ACTA issues

Sharpness results concerning finite differences in Fourier analysis on the circle group

Rodney Nillsen, Susumu Okada

Acta Sci. Math. (Szeged) 84:3-4(2018), 591-609

Abstract. Let $G$ denote the group ${\msbm R}$ or ${\msbm T}$, let $\iota $ denote the identity element of $G$, and let $s\in{{\msbm N}}$ be given. Then, a \emph{difference of order} $s$ is a function $f\in L^2(G)$ for which there are $a\in G$ and $g \in L^2({G})$ such that $f= (\delta_{\iota }-\delta_{a})^s\ast g$. Let ${{\cal D}}_s(L^2(G))$ be the vector space of functions that are finite sums of differences of order $s$. It is known that if $f\in L^2({{\msbm R}})$, $f\in{{\cal D}}_s(L^2({{\msbm R}}))$ if and only if $\int_{-\infty }^{\infty }|{\widehat f}(x)|^2|x|^{-2s}dx< \infty $. Also, if $f\in L^2({{\msbm T}})$, $f\in{{\cal D}}_s(L^2({{\msbm T}}))$ if and only if ${\widehat f}(0)=0$. Consequently, ${{\cal D}}_s(L^2(G))$ is a Hilbert space in a (possibly) weighted $L^2$-norm. It is known that every function in ${{\cal D}}_s(L^2(G))$ is a sum of $2s+1$ differences of order $s$. However, there are functions in ${{\cal D}}_s(L^2({{\msbm R}}))$ that are not a sum of $2s$ differences of order $s$, and we call the latter type of fact a \emph{sharpness result}. In ${{\cal D}}_1(L^2({{\msbm T}}))$, it is known that there are functions that are not a sum of two differences of order one. A main aim here is to obtain new sharpness results in the spaces ${{\cal D}}_s(L^2({{\msbm T}}))$ that complement the results known for ${{\msbm R}}$, but also to present new results in ${{\cal D}}_s(L^2({{\msbm T}}))$ that do not correspond to known results in ${{\cal D}}_s(L^2({{\msbm R}}))$. Some results are obtained using connections with Diophantine approximation. The techniques also use combinatorial estimates for potentials arising from points in the unit cube in Euclidean space, and make use of subtraction sets in arithmetic combinatorics.

DOI: 10.14232/actasm-017-522-y

AMS Subject Classification (1991): 42A16, 42A38

Keyword(s): Fourier transform, finite differences, subspaces of $L^2({{\msbm T}})$, combinatorial inequalities, badly approximable vectors in ${{\msbm R}}^n$, sharpness results, Sobolev spaces

Received April 7, 2017 and in final form May 22, 2018. (Registered under 22/2017.)