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:^[1]^[2] $c_{k}={\frac {1}{2\pi }}\int _{0}^{2\pi }e^{-ik\theta }\,d\mu (\theta ),$ with $c_{-k}={\overline {c}}_{k}$ for $k\geq 1$ . In case the sequence is finite, i.e., $\{c_{k}\}_{k=0}^{n<\infty }$ , it is referred to as the truncated trigonometric moment problem.^[3]

An affirmative answer to the problem means that $\{c_{k}\}_{k\in \mathbb {N} _{0}}$ are the Fourier-Stieltjes coefficients for some (consequently positive) Radon measure $\mu$ on $[0,2\pi ]$ .^[4]^[5]

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 $(n + 1) \times (n + 1)$ Hermitian Toeplitz matrix $T=\left({\begin{matrix}c_{0}&c_{1}&\cdots &c_{n}\\c_{-1}&c_{0}&\cdots &c_{n-1}\\\vdots &\vdots &\ddots &\vdots \\c_{-n}&c_{-n+1}&\cdots &c_{0}\\\end{matrix}}\right)$ with $c_{-k}={\overline {c_{k}}}$ for $k\geq 1$ , 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 $T$ defines a sesquilinear product on $\mathbb {C} ^{n+1}$ , resulting in a Hilbert space $({\mathcal {H}},\langle \;,\;\rangle )$ of dimensional at most $n + 1$ . The Toeplitz structure of $T$ means that a "truncated" shift is a partial isometry on ${\mathcal {H}}$ . More specifically, let $\{e_{0},\dotsc ,e_{n}\}$ be the standard basis of $\mathbb {C} ^{n+1}$ . Let ${\mathcal {E}}$ and ${\mathcal {F}}$ be subspaces generated by the equivalence classes $\{[e_{0}],\dotsc ,[e_{n-1}]\}$ respectively $\{[e_{1}],\dotsc ,[e_{n}]\}$ . Define an operator $V:{\mathcal {E}}\rightarrow {\mathcal {F}}$ by $V[e_{k}]=[e_{k+1}]\quad {\mbox{for}}\quad k=0\ldots n-1.$ Since $\langle V[e_{j}],V[e_{k}]\rangle =\langle [e_{j+1}],[e_{k+1}]\rangle =T_{j+1,k+1}=T_{j,k}=\langle [e_{j}],[e_{k}]\rangle ,$ $V$ can be extended to a partial isometry acting on all of ${\mathcal {H}}$ . Take a minimal unitary extension $U$ of $V$ , on a possibly larger space (this always exists). According to the spectral theorem,^[7] there exists a Borel measure $m$ on the unit circle $\mathbb {T}$ such that for all integer $k$ $\langle (U^{*})^{k}[e_{n+1}],[e_{n+1}]\rangle =\int _{\mathbb {T} }z^{k}dm.$ For $k=0,\dotsc ,n$ , the left hand side is $\langle (U^{*})^{k}[e_{n+1}],[e_{n+1}]\rangle =\langle (V^{*})^{k}[e_{n+1}],[e_{n+1}]\rangle =\langle [e_{n+1-k}],[e_{n+1}]\rangle =T_{n+1,n+1-k}=c_{-k}={\overline {c_{k}}}.$ As such, there is a $j$ -atomic measure $m$ on $\mathbb {T}$ , with $j\leq 2n+1<\infty$ (i.e. the set is finite), such that^[8] $c_{k}=\int _{\mathbb {T} }z^{-k}dm=\int _{\mathbb {T} }{\bar {z}}^{k}dm,$ which is equivalent to $c_{k}={\frac {1}{2\pi }}\int _{0}^{2\pi }e^{-ik\theta }d\mu (\theta ).$

for some suitable measure $\mu$ .

Parametrization of solutions

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 $V$ .

Notes

^ Geronimus 1946.
^ Akhiezer 1965, pp. 180–181.
^ Schmüdgen 2017, p. 257.
^ Edwards 1982, pp. 72–73.
^ Zygmund 2002, p. 11.
^ Schmüdgen 2017, p. 260.
^ Simon 2005, pp. 26, 42.
^ 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.
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.

[FOOTNOTEGeronimus1946-1] Geronimus 1946.

[FOOTNOTEAkhiezer1965180–181-2] Akhiezer 1965, pp. 180–181.

[FOOTNOTESchmüdgen2017257-3] Schmüdgen 2017, p. 257.

[FOOTNOTEEdwards198272–73-4] Edwards 1982, pp. 72–73.

[FOOTNOTEZygmund200211-5] Zygmund 2002, p. 11.

[FOOTNOTESchmüdgen2017260-6] Schmüdgen 2017, p. 260.

[FOOTNOTESimon200526,_42-7] Simon 2005, pp. 26, 42.

[FOOTNOTESchmüdgen2017261-8] Schmüdgen 2017, p. 261.

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