Trigonometric moment problem


In mathematics, the trigonometric moment problem is formulated as follows: given a finite sequence, does there exist a positive Borel measure μ on the interval such that
In other words, an affirmative answer to the problems means that are the first n + 1 Fourier coefficients of some positive Borel measure μ on .

Characterization

The trigonometric moment problem is solvable, that is, is a sequence of Fourier coefficients, if and only if the × Toeplitz matrix
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 Cn + 1, resulting in a Hilbert space
of dimensional at most n + 1, a typical element of which is an equivalence class denoted by . The Toeplitz structure of A means that a "truncated" shift is a partial isometry on. More specifically, let be the standard basis of Cn + 1. Let be the subspace generated by and be the subspace generated by. Define an operator
by
Since
V can be extended to a partial isometry acting on all of. Take a minimal unitary extension U of V, on a possibly larger space. According to the spectral theorem, there exists a Borel measure m on the unit circle T such that for all integer k
For k = 0,...,n, the left hand side is
So
Finally, parametrize the unit circle T by eit on gives
for some suitable measure μ.

Parametrization of solutions

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