---
tag: 765P54K9BR
kind: definition
name: Trigonometric interpolation
msc: [42A15, 65T40]
defines: SVABG134WE
uses: [J3NJQ5FZJ5]
sources:
  - url: https://en.wikipedia.org/wiki/Trigonometric_interpolation
---

## Definition

Let $n \ge 0$ and let $x_0, \ldots, x_{2n}$ be $2n + 1$ real numbers that are pairwise distinct modulo $2\pi$.
Let $y_0, \ldots, y_{2n}$ be complex numbers.
The *trigonometric interpolation polynomial* for these data is the unique [[trigonometric polynomial|J3NJQ5FZJ5]]
$$p(x) = \sum_{k=-n}^{n} c_k e^{ikx}$$
of degree at most $n$ such that $p(x_j) = y_j$ for $j = 0, \ldots, 2n$.
*Trigonometric interpolation* is the construction of $p$ from the data.

## Remarks

Existence and uniqueness: the conditions form a square linear system in $c_{-n}, \ldots, c_n$.
Let $p$ have degree at most $n$ and vanish at all $x_j$.
With $z = e^{ix}$, the function $z^n p$ is an ordinary polynomial of degree at most $2n$ in $z$.
It has the $2n + 1$ distinct roots $e^{ix_j}$, so $p = 0$.
Hence the system has a unique solution.
If all $y_j$ are real, then $c_{-k} = \overline{c_k}$ and $p$ is real-valued.

For equally spaced nodes $x_j = 2\pi j / (2n+1)$, the coefficients are
$$c_k = \frac{1}{2n+1} \sum_{j=0}^{2n} y_j e^{-ik x_j},$$
a [[discrete Fourier transform|TST0GH66A5]] of the data.

Example: $n = 1$, nodes $0, 2\pi/3, 4\pi/3$ and values $1, 0, 0$.
The formula gives $c_0 = c_1 = c_{-1} = 1/3$, so $p(x) = \frac{1}{3} + \frac{2}{3} \cos x$.
Check: $p(0) = 1$ and $p(2\pi/3) = p(4\pi/3) = \frac{1}{3} - \frac{1}{3} = 0$.

Non-example: the nodes $0$ and $2\pi$ are equal modulo $2\pi$, so they are not allowed.
The data $p(0) = 1$, $p(2\pi) = 0$ cannot be met by any $2\pi$-periodic function.

With an even number $2n$ of nodes one uses trigonometric polynomials of degree $n$ with a constraint on the top frequency, such as a cosine term only.