Trigonometric moment problem

In mathematics, the trigonometric moment problem is formulated as follows: given a finite sequence ${\displaystyle \{c_{0},\dotsc ,c_{n}\}}$, does there exist a distribution function ${\displaystyle \mu }$ on the interval ${\displaystyle [0,2\pi ]}$ such that:^[1]

${\displaystyle c_{k}={\frac {1}{2\pi }}\int _{0}^{2\pi }e^{-ik\theta }\,d\mu (\theta ).}$

In other words, an affirmative answer to the problems means that ${\displaystyle \{c_{0},\dotsc ,c_{n}\}}$ are the first n + 1 Fourier coefficients of some 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.^[2]

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

${\displaystyle \langle (U^{*})^{k}[e_{n+1}],[e_{n+1}]\rangle =\langle (V^{*})^{k}[e_{n+1}],[e_{n+1}]\rangle =\langle [e_{n+1-k}],[e_{n+1}]\rangle =T_{n+1,n+1-k}=c_{-k}={\overline {c_{k}}}.}$

So

${\displaystyle c_{k}=\int _{\mathbb {T} }z^{-k}dm=\int _{\mathbb {T} }{\bar {z}}^{k}dm}$

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

• Moment problem
• Orthogonal polynomials on the unit circle
• Schur class
• Szegő limit theorems

Notes

1. Geronimus 1946.
2. Schmüdgen 2017, p. 260.

References

• Akhiezer, Naum I. (1965). The classical moment problem and some related questions in analysis. New York: Hafner Publishing Co. (translated from the Russian by N. Kemmer)
• Akhiezer, N.I.; Kreĭn, M.G. (1962). Some Questions in the Theory of Moments. Translations of mathematical monographs. American Mathematical Society. ISBN 978-0-8218-1552-6.
• Geronimus, J. (1946). "On the Trigonometric Moment Problem". Annals of Mathematics. 47 (4): 742–761. doi:10.2307/1969232. ISSN 0003-486X. JSTOR 1969232.
• Schmüdgen, Konrad (2017). The Moment Problem. Graduate Texts in Mathematics. Vol. 277. Cham: Springer International Publishing. doi:10.1007/978-3-319-64546-9. ISBN 978-3-319-64545-2. ISSN 0072-5285.