Rodney Nillsen and Susumu Okada

Let $\ast$ denote convolution and let $\delta_x$ denote the Dirac measure at a point $x$. A function in $L^2({\mathbb{R}})$ is called a difference of order $1$ if it is of the form $ g- \delta_{x} \ast g$ for some $x \in {\mathbb{R}}$ and $g \in L^2 ({\mathbb{R}})$. Also, a difference of order $2$ is a function of the form $ g- 2^{-1}( \delta_{x} \ast g + \delta_{-x} \ast g)$ for some $x \in {\mathbb{R}}$ and $g \in L^2 ({\mathbb{R}})$. In fact, the concept of a 'difference of order $s$' may be defined in a similar manner for each $s > 0$. If $\widehat{f}$ denotes the Fourier transform of $f$, it is known that a function $f$ in $L^2 ({\mathbb{R}})$ is a finite sum of differences of order $s$ if and only if $\int_{-\infty}^{\infty} |\widehat {f}(x)|^2 |x|^{-2s} dx < \infty,$ and the vector space of all such functions is denoted by ${\mathcal{D}}_{s} (L^2 ({\mathbb{R}}) )$. Every function in ${\mathcal{D}}_{s} (L^2 ({\mathbb{R}}) )$ is a sum of ${\mathrm{int}} (2s) + 1$ differences of order $s$, where ${\mathrm{int}} (t)$ denotes the integer part of $t$. Thus, every function in ${\mathcal{D}}_{1} (L^2 ({\mathbb{R}}) )$ is a sum of three first order differences, but it was proved in 1994 that there is a function in ${\mathcal{D}}_1 (L^2 ({\mathbb{R}}) )$ which is never the sum of two first order differences. This complemented, for the group ${\mathbb{R}}$, the corresponding result for first order differences obtained by Meisters and Schmidt in 1972 for the circle group. The results show that there is a function in $L^2 ({\mathbb{R}})$ such that, for each $s \ge 1/2$, this function is a sum of ${\mathrm{int}} (2s) + 1$ differences of order $s$ but it is never the sum of ${\mathrm{int}} (2s)$ differences of order $s$. The proof depends upon extending to higher dimensions the following result in two dimensions obtained by Schmidt in 1972 in connection with Heilbronn's problem: if $x_1, \ldots, x_n$ are points in the unit square, then

$$\sum_{1 \le i < j \le n} |x_i - x_j|^{-2} \ge 200^{-1} n^2 \ln n.$$

Following on from the work of Meisters and Schmidt, this work further develops a connection between certain estimates in combinatorial geometry and some questions of sharpness in harmonic analysis.

2000 Mathematical Subject Classification: 42A38 (primary), 52A40 (secondary).

Keywords: Fourier transform, finite difference.

E-mail:
rodney_nillsen@uow.edu.au
okada@ics.mq.edu.au

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