Tim N. T. Goodman

Wavelets in wandering subspaces

Mallats construction, via a multiresolution approximation, of orthonormal wavelets generated by a single function is extended to wavelets generated by a finite set of functions. The connection between multiresolution approximation and the concept of wandering subspaces of unitary operators in Hilbert space is exploited in the general setting. An example of multiresolution approximation generated by cardinal Hermite B-splines is constructed

On the optimal stability of the Bernstein basis

We show that the Bernstein polynomial basis on a given interval is optimally stable, in the sense that no other nonnegative basis yields systematically smaller condition numbers for the values or roots of arbitrary polynomials on that interval. This result follows from a partial ordering of the set of all nonnegative bases that is induced by nonnegative basis transformations. We further show, by means of some low-degree examples, that the Bernstein form is not uniquely optimal in this respect. However, it is the only optimally stable basis whose elements have no roots on the interior of the chosen interval. These ideas are illustrated by comparing the stability properties of the power, Bernstein, and generalized Ball bases.

Total Positivity and the Shape of Curves

We discuss why the variation diminishing property is useful when designing curves or constructing approximation operators from bases with totally positive collocation matrices. Various such bases are considered and a generalisation of the variation diminishing property is presented and applied.

Spectral factorization of Laurent polynomials

We analyse the performance of five numerical methods for factoring a Laurent polynomial, which is positive on the unit circle, as the modulus squared of a real algebraic polynomial. It is found that there is a wide disparity between the methods, and all but one of the methods are significantly influenced by the variation in magnitude of the coefficients of the Laurent polynomial, by the closeness of the zeros of this polynomial to the unit circle, and by the spacing of these zeros.

Shape preserving properties of the generalised Ball basis

Abstract We show that the generalised Ball basis for odd degree polynomials on a finite interval has a totally positive collocation matrix and thus possesses the same kind of shape preserving properties as the Bernstein basis, though to a lesser degree. This is proved by constructing a corner cutting algorithm for obtaining the Bezier polygon of a polynomial curve from its control polygon with respect to the generalised Ball basis.

Local derivative estimation for scattered data interpolation

Abstract In scattered data interpolation a surface through the given data points is constructed. A class of methods requires triangulation of the domain with the data points at the vertices and definition of a local interpolant over each triangle. In order to construct a smooth surface, it is usual to employ certain derivative values at the vertices. If these are not given, they can be prescribed by estimating the derivatives using the data points. We present here a method of derivative estimation by using a convex combination of all derivatives on related triangular planes. The method has comparable accuracy to the existing method of least-squares minimization but with less computation.

Properties of generalized Ball curves and surfaces

Abstract It is shown that the generalized Ball representation for a polynomial curve is much better suited to degree raising and lowering than the Bezier representation. The generalized Ball basis is then extended to polynomial surfaces over a triangle, and recursive algorithms for evaluation and degree raising are given.

Compactly supported fundamental functions for spline interpolation

SummaryIn this paper various ways of constructing locally supported fundamental splines leading to highly accurate local interpolation schemes are proposed and analyzed.

Construction of orthogonal bases for polynomials in Bernstein form on triangular and simplex domains

A scheme for constructing orthogonal systems of bivariate polynomials in the Bernstein-Bezier form over triangular domains is formulated. The orthogonal basis functions have a hierarchical ordering by degree, facilitating computation of least-squares approximations of increasing degree (with permanence of coefficients) until the approximation error is subdued below a prescribed tolerance. The orthogonal polynomials reduce to the usual Legendre polynomials along one edge of the domain triangle, and within each fixed degree are characterized by vanishing Bernstein coefficients on successive rows parallel to that edge. Closed-form expressions and recursive algorithms for computing the Bernstein coefficients of these orthogonal bivariate polynomials are derived, and their application to surface smoothing problems is sketched. Finally, an extension of the scheme to the construction of orthogonal bases for polynomials over higher-dimensional simplexes is also presented.

On refinement equations determined by Po´lya frequency sequences

The refinement equation \[ \phi (x) = \sum_{i \in \mathbb{Z}} {a_i \phi (2x - i)} ,\quad x \in \mathbb{R}\] for a given sequence