Abstract :
We solve the truncated complex moment problem for measures supported on the variety K(identical to){ z (element of) C: z z = A+Bz+C z +Dz^ 2 ,D (not equal to) 0. Given a doubly indexed finite sequence of complex numbers(gamma)(identical to)gamma^(2n)(gamma)_00,(gamma)_01,(gamma)_10,…,(gamma)_0,2n,(ga mma)_1,2n-1,…,(gamma)_2n-1,1,(gamma)_2n,0, there exists a positive Borel measure (mu) supported in K such that (gamma)_ij=(integral)z^iz^j,d(mu),(0<=i+j<=2n) if and only if the moment matrix M(n)( (gamma) ) is positive, recursively generated, with a column dependence relation Z Z= A1+BZ +C Z +DZ ^2, and card V(gamma)>= rank M(n), where V(gamma) is the variety associated to (gamma) . The last condition may be replaced by the condition that there exists a complex number (gamma)_n,n+1 satisfying (gamma)_n+1,n(identical to)(gamma)_n,n+1=A(gamma)_n,n1+B(gamma)_n,n+C(gamma)_n+1,n1+D(gamma)_n,n+1. We combine these results with a recent theorem of J. Stochel to solve the full complex moment problem for K , and we illustrate the connection between the truncated and full moment problems for other varieties as well, including the variety z^ k = p(z, Z ), deg p < k.
