Trigonometric Moment Problem

In mathematics, the trigonometric moment problem is formulated as follows: given a sequence $\_$, does there exist a distribution function $\mu$ on the interval ,2\pi/math> such that:
$c_k = \frac\int_0 ^ e^\,d \mu(\theta),$
with $c_ = \overline_k$ for $k \geq 1$. In case the sequence is finite, i.e., $\_^$, it is referred to as the truncated trigonometric moment problem.

An affirmative answer to the problem means that $\_$ are the Fourier-Stieltjes coefficients for some (consequently positive) Radon measure $\mu$ on ,2\pi/math>.

Characterization

The trigonometric moment problem is solvable, that is, $\_^$ is a sequence of Fourier coefficients, if and only if the Hermitian Toeplitz matrix
$T = \left(\begin c_0 & c_1 & \cdots & c_n \\ c_ & c_0 & \cdots & c_ \\ \vdots & \vdots & \ddots & \vdots \\ c_ & c_ & \cdots & c_0 \\ \end\right)$ with $c_=\overline$ for $k \geq 1$,
is positive semi-definite.

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 $\mathbb^$, resulting in a Hilbert space
$(\mathcal, \langle \;,\; \rangle)$
of dimensional at most . The Toeplitz structure of $T$ means that a "truncated" shift is a partial isometry on $\mathcal$. More specifically, let $\$ be the standard basis of $\mathbb^$. Let $\mathcal$ and $\mathcal$ be subspaces generated by the equivalence classes $\$ respectively $\$. Define an operator
$V: \mathcal \rightarrow \mathcal$
by
V _k= _\quad \mbox \quad k = 0 \ldots n-1.
Since
\langle V _j V _k\rangle = \langle _ _\rangle = T_ = T_ = \langle _ _\rangle,
$V$ can be extended to a partial isometry acting on all of $\mathcal$. Take a minimal unitary extension $U$ of $V$, on a possibly larger space (this always exists). According to the spectral theorem, there exists a Borel measure $m$ on the unit circle $\mathbb$ such that for all integer
\langle (U^*)^k e_ e_ \rangle = \int_ z^ dm .
For $k = 0,\dotsc,n$, the left hand side is

\langle (U^*)^k e_ e_ \rangle
= \langle (V^*)^k e_ e_ \rangle
= \langle _ e_ \rangle
= T_
= c_=\overline.

As such, there is a $j$-atomic measure $m$ on $\mathbb$, with $j \leq 2n + 1 < \infty$ (i.e. the set is finite), such that
$c_k = \int_ z^ dm = \int_ \bar^k dm,$
which is equivalent to
$c_k = \frac \int_0 ^ e^ d\mu(\theta).$

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 $V$.

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

References

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* {{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

Probability problems
Measure theory
Functional analysis