Publication Details

This article was originally published as: Nillsen, R & Okada, S, Sharp results concerning the expression of functions as sums of finite differences, Proceedings of the London Mathematical Society, 2004, 88(2), 479-504. Copyright 2004 Cambridge University Press. Original journal can be found here.


A square integrable function on the real line can be written as a finite sum of differences of order s if and only if the Fourier transform of the function vanishes at the origin at a rate comparable with |x|^{-s}. This enables us to see that the space of all such functions becomes a Hilbert space D^s(R) and that this space is the range of the derivative of order s on the Sobolev space of order s. It was proved by the author (Journal of Functional Analysis 110 (1992), 73-95) that each function in D^s(R) is a sum of 2s+1 differences of order s. It was also known (Springer Lecture Notes in Mathematics vol. 1586, 1994) that in the case s=1, although every function in D^1 (R) is a sum of 3 first order differences, there are functions in D^1(R) that cannot be written as the sum of 2 first order differences – that is, 3 is sharp as an estimate of the minimum number of differences required. It was an open problem as to whether in the case of D^s(R), the number 2s+1 is sharp in this same sense. The main result in this paper shows that 2s+1 is sharp. That is, although every function in D^s (R) is a sum of 2s+1 differences of order s, there are functions in D^s(R) that cannot be written as the sum of 2s differences of order s. In fact, substantially stronger results are proved, and results in related spaces of distributions are discussed. The main techniques involve the Fourier transform and combinatorial methods in harmonic analysis, in particular an estimate of the minimum potential of n points scattered in the unit cube in R^s. This aspect generalizes work of Wolfgang Schmidt. Further ideas related to this work, the behaviour of the Fourier transform near the origin, and differential operators are on the author’s website at http://www.uow.edu.au/~nillsen



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