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 ${\displaystyle \sigma }$ on the interval ${\displaystyle [0,2\pi ]}$ such that:^[1][2] $${\displaystyle c_{k}={\frac {1}{2\pi }}\int _{0}^{2\pi }e^{-ik\theta }\,d\sigma (\theta ),}$$ 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.^[3]

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 ${\displaystyle [0,2\pi ]}$ as distribution function.^[4][5][6]

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.^[7]

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,^[8][9] 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^[10] $${\displaystyle c_{k}=\int _{\mathbb {T} }z^{-k}dm=\int _{\mathbb {T} }{\bar {z}}^{k}dm,}$$ 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 ${\displaystyle \mu }$.

The above discussion shows that the truncated trigonometric moment problem has infinitely many solutions if the Toeplitz matrix ${\displaystyle T}$ is invertible.^[11][12]

In that case, the solutions to the problem are in bijective correspondence with minimal unitary extensions of the partial isometry ${\displaystyle V}$.

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

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