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:[1][2] $${\displaystyle c_{k}={\frac {1}{2\pi }}\int _{0}^{2\pi }e^{-ik\theta }\,d\mu (\theta ),}$$ 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.[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 ${\displaystyle [0,2\pi ]}$.[4][5]

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.[6]

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,[7][8] 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[9] $${\displaystyle c_{k}=\int _{\mathbb {T} }z^{-k}dm=\int _{\mathbb {T} }{\bar {z}}^{k}dm,}$$ 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

See also: Spectrum (functional analysis) § Classification of points in the spectrum

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

1. Geronimus 1946.
2. Akhiezer 1965, pp. 180–181.
3. Schmüdgen 2017, p. 257.
4. Edwards 1982, pp. 72–73.
5. Zygmund 2002, p. 11.
6. Schmüdgen 2017, p. 260.
7. Simon 2005, pp. 26, 42.
8. Katznelson 2004, pp. 38–45.
9. Schmüdgen 2017, p. 261.

References

• Akhiezer, N. I. (1965). The Classical Moment Problem and Some Related Questions in Analysis. Philadelphia, PA: Society for Industrial and Applied Mathematics. doi:10.1137/1.9781611976397. ISBN 978-1-61197-638-0.
• 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.
• Edwards, R. E. (1982). Fourier Series. Vol. 85. New York, NY: Springer New York. doi:10.1007/978-1-4613-8156-3. ISBN 978-1-4613-8158-7.
• Geronimus, J. (1946). "On the Trigonometric Moment Problem". Annals of Mathematics. 47 (4): 742–761. doi:10.2307/1969232. ISSN 0003-486X. JSTOR 1969232.
• Katznelson, Yitzhak (2004). An Introduction to Harmonic Analysis. Cambridge University Press. doi:10.1017/cbo9781139165372. ISBN 978-0-521-83829-0.
• 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.
• Simon, Barry (2005). Orthogonal polynomials on the unit circle. Part 1. Classical theory. American Mathematical Society Colloquium Publications. Vol. 54. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-3446-6. MR 2105088.
• Zygmund, A. (2002). Trigonometric Series (third ed.). Cambridge: Cambridge University Press. ISBN 0-521-89053-5.