In mathematics, the trigonometric moment problem is formulated as follows: given a finite sequence, does there exist a distribution function \mu on the interval [0,2\pi] such that: In other words, an affirmative answer to the problems means that are the first n + 1 Fourier coefficients of some measure \mu on [0,2\pi].

Characterization

The trigonometric moment problem is solvable, that is, is a sequence of Fourier coefficients, if and only if the (n + 1) × (n + 1) Hermitian Toeplitz matrix is positive semi-definite. 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 T defines a sesquilinear product on, resulting in a Hilbert space of dimensional at most n + 1 . The Toeplitz structure of T means that a "truncated" shift is a partial isometry on \mathcal{H}. More specifically, let be the standard basis of. Let \mathcal{E} and \mathcal{F} be subspaces generated by the equivalence classes respectively. Define an operator by Since V can be extended to a partial isometry acting on all of \mathcal{H}. 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 \mathbb{T} such that for all integer k For, the left hand side is So which is equivalent to for some suitable measure \mu.

Parametrization of solutions

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