In mathematics, the trigonometric moment problem is formulated as follows: given a sequence
, does there exist a distribution function
d μ ( θ ) ,
In case the sequence is finite, i.e.,
, it is referred to as the truncated trigonometric moment problem.
[3] An affirmative answer to the problem means that
are the Fourier-Stieltjes coefficients for some (consequently positive) Radon measure
[4][5] The trigonometric moment problem is solvable, that is,
is a sequence of Fourier coefficients, if and only if the (n + 1) × (n + 1) Hermitian Toeplitz matrix
, is positive semi-definite.
[6] 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
defines a sesquilinear product on
, resulting in a Hilbert space
means that a "truncated" shift is a partial isometry on
be the standard basis of
be subspaces generated by the equivalence classes
Define an operator
can be extended to a partial isometry acting on all of
Take a minimal unitary extension
, on a possibly larger space (this always exists).
According to the spectral theorem,[7][8] there exists a Borel measure
on the unit circle
, the left hand side is
(i.e. the set is finite), such that[9]
d μ ( θ ) .
for some suitable measure
The above discussion shows that the trigonometric moment problem has infinitely many solutions if the Toeplitz matrix
In that case, the solutions to the problem are in bijective correspondence with minimal unitary extensions of the partial isometry