Vanishing Fourier coefficients and the expression of functions in L2(T) as sums of generalised differences

If g∈L2([0,2π]) let gˆ be the sequence of Fourier coefficients of g, let D denote differentiation and let I denote the identity operator. Given α,β∈Z, we consider the operator D2−i(α+β)D−αβI on the second order Sobolev space of L2([0,2π]). The multiplier of this operator is −(n−α)(n−β) considered as a function of n∈Z, so that gˆ(α)=gˆ(β)=0 for any function g in the range of the operator. Let δx denote the Dirac measure at x, and let ⁎ denote convolution. If b∈[0,2π] let λb be the measure2−1[(eib(α−β2)+e−ib(α−β2)]δ0−2−1[(eib(α+β2)δb+e−ib(α+β2)δ−b]. A function of the form λb⁎f is called a generalised difference, and we let F be the family of functions h such that h is a sum of five generalised differences. It is shown that for g∈L2([0,2π]), g∈F if and only if gˆ(α)=gˆ(β)=0. Consequently, F is a Hilbert subspace of L2([0,2π]) and it is the range of D2−i(α+β)D−αβI. The methods use partitions of intervals and estimates of integrals in Euclidean space. There are applications to the automatic continuity of linear forms in abstract harmonic analysis.