---
tag: YJFP002B62
kind: problem
name: Trigonometric moment problem
msc: [42A70, 44A60, 47B35]
uses: [DYR1QFNGNJ, GKY2WSHE26, 01MMVW8CKC, 04EAFVFTF7]
about: [DYR1QFNGNJ, GKY2WSHE26]
sources:
  - url: https://en.wikipedia.org/wiki/Trigonometric_moment_problem
same_as:
  wikidata: Q7841830
  wikipedia: Trigonometric moment problem
history:
  posed_by: "Constantin Carathéodory"
  first_stated: "1907"
---

## Problem

Let $(c_k)_{k \in \Z}$ be complex numbers with $c_{-k} = \overline{c_k}$ for all $k$.
Decide whether there is a positive finite [[Borel measure|DYR1QFNGNJ]] $\mu$ on the [[circle|04EAFVFTF7]] $\mathbb{T} = \R / 2\pi\Z$ such that
$$c_k = \frac{1}{2\pi} \int_{\mathbb{T}} e^{-ik\theta} \, d\mu(\theta) \quad \text{for all } k \in \Z.$$
The numbers $c_k$ are then the trigonometric moments, or Fourier–Stieltjes coefficients, of $\mu$.

## Remarks

The problem is solved.
Such a measure exists if and only if, for every $n \ge 0$, the Hermitian [[Toeplitz matrix|GKY2WSHE26]] $T_n = (c_{j-k})_{j,k=0}^{n}$ is [[positive semidefinite|01MMVW8CKC]].
This is the Carathéodory–Toeplitz theorem, also known in the form of Herglotz's theorem on positive-definite sequences.
Necessity is a short computation: for complex numbers $\xi_0, \ldots, \xi_n$,
$$\sum_{j,k=0}^{n} c_{j-k} \xi_j \overline{\xi_k} = \frac{1}{2\pi} \int_{\mathbb{T}} \Big\lvert \sum_{j=0}^{n} \xi_j e^{-ij\theta} \Big\rvert^2 d\mu(\theta) \ge 0.$$
When the measure exists it is unique, because trigonometric polynomials are dense in the continuous functions on $\mathbb{T}$; this contrasts with the [[Hamburger moment problem|TBNCFXTMMW]].
The truncated problem, with only $c_0, \ldots, c_n$ given, is also solvable exactly when $T_n$ is positive semidefinite.

Example: $c_0 = 1$ and $c_k = 0$ for $k \ne 0$ give $T_n = I$, and $\mu$ is Lebesgue measure $d\theta$.
The sequence $c_0 = 1$, $c_{\pm 1} = 1$, $c_k = 0$ otherwise is not a moment sequence: the matrix $T_1 = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ is positive semidefinite, but $T_2$ has determinant $-1$.

## Explanations

### L1 human

A measure on the circle can be summarized by its Fourier coefficients, one complex number for each integer frequency.
The trigonometric moment problem asks which sequences of numbers arise in this way from a positive measure.
The answer is a positivity test on matrices built from the sequence, the Toeplitz matrices, which have constant diagonals.