  Recent results from real algebraic geometry and the theory of polynomial
optimization are related in a new framework to the existence question of
multivariate tight wavelet frames whose generators have at least one vanishing
moment. Namely, several equivalent formulations of the so-called Unitary
Extension Principle by Ron and Shen are interpreted in terms of hermitian sums
of squares of certain nonnegative trigonometric polynomials and in terms of
semi-definite programming. The latter together with the recent results in
algebraic geometry and semi-definite programming allow us to answer
affirmatively the long standing open question of the existence of such tight
wavelet frames in dimension $d=2$; we also provide numerically efficient
methods for checking their existence and actual construction in any dimension.
We exhibit a class of counterexamples in dimension $d=3$ showing that, in
general, the UEP property is not sufficient for the existence of tight wavelet
frames. On the other hand we provide stronger sufficient conditions for the
existence of tight wavelet frames in dimension $d > 3$ and illustrate our
results by several examples.
