In mathematics, the trigonometric moment problem is formulated as follows: given a sequence { c k } k ∈ N 0 {\displaystyle \{c_{k}\}_{k\in \mathbb {N} _{0}}} , does there exist a distribution function σ {\displaystyle \sigma } on the interval [ 0 , 2 π ] {\displaystyle [0,2\pi ]} such that:
c k = 1 2 π ∫ 0 2 π e − i k θ d σ ( θ ) , {\displaystyle c_{k}={\frac {1}{2\pi }}\int _{0}^{2\pi }e^{-ik\theta }\,d\sigma (\theta ),}
with c − k = c ¯ k {\displaystyle c_{-k}={\overline {c}}_{k}} for k ≥ 1 {\displaystyle k\geq 1} . An affirmative answer to the problem means that { c k } k ∈ N 0 {\displaystyle \{c_{k}\}_{k\in \mathbb {N} _{0}}} are the Fourier-Stieltjes coefficients for some (consequently positive) unique Radon measure μ {\displaystyle \mu } on [ 0 , 2 π ] {\displaystyle [0,2\pi ]} as distribution function. In case the sequence is finite, i.e., { c k } k = 0 n < ∞ {\displaystyle \{c_{k}\}_{k=0}^{n<\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 {\displaystyle \{c_{k}\}_{k=0}^{n}} is a sequence of Fourier coefficients, if and only if the (n + 1) × (n + 1) Hermitian Toeplitz matrix
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