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:{{sfn | Geronimus | 1946}}{{sfn|Akhiezer|1965|pp=180-181}}

$c_k = \frac{1}{2 \pi}\int_0 ^{2 \pi} e^{-ik\theta}\,d \mu(\theta),$

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.{{sfn|Schmüdgen|2017|page=257}}

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 $[0,2\pi]$ .{{sfn|Edwards|1982|pp=72-73}}{{sfn|Zygmund|2002|p=11}}

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 {{math |(n + 1) × (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.{{sfn|Schmüdgen|2017|page=260}}

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 {{math|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

$V: \mathcal{E} \rightarrow \mathcal{F}$

$V[e_k] = [e_{k+1}] \quad \mbox{for} \quad k = 0 \ldots n-1.$

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,{{sfn|Simon|2005|pp=26,42}}{{sfn|Katznelson |2004|pp=38-45}} there exists a Borel measure $m$ on the unit circle $\mathbb{T}$ such that for all integer {{math|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{{sfn|Schmüdgen|2017|page=261}}

$c_k = \int_{\mathbb{T}} z^{-k} dm
= \int_{\mathbb{T}} \bar{z}^k dm,$

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 =

{{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 $T$ is invertible. In that case, the solutions to the problem are in bijective correspondence with minimal unitary extensions of the partial isometry $V$ .

Notes

References

{{cite book | last=Akhiezer | first=N. I. | author-link = Naum Akhiezer|title=The Classical Moment Problem and Some Related Questions in Analysis | publisher=Society for Industrial and Applied Mathematics | publication-place=Philadelphia, PA | date=1965 | isbn=978-1-61197-638-0 | doi=10.1137/1.9781611976397 | doi-access=free | url=https://epubs.siam.org/doi/pdf/10.1137/1.9781611976397.fm }}
{{cite book | last1=Akhiezer | first1=N.I. | last2=Kreĭn | first2=M.G. | title=Some Questions in the Theory of Moments | publisher=American Mathematical Society | series=Translations of mathematical monographs | year=1962 | isbn=978-0-8218-1552-6 | url=https://books.google.com/books?id=9WLLCwAAQBAJ }}
{{cite book | last=Edwards | first=R. E. | title=Fourier Series | publisher=Springer New York | publication-place=New York, NY | volume=85 | date=1982 | isbn=978-1-4613-8158-7 | doi=10.1007/978-1-4613-8156-3}}
{{cite journal | last=Geronimus | first=J. | title=On the Trigonometric Moment Problem | journal=Annals of Mathematics | volume=47 | issue=4 | year=1946 | issn=0003-486X | jstor=1969232 | pages=742–761 | doi=10.2307/1969232 | url=http://www.jstor.org/stable/19692328| url-access=subscription }}
{{cite book | last=Katznelson | first=Yitzhak |authorlink=Yitzhak Katznelson| title=An Introduction to Harmonic Analysis | publisher=Cambridge University Press | date=2004 | isbn=978-0-521-83829-0 | doi=10.1017/cbo9781139165372}}
{{cite book | last=Schmüdgen | first=Konrad | series= Graduate Texts in Mathematics|title=The Moment Problem | publisher=Springer International Publishing | publication-place=Cham | year=2017 | volume=277 | isbn=978-3-319-64545-2 | issn=0072-5285 | doi=10.1007/978-3-319-64546-9}}
{{cite book | last1=Simon | first1=Barry | author1-link=Barry Simon | title=Orthogonal polynomials on the unit circle. Part 1. Classical theory | url=https://books.google.com/books?id=d94r7kOSnKcC | publisher=American Mathematical Society | location=Providence, R.I. | series=American Mathematical Society Colloquium Publications | isbn=978-0-8218-3446-6 | mr=2105088 | year=2005 | volume=54}}
{{cite book | last= Zygmund |first= A. | author-link=Antoni Zygmund | title= Trigonometric Series | title-link = Trigonometric Series | edition=third | publisher = Cambridge University Press | location=Cambridge | year=2002 | isbn=0-521-89053-5}}

Category:Probability problems

Category:Measure theory

Category:Functional analysis

Trigonometric moment problem

Characterization

= Parametrization of solutions =

See also

Notes

References