Costanza Conti

Full length article: Polynomial reproduction for univariate subdivision schemes of any arity

In this paper, we study the ability of convergent subdivision schemes to reproduce polynomials in the sense that for initial data, which is sampled from some polynomial function, the scheme yields the same polynomial in the limit. This property is desirable because the reproduction of polynomials up to some degree d implies that a scheme has approximation order d+1. We first show that any convergent, linear, uniform, and stationary subdivision scheme reproduces linear functions with respect to an appropriately chosen parameterization. We then present a simple algebraic condition for polynomial reproduction of higher order. All results are given for subdivision schemes of any arity m>=2 and we use them to derive a unified definition of general m-ary pseudo-splines. Our framework also covers non-symmetric schemes and we give an example where the smoothness of the limit functions can be increased by giving up symmetry.

Reproduction of exponential polynomials by multivariate non-stationary subdivision schemes with a general dilation matrix

We study scalar multivariate non-stationary subdivision schemes with a general integer dilation matrix. We characterize the capability of such schemes to reproduce exponential polynomials in terms of simple algebraic conditions on their symbols. These algebraic conditions provide a useful theoretical tool for checking the reproduction properties of existing schemes and for constructing new schemes with desired reproduction capabilities and other enhanced properties. We illustrate our results with several examples.

Polynomial reproduction of multivariate scalar subdivision schemes

A stationary subdivision scheme generates the full space of polynomials of degree up to k if and only if its mask satisfies sum rules of order k+1, or its symbol satisfies zero conditions of order k+1. This property is often called the polynomial reproduction property of the subdivision scheme. It is a well-known fact that this property is, in general, only necessary for the associated refinable function to have approximation order k+1. In this paper we study a different polynomial reproduction property of a multivariate scalar subdivision scheme with dilation matrix mI,|m|>=2. Namely, we are interested in capability of a subdivision scheme to reproduce in the limit exactly the same polynomials from which the data is sampled. The motivation for this paper are the results in Levin (2003) [9] that state that such a reproduction property of degree k of the subdivision scheme is sufficient for having approximation order k+1. Our main result yields simple algebraic conditions on the subdivision symbol for computing the exact degree of such polynomial reproduction and also for determining the associated parametrization. The parametrization determines the grid points to which the newly computed values are attached at each subdivision iteration to ensure the higher degree of polynomial reproduction. We illustrate our results with several examples.

From approximating to interpolatory non-stationary subdivision schemes with the same generation properties

In this paper we describe a general, computationally feasible strategy to deduce a family of interpolatory non-stationary subdivision schemes from a symmetric non-stationary, non-interpolatory one satisfying quite mild assumptions. To achieve this result we extend our previous work (Conti et al., Linear Algebra Appl 431(10):1971–1987, 2009) to full generality by removing additional assumptions on the input symbols. For the so obtained interpolatory schemes we prove that they are capable of reproducing the same space of exponential polynomials as the one generated by the original approximating scheme. Moreover, we specialize the computational methods for the case of symbols obtained by shifted non-stationary affine combinations of exponential B-splines, that are at the basis of most non-stationary subdivision schemes. In this case we find that the associated family of interpolatory symbols can be determined to satisfy a suitable set of generalized interpolating conditions at the set of the zeros (with reversed signs) of the input symbol. Finally, we discuss some computational examples by showing that the proposed approach can yield novel smooth non-stationary interpolatory subdivision schemes possessing very interesting reproduction properties.

Regularity of multivariate vector subdivision schemes

Abstract In this paper we discuss methods for investigating the convergence of multivariate vector subdivision schemes and the regularity of the associated limit functions. Specifically, we consider difference vector subdivision schemes whose restricted contractivity determines the convergence of the original scheme and describes the connection between the regularity of the limit functions of the difference subdivision scheme and the original subdivision scheme.

Scalar multivariate subdivision schemes and box splines

We study scalar d-variate subdivision schemes, with dilation matrix 2I, satisfying the sum rules of order k. Using the results of Moller and Sauer, stated for general expanding dilation matrices, we characterize the structure of the mask symbols of such schemes by showing that they must be linear combinations of shifted box spline generators of some polynomial ideal. The directions of the corresponding box splines are columns of certain unimodular matrices. The ideal is determined by the given order of the sum rules or, equivalently, by the order of the zero conditions. The results presented in this paper open a way to a systematic study of subdivision schemes, since box spline subdivisions turn out to be the building blocks of any reasonable multivariate subdivision scheme. As in the univariate case, the characterization we give is the proper way of matching the smoothness of the box spline building blocks with the order of polynomial reproduction of the corresponding subdivision scheme. However, due to the interaction of the building blocks, convergence and smoothness properties may change, if several convergent schemes are combined. The results are illustrated with several examples.

Convergence of univariate non-stationary subdivision schemes via asymptotic similarity

A new equivalence notion between non-stationary subdivision schemes, termed asymptotical similarity, which is weaker than asymptotical equivalence, is introduced and studied. It is known that asymptotical equivalence between a non-stationary subdivision scheme and a convergent stationary scheme guarantees the convergence of the non-stationary scheme. We show that for non-stationary schemes reproducing constants, the condition of asymptotical equivalence can be relaxed to asymptotical similarity. This result applies to a wide class of non-stationary schemes of importance in theory and applications.

A new subdivision method for bivariate splines on the four-directional mesh

We present a new bivariate subdivision scheme based on two generators of a four-directional spline space. In particular, we are dealing with piecewise linear, continuous splines, and with piecewise cubic, continuously differentiable splines. The subdivision schemes rely on a matrix mask of small support. In addition, we present some results on related quasi-interpolating operators, and on approximation order of the underlying shift-invariant spaces.

Interpolatory rank-1 vector subdivision schemes

The concept of stationary scalar subdivision being interpolatory does not always extend immediately to the vector valued case. We introduce the concepts of interpolating and data preserving vector subdivision schemes and discuss how these concepts are related. We also present two examples.

Full rank interpolatory subdivision schemes: Kronecker, filters and multiresolution

In this extension of earlier work, we point out several ways how a multiresolution analysis can be derived from a finitely supported interpolatory matrix mask which has a positive definite symbol on the unit circle except at -1. A major tool in this investigation will be subdivision schemes that are obtained by using convolution or correlation operations based on replacing the usual matrix multiplications by Kronecker products.