trigonometric interpolation
Polynomials are the wrong tool for data that REPEATS — a daily temperature cycle, a sound wave, a planet's orbit. For periodic data you want building blocks that themselves repeat: sines and cosines. Trigonometric interpolation fits a sum of sines and cosines (a trigonometric polynomial) exactly through periodic samples, so the interpolant naturally wraps around with the same period as the data.
Given N equally spaced samples over one period, you seek coefficients so that a sum like a_0 + sum_k (a_k cos(k x) + b_k sin(k x)) hits every sample point. Equivalently, using complex exponentials, you write the interpolant as sum_k c_k e^(i k x) and the coefficients c_k are exactly the discrete Fourier transform (DFT) of the sample values. This is the beautiful link: trigonometric interpolation at equispaced points IS the DFT, so the coefficients are computed not by solving any linear system but by the fast Fourier transform (FFT) in O(N log N) time — astonishingly fast. Evaluating the interpolant back at the sample grid is the inverse FFT. For smooth periodic functions the accuracy is spectacular: the error decays faster than any power of 1/N (spectral accuracy), far beyond what equispaced polynomial interpolation could ever achieve.
Trigonometric interpolation underlies signal processing, spectral PDE solvers, and any analysis of periodic phenomena, precisely because the FFT makes it cheap and its accuracy for smooth periodic data is unmatched. Two honest cautions. First, it requires data sampled UNIFORMLY over a full period and assumes genuine periodicity — feeding it a non-periodic function (or a period mismatch) produces spurious wiggles at the seam, a leakage artifact. Second, sampling too coarsely makes high frequencies masquerade as low ones (aliasing): you must sample above the Nyquist rate to recover the true frequencies.
Sample a periodic signal at 8 equally spaced points over [0, 2pi). Taking the FFT of those 8 values yields 8 Fourier coefficients; reading them back as a sum of sines and cosines gives a trigonometric polynomial that passes through all 8 samples and smoothly repeats with period 2pi — all computed in a handful of FFT operations.
On a uniform grid, the interpolation coefficients ARE the FFT.
Trigonometric interpolation assumes genuinely PERIODIC, uniformly sampled data. Applied to non-periodic data it produces seam wiggles (leakage), and too-coarse sampling causes aliasing — high frequencies disguised as low ones. Sample above the Nyquist rate.