\begingroup$ There are several numerical integration techniques and probably several for computing Fourier coefficients. What references have you looked at? $\endgroup$ – Somos Mar 28 at 2:34
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    $\begingroup$ 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 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 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 at 21:34

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