Browsing by Author "Charina, Maria"

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
    Stationary multivariate subdivision: Joint spectral radius and asymptotic similarity
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Charina, Maria; Conti, Costanza; Guglielmi, Nicola; Protasov, Vladimir
    In this paper we study scalar multivariate non-stationary subdivision schemes with a general integer dilation matrix. We present a new numerically efficient method for checking convergence and H ̈older regularity of such schemes. This method relies on the concepts of approximate sum rules, asymptotic similarity and the so-called joint spectral radius of a finite set of square matrices. The combination of these concepts allows us to employ recent advances in linear algebra for exact computation of the joint spectral radius that have had already a great impact on studies of stationary subdivision schemes. We also expose the limitations of non-stationary schemes in their capability to reproduce and generate certain function spaces. We illustrate our results with several examples.