Johan de Villiers

Approximation of Periodic Functions

In the study of approximation of functions in Chaps. 1— 8, the emphasis is on (algebraic) polynomial approximation on the bounded interval \( [a,b] \). Since algebraic polynomials are not periodic functions, they are not suitable basis functions for representing and approximating periodic functions on the entire real line \( {\text{R}} \). On the other hand, many natural phenomena can only be represented by periodic functions. It is therefore essential to study approximation of periodic continuous functions \( f:{\text{R}} \to {\text{R}} \), by the linear span of some elementary periodic functions. This chapter is devoted to the study of this topic by considering basis functions that are formulated in terms of the sine and cosine functions.

Orthogonal polynomials for refinable linear functionals

A refinable linear functional is one that can be expressed as a convex combination and defined by a finite number of mask coefficients of certain stretched and shifted replicas of itself. The notion generalizes an integral weighted by a refinable function. The key to calculating a Gaussian quadrature formula for such a functional is to find the three-term recursion coefficients for the polynomials orthogonal with respect to that functional. We show how to obtain the recursion coefficients by using only the mask coefficients, and without the aid of modified moments. Our result implies the existence of the corresponding refinable functional whenever the mask coefficients are nonnegative, even when the same mask does not define a refinable function. The algorithm requires O(n 2 ) rational operations and, thus, can in principle deliver exact results. Numerical evidence suggests that it is also effective in floating-point arithmetic.

Wavelet Subdivision Methods: GEMS for Rendering Curves and Surfaces

Prevalent in animation movies and interactive games, subdivision methods allow users to design and implement simple but efficient schemes for rendering curves and surfaces. Adding to the current subdivision toolbox, Wavelet Subdivision Methods: GEMS for Rendering Curves and Surfaces introduces geometry editing and manipulation schemes (GEMS) and covers both subdivision and wavelet analysis for generating and editing parametric curves and surfaces of desirable geometric shapes. The authors develop a complete constructive theory and effective algorithms to derive synthesis wavelets with minimum support and any desirable order of vanishing moments, along with decomposition filters. Through numerous examples, the book shows how to represent curves and construct convergent subdivision schemes. It comprehensively details subdivision schemes for parametric curve rendering, offering complete algorithms for implementation and theoretical development as well as detailed examples of the most commonly used schemes for rendering both open and closed curves. It also develops an existence and regularity theory for the interpolatory scaling function and extends cardinal B-splines to box splines for surface subdivision. Keeping mathematical derivations at an elementary level without sacrificing mathematical rigor, this book shows how to apply bottom-up wavelet algorithms to curve and surface editing. It offers an accessible approach to subdivision methods that integrates the techniques and algorithms of bottom-up wavelets.

On refinable functions and subdivision with positive masks

We consider aspects of the analysis of refinement equations with positive mask coefficients. First we derive, explicitly in terms of the mask, estimates for the geometric convergence rate of both the cascade algorithm and the corresponding subdivision scheme, as well as the Hölder continuity exponent of the resulting refinable function. Moreover, we show that the subdivision scheme converges for a class of unbounded initial sequences. Finally, we present a regularity result containing sufficient conditions on the mask for the refinable function to possess continuous derivatives up to a given order.

Error Analysis for Polynomial Interpolation

As a continuation of Chap. 1, the notion of divided difference is applied to deduce the uniform error bound for polynomial interpolation for any given finite sample point set. In addition, an optimal sample point set, on which the minimum uniform error bound is achieved among all sample point sets with the same cardinality, is derived

On Bivariate Interpolatory Mask Symbols, Subdivision and Refinable Functions

We present a full characterisation of interpolatory mask symbols where the dilation matrix is M = 2I. The characterization involves the analysis of polyno- mial identities in two variables by means of the Bezout theorem and the Euclidean algorithm. The convergence of the associated interpolatory subdivision scheme is closely related to the existence of a corresponding interpolatory refinable function. As a special case of our theory, we present the mask symbol corresponding to the Butterfly subdivision scheme.

On interpolatory subdivision symbol formulation and parameter convergence intervals

Abstract This paper is concerned with general symmetric 2 n -point interpolatory subdivision scheme (ISS) with polynomial reproduction of arbitrary order m ≤ n . An explicit formulation is derived for the corresponding refinement symbol, and convergence intervals are obtained for the one-parameter case, the left hand endpoints of which improve on previous such lower parameter convergence bounds.

On Hermite vector splines and multi-wavelets

Abstract Explicit and recursive formulations are derived for the computation of refinable interpolatory Hermite vector splines, of arbitrary odd degree, and supported on [ − 1 , 1 ] , as well as for the corresponding refinement matrix sequences. It is moreover shown that a contracted and shifted version of these Hermite vector splines is a minimally supported Hermite spline multi-wavelet, with an explicitly calculated decomposition relation.

Polynomial Interpolation Formulas

When an (algebraic) polynomial P is used to approximate a certain function f , the polynomial P is said to interpolate the function f on a given finite sample set of distinct points in the domain of f , if P is obtained to satisfy the condition that P agrees with f on the sample set. The objective of this chapter is to establish a fundamental existence and uniqueness result for polynomial interpolation as well as to derive explicit formulations of the interpolation polynomial P.

Best Uniform Polynomial Approximation

This chapter is a continuation of Chapter 4, in that the best uniform polynomial approximation \( P^{*} \in \pi_{n} \,{\text{of}}\,f \in C[a,\,b], \) with existence of \( P^{*} \) guaranteed by Theorem 4.1.2, will be characterized in terms of the alternation properties of the error function \( f - P^{*} . \) As an application, the uniqueness of \( P^{*} \in \pi_{n} \) as the only best uniform polynomial approximant of \( f \in C[a,b] \) is assured