Truncated Trigonometric Moment Problems and Determinate Measures

We consider the problem of deciding whether or not a given positive Borel measure m on the n-dimensional torus Tn is S-determinate, which means that m is entirely determined by the subset of its Fourier coefficients mˆ k., kgS4, where S is an arbitrary subset of Zn containing 0 and with the property that ykgS whenever kgS. We give necessary and sufficient conditions for S-determinacy and show that, when S is a finite set, the validity of this property can be verified algebraically by an algorithmic procedure involving trigonometric polynomials with spectrum in S vanishing on the support of m and satisfying certain positivity properties.