Trigonometric moment problem

In mathematics, the trigonometric moment problem is formulated as follows: given a sequence $\{c_k{\}}_{k\in {\mathbb{N}}_0}$, does there exist a distribution function $\mu$ on the interval $[0,2\pi ]$ such that:Template:SfnTemplate:Sfn $$c_k=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}{e}^{-ik\theta }d\mu (\theta ),$$ with $c_{-k}={\bar{c}}_k$ for $k\ge 1$. In case the sequence is finite, i.e., $\{c_k{\}}_{k=0}^{n<\mathrm{\infty }}$, it is referred to as the truncated trigonometric moment problem.Template:Sfn

An affirmative answer to the problem means that $\{c_k{\}}_{k\in {\mathbb{N}}_0}$ are the Fourier-Stieltjes coefficients for some (consequently positive) Radon measure $\mu$ on $[0,2\pi ]$.Template:SfnTemplate:Sfn

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 Template:Math Hermitian Toeplitz matrix $$T=\left(\begin{array}{cccc}{c}_0& c_1& \cdots & c_n\\ c_{-1}& c_0& \cdots & c_{n-1}\\ ⋮& ⋮& \ddots & ⋮\\ c_{-n}& c_{-n+1}& \cdots & c_0\end{array}\right)$$ with $c_{-k}=\overline{c_k}$ for $k\ge 1$, is positive semi-definite.Template:Sfn

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 ${ℂ}^{n+1}$, resulting in a Hilbert space $$(\mathscr{H},⟨,⟩)$$ of dimensional at most Template:Math. The Toeplitz structure of $T$ means that a "truncated" shift is a partial isometry on $\mathscr{H}$. More specifically, let $\{e_0,\dots ,e_n\}$ be the standard basis of ${ℂ}^{n+1}$. Let $ℰ$ and $ℱ$ be subspaces generated by the equivalence classes $\{[e_0],\dots ,[e_{n-1}]\}$ respectively $\{[e_1],\dots ,[e_n]\}$. Define an operator $$V:ℰ\to ℱ$$ by $$V[e_k]=[e_{k+1}]\text{for}k=0\dots n-1.$$ Since $$⟨V[e_j],V[e_k]⟩=⟨[e_{j+1}],[e_{k+1}]⟩=T_{j+1,k+1}=T_{j,k}=⟨[e_j],[e_k]⟩,$$ $V$ can be extended to a partial isometry acting on all of $\mathscr{H}$. Take a minimal unitary extension $U$ of $V$, on a possibly larger space (this always exists). According to the spectral theorem,Template:SfnTemplate:Sfn there exists a Borel measure $m$ on the unit circle $𝕋$ such that for all integer Template:Math $$⟨(U^{*}{)}^k[e_{n+1}],[e_{n+1}]⟩=\underset{𝕋}{\int }{z}^{k}dm.$$ For $k=0,\dots ,n$, the left hand side is $$⟨(U^{*}{)}^k[e_{n+1}],[e_{n+1}]⟩=⟨(V^{*}{)}^k[e_{n+1}],[e_{n+1}]⟩=⟨[e_{n+1-k}],[e_{n+1}]⟩=T_{n+1,n+1-k}=c_{-k}=\overline{c_k}.$$ As such, there is a $j$-atomic measure $m$ on $𝕋$, with $j\le 2n+1<\mathrm{\infty }$ (i.e. the set is finite), such thatTemplate:Sfn $$c_k=\underset{𝕋}{\int }{z}^{-k}dm=\underset{𝕋}{\int }{\overline{z}}^{k}dm,$$ which is equivalent to $$c_k=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}{e}^{-ik\theta }d\mu (\theta ).$$

for some suitable measure $\mu$.

Parametrization of solutions

Template:See also 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 $V$.

See also

• Bochner's theorem
• Hamburger moment problem
• Moment problem
• Orthogonal polynomials on the unit circle
• Spectral measure
• Schur class
• Szegő limit theorems
• Wiener's lemma

References

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