Trigonometric interpolation

The trigonometric interpolation is a term from the mathematical subfield of numerical analysis. One seeks thereby to predetermined points a trigonometric polynomial ( a sum of sine and cosine given period lengths ) that passes through all these points. It is a special interpolation, which is especially suitable for the interpolation of periodic functions.

The distances between the predetermined points are equal, there is an important special case. In this the solution can be calculated by means of discrete Fourier transformation.

Formulation of Interpolationsproblems

A trigonometric polynomial of degree has the form

This expression has coefficients, so we assume interpolation conditions:

Since the trigonometric polynomial is periodic with the period, one can without loss of generality

Presuppose. In general it is not necessary that these points are equidistant. The interpolation problem now is to find coefficients such that the trigonometric polynomial satisfies the interpolation conditions.

Solution of the problem

Under the above conditions there exists a unique solution of the problem. This solution can be specified in a similar to the interpolation formula of Lagrange form:

It can be shown that this is a trigonometric polynomial by. The formula for multiple angles and other identities for applying

Formulation in the complex plane,

The problem is easier if we describe it in the complex plane. We can use the formula for a trigonometric polynomial rewritten as

Where is the imaginary unit. Let, then is it

Thus, the problem of the trigonometric interpolation is reduced to a polynomial interpolation on the unit circle. The existence and uniqueness of trigonometric interpolation follow immediately from the corresponding results for the polynomial interpolation.

Equidistant points and the discrete Fourier transform

The special case, when the points are distributed equidistantly is particularly important. In which case

The transformation that is applied to the support points, the mapping coefficients and discrete Fourier transform (DFT) of the called procedure.

The case of a pure cosine -based interpolation for equidistant distributed nodes, which leads to a trigonometric polynomial when the interpolation points are odd symmetric, was treated in 1754 by Alexis Clairaut. In this case the solution is equivalent to a discrete cosine transform. The pure sine -based interpolation for equidistant distributed nodes, the discrete sine transform. The full cosine and sine interpolation, which led to the DFT, has been solved by Carl Friedrich Gauss around 1805 in an unpublished work, in which he has also derived an algorithm for the fast Fourier transform to compute the polynomials. Clairaut, Lagrange and Gauss all concerned with the problem of deriving the orbit of planets, asteroids, etc. of a finite set of observation points. Since the tracks are even periodically, a trigonometric polynomial was the natural choice.