Trigonometric moment problem

In mathematics, the trigonometric moment problem is formulated as follows: given a sequence $\{c_k{\}}_{k\in {\mathbb{\mathbb{N}}}_0}$, does there exist a distribution function $\sigma$ on the interval $[0,2\pi ]$ such that: $$c_k=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}{e}^{-ik\theta }d\sigma (\theta ),$$ with $c_{-k}={\bar{c}}_k$ for $k\ge 1$. An affirmative answer to the problem means that $\{c_k{\}}_{k\in {\mathbb{\mathbb{N}}}_0}$ are the Fourier-Stieltjes coefficients for some (consequently positive) unique Radon measure $\mu$ on $[0,2\pi ]$ as distribution function.

In case the sequence is finite, i.e., $\{c_k{\}}_{k=0}^{n<\mathrm{\infty }}$, it is referred to as the truncated trigonometric moment problem.

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) × (n + 1) Hermitian Toeplitz matrix $$T=\left(\begin{array}{cccc}{c}_0& c_1& \cdots & c_n\\ c_{-1}& c_0& \cdots & c_{n-1}\\ ⋮& ⋮& \ddots & ⋮\\ c_{-n}& c_{-n+1}& \cdots & c_0\\ \end{array}\right)$$ with $c_{-k}=\overline{c_k}$ for $k\ge 1$, 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 ${\mathbb{ℂ}}^{n+1}$, resulting in a Hilbert space $$({\mathscr{H}},⟨,⟩)$$ of dimensional at most n + 1. The Toeplitz structure of $T$ means that a "truncated" shift is a partial isometry on ${\mathscr{H}}$. More specifically, let $\{e_0,\dots ,e_n\}$ be the standard basis of ${\mathbb{ℂ}}^{n+1}$. Let ${ℰ}$ and ${ℱ}$ be subspaces generated by the equivalence classes $\{[e_0],\dots ,[e_{n-1}]\}$ respectively $\{[e_1],\dots ,[e_n]\}$. Define an operator $$V:{ℰ}\to {ℱ}$$ by $$V[e_k]=[e_{k+1}]\text{for}k=0\dots n-1.$$ Since $$⟨V[e_j],V[e_k]⟩=⟨[e_{j+1}],[e_{k+1}]⟩=T_{j+1,k+1}=T_{j,k}=⟨[e_j],[e_k]⟩,$$ $V$ can be extended to a partial isometry acting on all of ${\mathscr{H}}$. 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 $\mathbb{𝕋}$ such that for all integer k $$⟨(U^{*}{)}^k[e_{n+1}],[e_{n+1}]⟩=\underset{\mathbb{𝕋}}{\int }{z}^{k}dm.$$ For $k=0,\dots ,n$, the left hand side is $$⟨(U^{*}{)}^k[e_{n+1}],[e_{n+1}]⟩=⟨(V^{*}{)}^k[e_{n+1}],[e_{n+1}]⟩=⟨[e_{n+1-k}],[e_{n+1}]⟩=T_{n+1,n+1-k}=c_{-k}=\overline{c_k}.$$ As such, there is a $j$-atomic measure $m$ on $\mathbb{𝕋}$, with $j\le 2n+1<\mathrm{\infty }$ (i.e. the set is finite), such that $$c_k=\underset{\mathbb{𝕋}}{\int }{z}^{-k}dm=\underset{\mathbb{𝕋}}{\int }{\overline{z}}^{k}dm,$$ which is equivalent to $$c_k=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}{e}^{-ik\theta }d\mu (\theta ).$$

for some suitable measure $\mu$.

Parametrization of solutions

The above discussion shows that the truncated 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

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. https://epubs.siam.org/doi/pdf/10.1137/1.9781611976397.fm. 
• 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. https://books.google.com/books?id=9WLLCwAAQBAJ. 
• Curto, Raúl E.; Fialkow, Lawrence A. (1991). "Recursiveness, positivity, and truncated moment problems". Houston Journal of Mathematics 17 (4): 603–635. https://www.math.uh.edu/~hjm/v017n4/0603CURTO.pdf. Retrieved 2025-09-10. 
• Edwards, R. E. (1982). Fourier Series. 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. http://www.jstor.org/stable/19692328. 
• Iokhvidov, Iosif Semenovich (1982). Hankel and Toeplitz Matrices and Forms. Boston: Birkhäuser. ISBN 978-3-7643-3090-3. 
• 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. 277. Cham: Springer International Publishing. doi:10.1007/978-3-319-64546-9. ISBN 978-3-319-64545-2. 
• Simon, Barry (2005). Orthogonal polynomials on the unit circle. Part 1. Classical theory. American Mathematical Society Colloquium Publications. 54. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-3446-6. https://books.google.com/books?id=d94r7kOSnKcC. 
• Zygmund, A. (2002). Trigonometric Series (third ed.). Cambridge: Cambridge University Press. ISBN 0-521-89053-5.