Truncated trigonometric moment problems and determinate measures (Q1818994)

A positive Borel measure \(\mu\) on the \(n\)-dimensional torus \(T^n\) is called \(S\)-determinate if \(\mu\) is entirely determined by the subset of its Fourier coefficients \(\{\widehat\mu(k), k\in S\}\), where \(S\) is an arbitrary subset of \(\mathbb{Z}^n\) containing \(0\) and with the property that \(-k\in S\) whenever \(k\in S\). Necessary and sufficient conditions for \(S\)-determinacy are given. It is shown that, when \(S\) is a finite set, the validity of this property can be verified by an algorithmic procedure involving trigonometric polynomials with spectrum in \(S\) vanishing on the support of \(\mu\) and satisfying certain positivity properties.