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Harmonic analysis
In 1974 Gary H Meisters and Wolfgang Schmidt showed that on the circle group, a square integrable function whose Fourier transform vanished at the origin could be written as a sum of three first order differences. Further related results were found by J. Bourgain, A. Connes, B. Johnson, G. Meisters, and others. The work below is concerned with the non-compact case, in particular the real line. Hilbert spaces arise from considering all functions whose Fourier transforms have a specified behaviour near the origin in, say, Euclidean spaces of finite dimension. Functions in these spaces may be characterised by whether they can be expressed as a sum of differences of functions. There are connections with multiplier theory, Sobolev spaces, Heilbronn's problem, Liouville numbers and automatic continuity of linear forms. You may read or download the following.
(with Susumu Okada) Sharp results concerning the expression of functions as sums of finite differences Journ. London Math. Soc., 88 (2004), 479-504. (with Susumu Okada) Function spaces and multiplier operators, Math. Annal. 321 (2001), 615-644. Differentiate and make waves, Expositiones Mathematicae 14 (1996), 57-84. Difference spaces and invariant linear forms, Lecture notes in Mathematics volume 1586, Springer-Verlag 1994. Banach spaces of functions and distributions characterized by singular integrals involving the Fourier transform, Journ. Func. Anal., 110 (1992), 73-95. Adapted sets of measures and invariant functionals on Lp(G), Trans. Amer. Math. Soc., 325 (1991), 345-362. Dynamical systems and measure theory Selected publications Randomness and Recurrence in Dynamical Systems, Carus Mathematical Monographs, Mathematical Association of America, Washington DC, USA, 2010 [357 pages]. (with A. Koeller and G. Williams) Weakly mixing transformations and the Carathéodory definition of measurable sets, Colloquium Math. 108 (2007), 317-328. Irrational rotations motivate measurable sets, Elem. Der Math. 56 (2001),1-17. Normal numbers without measure theory, Amer. Math. Monthly, 107 (2000), 639-644. Chaos and one-to-oneness, Math. Mag. 72 (1999), 14-21. A Cause of Chaos, The Mathematical Scientist 22 (1997), 58-61. Extreme points and Banach limits Banach limits are invariant linear forms on the Banach space of bounded sequences. They form a compact convex set in the weak*-topology of the dual of this space. This research proved that the set of extreme points of this set is non-compact. Some research carried out on the Stone-Čech compactification of the natural numbers is closely related to this work. Publications Nets of extreme Banach limits, Proc. Amer. Math. Soc. 55 (1976), 347-352. Discrete orbits in βN-N, Colloq. Math. 33 (1975), 71-81. Differential and difference equations research This work is concerned with factorizing ordinary differential operators and difference operators. Publications (with P. J. Browne) On difference operators and their factorization, Canadian Journ. Math., XXXV (1983), 873-897. (with P. J. Browne) The two-sided factorization of ordinary differential operators, Canadian Journ. Math., XXXII (1980), 1045-1057. (with J. M. Hill) Mutually conjugate solutions of formally self adjoint differential equations, Proc. Roy. Soc. Edinburgh (Series A), 83 (1979), 93-101. Functional analysis research Selected publications Function spaces and n-dimensional wavelet transforms, Israel Mathematical Conference Proceedings, 13 (1999), 150--160. Functions with piecewise linear Fourier transforms, Journ. Math. Anal. Appl., 157 (1991), 301-317. Piecewise linear functions and series expansions in terms of Dirichlet and Féjer kernels, Proc. Centre Math. Anal., Australian National University, 16(1988) (Miniconferences on Harmonic Analysis and Operator algebras, 5-8 August and 2-3 December 1987), 223-225. Some basic sequences and their moment operators, Proc. Centre Math. Anal, Australian National University, 14 (1986) (Miniconference on Operator Theory and Partial Differential Equations, held at Macquarie University, September 8-10, 1986), 246-271. Invariant subspaces of some function spaces on a locally compact group, Journ. Func. Anal., 64 (1985), 338-357. Ideas about the Golden Ratio and irrationality  Go to The Golden Ratio Page. The figure illustrates a regular pentagon together with successive constructions of smaller pentagons. The constructions go on forever, and from this fact one can deduce that the square root of five is irrational. The Golden Ratio is the ratio of the length of the diagonal of a regular pentagon to the length of its side.
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