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In mathematics, the trigonometric moment problem is formulated as follows: given a finite sequence {α₀, ... α_n }, does there exist a positive Borel measure μ on the interval [0, 2π] such that

$\alpha _{k}={\frac {1}{2\pi }}\int _{0}^{2\pi }e^{-ikt}\,d\mu (t).$

In other words, an affirmative answer to the problems means that {α₀, ... α_n } are the first n + 1 Fourier coefficients of some positive Borel measure μ on [0, 2π].

Characterization

The trigonometric moment problem is solvable, that is, {α_k} is a sequence of Fourier coefficients, if and only if the (n + 1) × (n + 1) Toeplitz matrix

$A=\left({\begin{matrix}\alpha _{0}&\alpha _{1}&\cdots &\alpha _{n}\\{\bar {\alpha _{1}}}&\alpha _{0}&\cdots &\alpha _{n-1}\\\vdots &\vdots &\ddots &\vdots \\{\bar {\alpha _{n}}}&{\bar {\alpha _{n-1}}}&\cdots &\alpha _{0}\\\end{matrix}}\right)$

is positive semidefinite.

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 A defines a sesquilinear product on C^{n + 1}, resulting in a Hilbert space

$({\mathcal {H}},\langle \;,\;\rangle )$

of dimensional at most n + 1, a typical element of which is an equivalence class denoted by [f]. The Toeplitz structure of A means that a "truncated" shift is a partial isometry on ${\mathcal {H}}$. More specificly, let { e₀, ...e_{n + 1} } be the standard basis of C^{n + 1}. Let ${\mathcal {E}}$ be the subspace generated by { [e₀], ... [e_{n - 1}] } and ${\mathcal {F}}$ be the subspace generated by { [e₁], ... [e_n] }. Define an operator

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

by

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

Since

$\langle V[e_{j}],V[e_{k}]\rangle =\langle [e_{j+1}],[e_{k+1}]\rangle =A_{j+1,k+1}=A_{j,k}=\langle [e_{j+1}],[e_{k+1}]\rangle ,$

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, there exists a Borel measure m on the unit circle T such that for all integer k

$\langle (U^{*})^{k}[e_{n+1}],[e_{n+1}]\rangle =\int _{\mathbf {T} }z^{k}dm.$

For k = 0,...,n, the right hand side is

$\langle (U^{*})^{k}[e_{n+1}],[e_{n+1}]\rangle =\langle (V^{*})^{k}[e_{n+1}],[e_{n+1}]\rangle =\langle [e_{n+1-k}],[e_{n+1}]\rangle =A_{n+1,n+1-k}={\bar {\alpha _{k}}}$

So

$\int _{\mathbf {T} }z^{-k}dm=\int _{\mathbf {T} }{\bar {z}}dm=\alpha _{k}$

Finally, parametrized the unit circle T by e^it on [0, 2π] gives

${\frac {1}{2\pi }}\int _{0}^{2\pi }e^{-ikt}d\mu (t)=\alpha _{k}$

for some suitable measure μ.

Parametrization of solutions

The above discussion shows that the solutions of the trigonometric moment problem has infinitely many solutions if the Toeplitz matrix A is invertible. In that case, the solutions to the problem is in bijective correspondence with minimal unitary extensions of the partial isometry V.

Krein's generalized coresolvent formula:

For a minimal unitary extension U and a complex number |w| ≤ 1, the generalized coresolvent of [U], the class of extensions equivalent to U, is

${\mathcal {C}}([U],w)=\langle (I+wU)(I-wU)^{-1}[e_{n+1}],[e_{n+1}]\rangle ={\frac {1}{2\pi }}\int _{0}^{2\pi }{\frac {e^{it}+w}{e^{it}-w}}d\mu (t).$

References

• N.I. Akhiezer, The Classical Moment Problem, Olivier and Boyd, 1965.
• N.I. Akhiezer, M.G. Krein, Some Questions in the Theory of Moments, Amer. Math. Soc., 1962.