Abstract. We consider finite subsets $\Lambda\subset {\msbm Z}^d$ which posses the extension property, namely that every collection $\{c_k\} _{k\in\Lambda -\Lambda }$ of complex numbers which is positive with respect to $\Lambda $ is the restriction to $\Lambda -\Lambda $ of the Fourier coefficients of some positive measure on ${\msbm T}^d$. Using matrix extension methods, we recover two recent results by J.P. Gabardo and introduce a new class of subsets which posses the extension property. The maximum entropy extensions are explicitly constructed for each class of finite index sets in ${\msbm Z}^2$ which posses the extension property.

AMS Subject Classification (1991): 42B99, 42A70, 47A57, 62M15, 94A17

Keyword(s): multidimensional trigonometric moment problem, positive definite extension, maximum entropy extension

Received November 3, 1997 and in revised form March 2, 1998. (Registered under 3308/2009.)