Prof. T.N.T. Goodman

Name:

Prof. Tim Goodman

Position:

Professor of Mathematics

Address:

Department of Mathematics

The University of Dundee

Dundee DD1 4HN

E-mail:

tgoodman@maths.dundee.ac.uk

Phone:

01382 384488

Fax:

01382 385516

Professional Duties

Fellow of the Royal Society of Edinburgh

Member of the EPSRC Peer Review College

Member of the editorial boards of the following journals:

Advances in Computational Mathematics

Computer Aided Geometric Design

Constructive Approximation

Research Interests:

These mostly fall into one of the following three areas:

1) Computer Aided Geometric Design

Work has focussed on spline functions, i.e. piecewise polynomials or rationals, particularly shape preserving interpolation and the theoretical study of shape properties. Recent collaborations include: the University of California, Davis; Universiti Sains Malaysia; University of Manitoba; and Lincoln University, New Zealand.

Some recent papers:

• Shape preserving interpolation by curves, Algorithms for Approximation IV, J.Levesley, I.J.Anderson, J.C.Mason (eds.), pp.24-35, University of Huddersfield, 2002.
• (with R.T.Farouki and T.Sauer),Construction of orthogonal bases for polynomials in Bernstein form on triangular and simplex domains, Computer Aided Geometric Design 20 (2003), 209-230.
• (with B.H.Ong), Shape preserving interpolation by splines using vector subdivision, Advances in Computational Mathematics 22 (2005), 49-77.
• (with D.S.Meek and D.J.Walton), An involute spiral that matches G^2 Hermite data in the plane, to appear Computer Aided Geometric Design.
• (with A.Piah and K.Unsworth), Positivity-preserving scattered data interpolation, Mathematics of Surfaces 2005, R.Martin, M.Bez, M.Sabin (eds.), pp.336-349, Springer Verlag, Berlin, 2005.
• (with D.S.Meek), Planar interpolation with a pair of rational spirals, J. Comp. Appl. Math. 201 (2007), 112-127.

2) Multiresolution and Wavelets

Refinable spline functions have been studied, including construction of orthogonal splines, wavelets, frames, and a new bivariate generalisation of B-splines. There is also work on more general refinable functions, in particular asymptotic and shape properties. Recent collaborations include: Justus Liebig University, Giessen; the Center for Constructive Approximation, Vanderbilt University; and the Centre for Wavelets, Approximation and Information Processing, the National University of Singapore.

Some recent papers:

• (with L.H.Y.Chen and S.L.Lee), Asymptotic normality of scaling functions, SIAM J. Math. Anal. 36 (2004), 323-346.
• A class of orthogonal refinable functions and wavelets, Constructive Approximation 19 (2003), 525-540.
• (with M.D.Buhmann and O.Davydov), Cubic spline prewavelets on the 4-direction mesh, Foundations of Comp. Math. 3 (2003), 113-133.
• (with Q.Y.Sun), Total positivity and refinable functions with general dilation, ACHA 16 (2004), 69-89.
• (with S.S.Goh and S.L.Lee), Causality properties of refinable functions and sequences, Adv. Comp. Math. 26 (2007), 231-250.
• (with D.Hardin), Refinable shift invariant spaces of spline functions, Mathematical Methods for Curves and Surfaces: Tromso 2004, H.Daehlen, K.Morken,L.L.Schumaker (eds.), p.179-197, Nashboro Press, Brentwood, 2005.
• (with Doug Hardin), Refinable multivariate spline functions, Topics in Multivariate Approximation and Interpolation, K.Jetter et al. (eds.), p.55-83, Elsevier, 2005.
• (with S.L.Lee), Asymptotic optimality in time-frequency localisation of scaling functions and wavelets, Frontiers in Interpolation and Approximation, N.K.Govil et al. (eds.), p.141-171, Chapman & Hall/CRC, Boca Raton, 2006.
• Multi-box splines, Constructive Approximation 25 (2007), 279-301.
• (with S.S.Goh and S.L.Lee), Hybrid spline frames, submitted Math. Comp.
• (with X.Gao and S.L.Lee), Foveated splines and wavelets, to appear Appl. Comp. Harmonic Anal.
• (with S.S.Goh and S.L.Lee), Constructing tight frames of multivariate functions, to appear special issue of J. Approximation Theory.

3) Uncertainty Principles

The collaboration on asymptotic properties of refinable functions at the National University of Singapore has led to further joint work there on uncertainty principles, in particular on spheres.

Some recent papers:

• (with S.S.Goh) Uncertainty principles and asymptotic behavior, ACHA 16 (2004), 19-43.
• (with S.S.Goh), Inequalities on time-concentrated or frequency-concentrated functions, Adv. Comp. Math. 24 (2006), 333-351.
• (with S.S.Goh), Uncertainty principles on circles and spheres, Advances in Constructive Approximation Vanderbilt 2003, M.Neamtu, E.B.Saff (eds.), p.207-218, Nashboro Press, Brentwood, 2004.
• (with S.S.Goh), Uncertainty principles in Banach spaces and signal recovery, J. Approximation Theory 143 (2006), 26-35.