2 research outputs found
An Algebraic Perspective on Multivariate Tight Wavelet Frames. II
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
A real algebra perspective on multivariate tight wavelet frames
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 ; we also provide numerically efficient
methods for checking their existence and actual construction in any dimension.
We exhibit a class of counterexamples in dimension 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 and illustrate our
results by several examples