Prof. T.N.T. Goodman
- Prof. Tim Goodman
- Professor of Mathematics
- University of Cambridge
- 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
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,
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
- (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,
- (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.
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.