Rod Nillsen
Mathematics research
The Fourier transform (1994-- )
Introduction
In the Journal of Functional Analysis in 1972, Gary Meisters and Wolfgang Schmidt showed that for a square integrable function on the circle group, the Fourier coefficient at 0 of the function is equal to 0 if and only if the function is equal to a sum of 3 finite differences. They deduced that a translation invariant linear form on the space of square integrable functions is automatically continuous. Further related results were found by J. Bourgain, A. Connes, B. Johnson, G. Meisters, and others. Rod Nillsen's research in this area extends some of these ideas to the non-compact case, including the real line in particular. This leads to a class of Hilbert spaces whose elements are sums of generalized differences and which have an inner product calculated from the behaviour of the Fourier transform near the origin. You may read or download the following:
Further details on the above ideas follow
The Fourier transform of a function on the real line displays the "frequencies" present in a function. If we differentiate a function, the new function has an increased level of high frequencies and a decreased level of low frequencies. In fact, when we differentiate a function, the frequencies towards infinity determine whether the function is in the first order Sobolev space, while the frequencies near 0 determine whether the function can be written as a sum of first order differences of functions. The precise connection for the latter situation comes from the facts that the behaviour of the Fourier transform of the differentiated function has a certain precise behaviour near 0, and that this behaviour is identical to the behaviour near 0 obtained by taking the Fourier transform of a first order difference of a function. A related observation is that the local approximation of derivatives by finite differences is a common technique in numerical methods, for getting approximate solutions to differential equations for example.
These observations suggest the question: what is the precise global relationship between first order derivatives and first order differences of functions? In the Lebesgue space of square integrable functions on the real line, the answer is that every function in the space, which is equal to the derivative of some function in the space, is the sum of 3 first order differences (the number 3 is sharp). Also, every function in the space which equals the second derivative of some function in the space is the sum of 5 second order differences (it has been proved recently that the number 5 is sharp). Such characterizations lead to a class of Hilbert spaces, each one consisting of all sums of differences of a given order and whose inner product is given in terms of the behaviour of the Fourier transform near 0. These spaces can be thought of as Sobolev spaces of negative index, characterized as consisting of sums of differences of a given order.
The above ideas have been systematically investigated in the research monograph "Difference Spaces and Invariant Linear Forms", which is volume 1586 of the Springer Lecture Notes in Mathematics series. Part of the Introduction to LNM 1586 can be downloaded, please click on the link above in the introduction.
An informal exposition is these ideas is on slides, given as part of a seminar at the University of Wollongong in 2005. Further aspects may be found in the invited talk, dedicated to Gary H. Meisters, presented at a conference held in Lincoln, Nebraska in 1997, in honour of his work. You can also read the Introduction to the expository paper "Differentiate and Make Waves", Expositiones Math. 1996, which deals with the phenomenon in the single real variable case only, and discusses connections with the theory of wavelets (click on the links above).
A recent paper with Susumu Okada in the Journal of the London Math. Soc. has shown that for any positive integer s, in the above space of order s, there are functions that can be expressed as the sum of 2s+1 differences of order s, but cannot be expressed as a sum of 2s differences of order s. You may read part of this paper (click on the link above). For a connection with the general theory of multiplier operators, see the paper with Susumu Okada "Function spaces and multiplier operators", Math. Annalen 321 (2001), 615-644.
Rod Nillsen, October 2005