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

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

by

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

Since

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

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

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

So

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

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.