Totally positive refinable functions with general dilation M

We construct a new class of approximating functions that are M-refinable and provide shape preserving approximations. The refinable functions in the class are smooth, compactly supported, centrally symmetric and totally positive. Moreover, their refinable masks are associated with convergent subdivision schemes. The presence of one or more shape parameters gives a great flexibility in the applications. Some examples for dilation M=4and M=5are also given

Bell-shaped nonstationary refinable ripplets

We study the approximation properties of the class of nonstationary refinable ripplets introduced in \cite{GP08}. These functions are solution of an infinite set of nonstationary refinable equations and are defined through sequences of scaling masks that have an explicit expression. Moreover, they are variation-diminishing and highly localized in the scale-time plane, properties that make them particularly attractive in applications. Here, we prove that they enjoy Strang-Fix conditions and convolution and differentiation rules and that they are bell-shaped. Then, we construct the corresponding minimally supported nonstationary prewavelets and give an iterative algorithm to evaluate the prewavelet masks. Finally, we give a procedure to construct the associated nonstationary biorthogonal bases and filters to be used in efficient decomposition and reconstruction algorithms. As an example, we calculate the prewavelet masks and the nonstationary biorthogonal filter pairs corresponding to the $C^2$ nonstationary scaling functions in the class and construct the corresponding prewavelets and biorthogonal bases. A simple test showing their good performances in the analysis of a spike-like signal is also presented. Keywords: total positivity, variation-dimishing, refinable ripplet, bell-shaped function, nonstationary prewavelet, nonstationary biorthogonal basisComment: 30 pages, 10 figure

Vector Subdivision Schemes for Arbitrary Matrix Masks

Employing a matrix mask, a vector subdivision scheme is a fast iterative averaging algorithm to compute refinable vector functions for wavelet methods in numerical PDEs and to produce smooth curves in CAGD. In sharp contrast to the well-studied scalar subdivision schemes, vector subdivision schemes are much less well understood, e.g., Lagrange and (generalized) Hermite subdivision schemes are the only studied vector subdivision schemes in the literature. Because many wavelets used in numerical PDEs are derived from refinable vector functions whose matrix masks are not from Hermite subdivision schemes, it is necessary to introduce and study vector subdivision schemes for any general matrix masks in order to compute wavelets and refinable vector functions efficiently. For a general matrix mask, we show that there is only one meaningful way of defining a vector subdivision scheme. Motivated by vector cascade algorithms and recent study on Hermite subdivision schemes, we shall define a vector subdivision scheme for any arbitrary matrix mask and then we prove that the convergence of the newly defined vector subdivision scheme is equivalent to the convergence of its associated vector cascade algorithm. We also study convergence rates of vector subdivision schemes. The results of this paper not only bridge the gaps and establish intrinsic links between vector subdivision schemes and vector cascade algorithms but also strengthen and generalize current known results on Lagrange and (generalized) Hermite subdivision schemes. Several examples are provided to illustrate the results in this paper on various types of vector subdivision schemes with convergence rates

Smoothness of multivariate refinable functions with infinitely supported masks

AbstractIn this paper, we investigate the smoothness of multivariate refinable functions with infinitely supported masks and an isotropic dilation matrix. Using some methods as in [R.Q. Jia, Characterization of smoothness of multivariate refinable functions in Sobolev spaces, Trans. Amer. Math. Soc. 351 (1999) 4089–4112], we characterize the optimal smoothness of multivariate refinable functions with polynomially decaying masks and an isotropic dilation matrix. Our characterizations extend some of the main results of the above mentioned paper with finitely supported masks to the case in which masks are infinitely supported