Trigonometric moment problem

In mathematics, the trigonometric moment problem is formulated as follows: given a finite sequence {α₀, ... α_n }, does there exist a positive Borel measure μ on the interval [0, 2π] 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 {α₀, ... α_n } are the first n + 1 Fourier coefficients of some positive Borel measure μ on [0, 2π].

Characterization

The trigonometric moment problem is solvable, that is, {α_k} is a sequence of Fourier coefficients, if and only if the (n + 1) × (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 C^{n + 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 specifically, let { e₀, ...e_n } be the standard basis of C^{n + 1}. Let $\mathcal{E}$ be the subspace generated by { [e₀], ... [e_{n - 1}] } and $\mathcal{F}$ be the subspace generated by { [e₁], ... [e_n] }. Define an operator

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

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 = 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 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 = A_{n+1, n+1-k} = \bar{\alpha_k}.

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

Finally, parametrize the unit circle T by e^it on [0, 2π] gives

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

for some suitable measure μ.

Parametrization of solutions

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.

References

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.

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