Trigonometric moment problem

In mathematics, the trigonometric moment problem is formulated as follows: given a finite sequence $$\{c_0,\dotsc,c_n\}$$, does there exist a distribution function $$\mu$$ on the interval $$[0,2\pi]$$ such that:


 * $$c_k = \frac{1}{2 \pi}\int_0 ^{2 \pi} e^{-ik\theta}\,d \mu(\theta).$$

In other words, an affirmative answer to the problems means that $$\{c_0,\dotsc,c_n\}$$ are the first $n + 1$ Fourier coefficients of some measure $$\mu$$ on $$[0,2\pi]$$.

Characterization
The trigonometric moment problem is solvable, that is, $$\{c_k\}_{k=0}^{n}$$ is a sequence of Fourier coefficients, if and only if the $(n + 1) &times; (n + 1)$ Hermitian Toeplitz matrix



T = \left(\begin{matrix} c_0      & c_1           & \cdots   & c_n     \\ c_{-1} & c_0           & \cdots   & c_{n-1} \\ \vdots         & \vdots             & \ddots   & \vdots       \\ c_{-n} & c_{-n+1} & \cdots   & c_0     \\ \end{matrix}\right)$$ with $$c_{-k}=\overline{c_{k}}$$ for $$k \geq 1$$,

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 $$\mathbb{C}^{n+1}$$, resulting in a Hilbert space


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

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 $$\{e_0,\dotsc,e_n\}$$ be the standard basis of $$\mathbb{C}^{n+1}$$. Let $$\mathcal{E}$$ and $$\mathcal{F}$$ be subspaces generated by the equivalence classes $$\{[e_0],\dotsc,[e_{n-1}]\}$$ respectively $$\{[e_1],\dotsc,[e_{n}]\}$$. Define an operator


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

by


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

Since


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

$$V$$ 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 $$\mathbb{T}$$ such that for all integer $k$


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

For $$k = 0,\dotsc,n$$, the left 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 = T_{n+1, n+1-k} = c_{-k}=\overline{c_k}. $$

So



c_k = \int_{\mathbb{T}} z^{-k} dm = \int_{\mathbb{T}} \bar{z}^k dm $$

which is equivalent to


 * $$ c_k = \frac{1}{2 \pi} \int_0 ^{2 \pi} e^{-ik\theta} d\mu(\theta) $$

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 $$V$$.