Trigonometric moment problem

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In mathematics, the trigonometric moment problem is formulated as follows: given a finite sequence {α0, ... αn }, does there exist a positive Borel measure μ on the interval [0, 2π] such that

In other words, an affirmative answer to the problems means that {α0, ... αn } are the first n + 1 Fourier coefficients of some positive Borel measure μ on [0, 2π].


The trigonometric moment problem is solvable, that is, {αk} is a sequence of Fourier coefficients, if and only if the (n + 1) × (n + 1) Toeplitz matrix

is positive semidefinite.

The "only if" part of the claims can be verified by a direct calculation.

We sketch an argument for the converse. The positive semidefinite matrix A defines a sesquilinear product on Cn + 1, resulting in a Hilbert space

of dimensional at most n + 1, a typical element of which is an equivalence class denoted by [f]. The Toeplitz structure of A means that a "truncated" shift is a partial isometry on . More specifically, let { e0, ...en } be the standard basis of Cn + 1. Let be the subspace generated by { [e0], ... [en - 1] } and be the subspace generated by { [e1], ... [en] }. Define an operator



V can be extended to a partial isometry acting on all of . Take a minimal unitary extension U of V, on a possibly larger space (this always exists). According to the spectral theorem, there exists a Borel measure m on the unit circle T such that for all integer k

For k = 0,...,n, the left hand side is


Finally, parametrize the unit circle T by eit on [0, 2π] gives

for some suitable measure μ.

Parametrization of solutions[edit]

The above discussion shows that the trigonometric moment problem has infinitely many solutions if the Toeplitz matrix A is invertible. In that case, the solutions to the problem are in bijective correspondence with minimal unitary extensions of the partial isometry V.


  • N.I. Akhiezer, The Classical Moment Problem, Olivier and Boyd, 1965.
  • N.I. Akhiezer, M.G. Krein, Some Questions in the Theory of Moments, Amer. Math. Soc., 1962.