Refinable function

Abstract: Using a base b and an even number of knots, we define a symmetric iterative interpolation process. The main properties of this process come from an associated function F. The basic functional equation for F is that F(t/b) = \([\sum olimits_n {F(n/b)F(t - n)} ]\). We prove that F is a continuous positive definite function. We find almost precisely in which Lipschitz classes derivatives of F belong. If a function y is defined only on integers, this process extends y continuously to the real axis as \([y(t) = \sum olimits_n {y(n)F(t - n)} ]\). Error bounds for this iterative interpolation are given.

Abstract: A two-scale difference equation is a functional equation of the form $f(x) = \sum _{n = 0}^N c_n f(\alpha x - \beta _n )$, where $\alpha > 1$ and $\beta _0 < \beta _1 <\cdots <\beta _n $, are real constants, and $c_n $ are complex constants. Solutions of such equations arise in spline theory, in interpolation schemes for constructing curves, in constructing wavelets of compact support, in constructing fractals, and in probability theory. This paper studies the existence and uniqueness of $L^1 $-solutions to such equations. In particular, it characterizes $L^1 $-solutions having compact support. A time-domain method is introduced for studying the special case of such equations where $\{ {\alpha ,\beta _0 , \cdots ,\beta _n } \}$ are integers, which are called lattice two-scale difference equations. It is shown that if a lattice two-scale difference equation has a compactly supported solution in $C^m (\mathbb{R})$, then $m < {{(\beta _n - \beta _0 )} / {(\alpha - 1)}} - 1$.

Abstract: It is well known that in applied and computational mathematics, cardinal B-splines play an important role in geometric modeling (in computer-aided geometric design), statistical data representation (or modeling), solution of differential equations (in numerical analysis), and so forth More recently, in the development of wavelet analysis, cardinal B-splines also serve as a canonical example of scaling functions that generate multiresolution analyses of L 2 (−∞,∞) However, although cardinal B-splines have compact support, their corresponding orthonormal wavelets (of Battle and Lemarie) have infinite duration To preserve such properties as self-duality while requiring compact support, the notion of tight frames is probably the only replacement of that of orthonormal wavelets In this paper, we study compactly supported tight frames Ψ={ψ 1 ,…,ψ N } for L 2 (−∞,∞) that correspond to some refinable functions with compact support, give a precise existence criterion of Ψ in terms of an inequality condition on the Laurent polynomial symbols of the refinable functions, show that this condition is not always satisfied (implying the nonexistence of tight frames via the matrix extension approach), and give a constructive proof that when Ψ does exist, two functions with compact support are sufficient to constitute Ψ, while three guarantee symmetry/anti-symmetry, when the given refinable function is symmetric

Abstract: Refinement equations play an important role in computer graphics and wavelet analysis. In this paper we investigate multivariate refinement equations associated with a dilation matrix and a finitely supported refinement mask. We characterize the Lp-convergence of a subdivision scheme in terms of the p-norm joint spectral radius of a collection of matrices associated with the refinement mask. In particular, the 2-norm joint spectral radius can be easily computed by calculating the eigenvalues of a certain linear operator on a finite dimensional linear space. Examples are provided to illustrate the general theory.