Trigonometric moment problem

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

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

Characterization

The trigonometric moment problem is solvable, that is, ${\displaystyle \{c_{k}\}_{k=0}^{n}}$ is a sequence of Fourier coefficients, if and only if the (n 1) × (n 1) Hermitian Toeplitz matrix $${\displaystyle T=\left({\begin{matrix}c_{0}&c_{1}&\cdots &c_{n}\\c_{-1}&c_{0}&\cdots &c_{n-1}\\\vdots &\vdots &\ddots &\vdots \\c_{-n}&c_{-n 1}&\cdots &c_{0}\\\end{matrix}}\right)}$$ with ${\displaystyle c_{-k}={\overline {c_{k}}}}$ for ${\displaystyle k\geq 1}$, 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 ${\displaystyle T}$ defines a sesquilinear product on ${\displaystyle \mathbb {C} ^{n 1}}$, resulting in a Hilbert space $${\displaystyle ({\mathcal {H}},\langle \;,\;\rangle )}$$ of dimensional at most n 1. The Toeplitz structure of ${\displaystyle T}$ means that a "truncated" shift is a partial isometry on ${\displaystyle {\mathcal {H}}}$. More specifically, let ${\displaystyle \{e_{0},\dotsc ,e_{n}\}}$ be the standard basis of ${\displaystyle \mathbb {C} ^{n 1}}$. Let ${\displaystyle {\mathcal {E}}}$ and ${\displaystyle {\mathcal {F}}}$ be subspaces generated by the equivalence classes ${\displaystyle \{[e_{0}],\dotsc ,[e_{n-1}]\}}$ respectively ${\displaystyle \{[e_{1}],\dotsc ,[e_{n}]\}}$. Define an operator $${\displaystyle V:{\mathcal {E}}\rightarrow {\mathcal {F}}}$$ by $${\displaystyle V[e_{k}]=[e_{k 1}]\quad {\mbox{for}}\quad k=0\ldots n-1.}$$ Since $${\displaystyle \langle V[e_{j}],V[e_{k}]\rangle =\langle [e_{j 1}],[e_{k 1}]\rangle =T_{j 1,k 1}=T_{j,k}=\langle [e_{j}],[e_{k}]\rangle ,}$$ ${\displaystyle V}$ can be extended to a partial isometry acting on all of ${\displaystyle {\mathcal {H}}}$. Take a minimal unitary extension ${\displaystyle U}$ of ${\displaystyle V}$, on a possibly larger space (this always exists). According to the spectral theorem, there exists a Borel measure ${\displaystyle m}$ on the unit circle ${\displaystyle \mathbb {T} }$ such that for all integer k $${\displaystyle \langle (U^{*})^{k}[e_{n 1}],[e_{n 1}]\rangle =\int _{\mathbb {T} }z^{k}dm.}$$ For ${\displaystyle k=0,\dotsc ,n}$, the left hand side is $${\displaystyle \langle (U^{*})^{k}[e_{n 1}],[e_{n 1}]\rangle =\langle (V^{*})^{k}[e_{n 1}],[e_{n 1}]\rangle =\langle [e_{n 1-k}],[e_{n 1}]\rangle =T_{n 1,n 1-k}=c_{-k}={\overline {c_{k}}}.}$$ As such, there is a ${\displaystyle j}$-atomic measure ${\displaystyle m}$ on ${\displaystyle \mathbb {T} }$, with ${\displaystyle j\leq 2n 1<\infty }$ (i.e. the set is finite), such that $${\displaystyle c_{k}=\int _{\mathbb {T} }z^{-k}dm=\int _{\mathbb {T} }{\bar {z}}^{k}dm,}$$ which is equivalent to $${\displaystyle c_{k}={\frac {1}{2\pi }}\int _{0}^{2\pi }e^{-ik\theta }d\mu (\theta ).}$$

for some suitable measure ${\displaystyle \mu }$.

Parametrization of solutions

The above discussion shows that the trigonometric moment problem has infinitely many solutions if the Toeplitz matrix ${\displaystyle T}$ is invertible. In that case, the solutions to the problem are in bijective correspondence with minimal unitary extensions of the partial isometry ${\displaystyle V}$.

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

Notes

References

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