  Continuing our recent work we study polynomial masks of multivariate tight
wavelet frames from two additional and complementary points of view: convexity
and system theory. We consider such polynomial masks that are derived by means
of the unitary extension principle from a single polynomial. We show that the
set of such polynomials is convex and reveal its extremal points as polynomials
that satisfy the quadrature mirror filter condition. Multiplicative structure
of such polynomial sets allows us to improve the known upper bounds on the
number of frame generators derived from box splines. In the univariate and
bivariate settings, the polynomial masks of a tight wavelet frame can be
interpreted as the transfer function of a conservative multivariate linear
system. Recent advances in system theory enable us to develop a more effective
method for tight frame constructions. Employing an example by S. W. Drury, we
show that for dimension greater than 2 such transfer function representations
of the corresponding polynomial masks do not always exist. However, for wavelet
masks derived from multivariate polynomials with non-negative coefficients, we
determine explicit transfer function representations. We illustrate our results
with several examples.
