trigonometric interpolation

(This fact was first noted by Gauss when solving the problem of trigonometric interpolation).

This reduces the problem of trigonometric interpolation to that of polynomial interpolation on the unit circle.

In mathematics, trigonometric interpolation is interpolation with trigonometric polynomials.

Existence and uniqueness for trigonometric interpolation now follows immediately from the corresponding results for polynomial interpolation.

Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions.

For trigonometric interpolation, this function has to be a trigonometric polynomial, that is, a sum of sines and cosines of given periods.

This can be seen as a form of polynomial interpolation with harmonic base functions, see trigonometric interpolation and trigonometric polynomial.

For instance, rational interpolation is interpolation by rational functions, and trigonometric interpolation is interpolation by trigonometric polynomials.

The case of the cosine-only interpolation for equally spaced points, corresponding to a trigonometric interpolation when the points have even symmetry, was treated by Alexis Clairaut in 1754.

In contrast, the most obvious trigonometric interpolation polynomial is the one in which the frequencies range from 0 to (instead of roughly to as above), similar to the inverse DFT formula.

While this method is traditionally attributed to a 1965 paper by J. W. Cooley and J. W. Tukey, Gauss developed it as a trigonometric interpolation method.

Clairaut, Lagrange, and Gauss were all concerned with studying the problem of inferring the orbit of planets, asteroids, etc., from a finite set of observation points; since the orbits are periodic, a trigonometric interpolation was a natural choice.

Equivalently, the are the amplitudes of the unique trigonometric interpolation polynomial with minimal mean-square slope passing through the N+1 points where f(cos θ) is evaluated, and we approximate the integral by the integral of this interpolation polynomial.

Technically, Clairaut's work was a cosine-only series (a form of discrete cosine transform), while Lagrange's work was a sine-only series (a form of discrete sine transform); a true cosine+sine DFT was used by Gauss in 1805 for trigonometric interpolation of asteroid orbits.