In mathematics, the trigonometric moment problem is formulated as follows: given a sequence , does there exist a distribution function on the interval such that:
with for . An affirmative answer to the problem means that are the Fourier-Stieltjes coefficients for some (consequently positive) unique Radon measure on as distribution function.
In case the sequence is finite, i.e., , it is referred to as the truncated trigonometric moment problem.
The trigonometric moment problem is solvable, that is, is a sequence of Fourier coefficients, if and only if the Hermitian Toeplitz matrix with for , 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 defines a sesquilinear product on , resulting in a Hilbert space
of dimensional at most . The Toeplitz structure of means that a "truncated" shift is a partial isometry on . More specifically, let be the standard basis of . Let and be subspaces generated by the equivalence classes respectively . Define an operator
by
Since
can be extended to a partial isometry acting on all of . Take a minimal unitary extension of , on a possibly larger space (this always exists). According to the spectral theorem, there exists a Borel measure on the unit circle such that for all integer
For , the left hand side is
As such, there is a -atomic measure on , with (i.e. the set is finite), such that
which is equivalent to
for some suitable measure .
The above discussion shows that the truncated trigonometric moment problem has infinitely many solutions if the Toeplitz matrix is invertible. In that case, the solutions to the problem are in bijective correspondence with minimal unitary extensions of the partial isometry .