  Using standard tools of harmonic analysis, we state and solve the problem of
moments for positive measures supported on the unit ball of a Sobolev space of
multivariate periodic trigonometric functions. We describe outer and inner
semidefinite approximations of the cone of Sobolev moments. They are the basic
components of an infinite-dimensional moment-sums of squares hierarchy,
allowing to solve numerically non-convex polynomial optimization problems on
infinite-dimensional Sobolev spaces, with global convergence guarantees.
