Browsing by Author "Stöckler, Joachim"

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    A real algebra perspective on multivariate tight wavelet frames
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2012) Charina, Maria; Putinar, Mihai; Scheiderer, Claus; Stöckler, Joachim
    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 (UEP) from [33] are interpreted in terms of hermitian sums of squares of certain nongenative trigonometric polynomials and in terms of semi-definite programming. The latter together with the results in [31, 35] answer affirmatively the long stading open question of the existence of such tight wavelet frames in dimesion 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.
  • Thumbnail Image
    Item
    Structured Function Systems and Applications
    (Zürich : EMS Publ. House, 2013) Lasserre, Jean-Bernard; Putinar, Mihai; Stöckler, Joachim
    Quite a few independent investigations have been devoted recently to the analysis and construction of structured function systems such as e.g. wavelet frames with compact support, Gabor frames, refinable functions in the context of subdivision and so on. However, difficult open questions about the existence, properties and general efficient construction methods of such structured function systems have been left without satisfactory answers. The goal of the workshop was to bring together experts in approximation theory, real algebraic geometry, complex analysis, frame theory and optimization to address key open questions on the subject in a highly interdisciplinary, unique of its kind, exchange.