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  "body": "\n## Problem\n\nLet $(c_k)_{k \\in \\Z}$ be complex numbers with $c_{-k} = \\overline{c_k}$ for all $k$.\nDecide whether there is a positive finite [[Borel measure|DYR1QFNGNJ]] $\\mu$ on the [[circle|04EAFVFTF7]] $\\mathbb{T} = \\R / 2\\pi\\Z$ such that\n$$c_k = \\frac{1}{2\\pi} \\int_{\\mathbb{T}} e^{-ik\\theta} \\, d\\mu(\\theta) \\quad \\text{for all } k \\in \\Z.$$\nThe numbers $c_k$ are then the trigonometric moments, or Fourier–Stieltjes coefficients, of $\\mu$.\n\n## Remarks\n\nThe problem is solved.\nSuch 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]].\nThis is the Carathéodory–Toeplitz theorem, also known in the form of Herglotz's theorem on positive-definite sequences.\nNecessity is a short computation: for complex numbers $\\xi_0, \\ldots, \\xi_n$,\n$$\\sum_{j,k=0}^{n} c_{j-k} \\xi_j \\overline{\\xi_k} = \\frac{1}{2\\pi} \\int_{\\mathbb{T}} \\Big\\lvert \\sum_{j=0}^{n} \\xi_j e^{-ij\\theta} \\Big\\rvert^2 d\\mu(\\theta) \\ge 0.$$\nWhen 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]].\nThe truncated problem, with only $c_0, \\ldots, c_n$ given, is also solvable exactly when $T_n$ is positive semidefinite.\n\nExample: $c_0 = 1$ and $c_k = 0$ for $k \\ne 0$ give $T_n = I$, and $\\mu$ is Lebesgue measure $d\\theta$.\nThe sequence $c_0 = 1$, $c_{\\pm 1} = 1$, $c_k = 0$ otherwise is not a moment sequence: the matrix $T_1 = \\begin{pmatrix} 1 \u0026 1 \\\\ 1 \u0026 1 \\end{pmatrix}$ is positive semidefinite, but $T_2$ has determinant $-1$.\n\n## Explanations\n\n### L1 human\n\nA measure on the circle can be summarized by its Fourier coefficients, one complex number for each integer frequency.\nThe trigonometric moment problem asks which sequences of numbers arise in this way from a positive measure.\nThe answer is a positivity test on matrices built from the sequence, the Toeplitz matrices, which have constant diagonals.\n",
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      "first_stated": "1907",
      "posed_by": "Constantin Carathéodory"
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