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In mathematics, the trigonometric moment problem is formulated as follows: given a finite sequence {&alpha;0, ... &alpha;n }, does there exist a positive Borel measure &mu; on the interval [0, 2&pi;] such that


 * $$\alpha_k = \frac{1}{2 \pi}\int_0 ^{2 \pi} e^{-ikt}\,d \mu(t).$$

In other words, an affirmative answer to the problems means that {&alpha;0, ... &alpha;n } are the first n + 1 Fourier coefficients of some positive Borel measure &mu; on [0, 2&pi;].

Characterization
The trigonometric moment problem is solvable, that is, {&alpha;k} is a sequence of Fourier coefficients, if and only if the (n + 1) &times; (n + 1) Toeplitz matrix



A = \left(\begin{matrix} \alpha_0      & \alpha_1           & \cdots   & \alpha_n     \\ \bar{\alpha_1} & \alpha_0           & \cdots   & \alpha_{n-1} \\ \vdots         & \vdots             & \ddots   & \vdots       \\ \bar{\alpha_n} & \bar{\alpha_{n-1}} & \cdots   & \alpha_0     \\ \end{matrix}\right)$$

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


 * $$(\mathcal{H}, \langle \;,\; \rangle)$$

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 $$\mathcal{H}$$. More specificly, let { e0, ...en + 1 } be the standard basis of Cn + 1. Let $$\mathcal{E}$$ be the subspace generated by { [e0], ... [en - 1] } and $$\mathcal{F}$$ be the subspace generated by { [e1], ... [en] }. Define an operator


 * $$V: \mathcal{E} \rightarrow \mathcal{F}$$

by


 * $$V[e_k] = [e_{k+1}] \quad \mbox{for} \quad k = 0 \ldots n.$$

Since


 * $$\langle V[e_j], V[e_k] \rangle = \langle [e_{j+1}], [e_{k+1}] \rangle = A_{j+1, k+1} = A_{j, k} = \langle [e_{j+1}], [e_{k+1}] \rangle,$$

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 T such that for all integer k


 * $$\langle (U^*)^k [ e_ {n+1} ], [ e_ {n+1} ] \rangle = \int_{\mathbf{T}} z^{k} dm .$$

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



\langle (U^*)^k [ e_ {n+1} ], [ e_ {n+1} ] \rangle = \langle (V^*)^k [ e_ {n+1} ], [ e_{n+1} ] \rangle = \langle [e_{n+1-k}], [ e_{n+1} ] \rangle = A_{n+1, n+1-k} = \bar{\alpha_k} $$

So



\int_{\mathbf{T}} z^{-k} dm = \int_{\mathbf{T}} \bar{z} dm = \alpha_k $$

Finally, parametrized the unit circle T by eit on [0, 2&pi;] gives


 * $$\frac{1}{2 \pi} \int_0 ^{2 \pi} e^{-ikt} d\mu(t) = \alpha_k$$

for some suitable measure ''&mu;.

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

Krein's generalized coresolvent formula:

For a minimal unitary extension U and a complex number |w| &le; 1, the generalized coresolvent of [U], the class of extensions equivalent to U, is



\mathcal{C}([U], w) = \langle (I + wU) (I - wU)^{-1} [ e_ {n+1} ], [ e_ {n+1} ] \rangle = \frac{1}{2 \pi} \int_0 ^{2 \pi} \frac{e^{it}+w}{e^{it}-w} d\mu(t). $$