If g∈L2([0,2π])g∈L2([0,2π]) let View the MathML sourcegˆ be the sequence of Fourier coefficients of g, let D denote differentiation and let I denote the identity operator. Given α,β∈Zα,β∈Z, we consider the operator D2−i(α+β)D−αβID2−i(α+β)D−αβI on the second order Sobolev space of L2([0,2π])L2([0,2π]). The multiplier of this operator is −(n−α)(n−β)−(n−α)(n−β) considered as a function of n∈Zn∈Z, so that View the MathML sourcegˆ(α)=gˆ(β)=0 for any function g in the range of the operator. Let δxδx denote the Dirac measure at x , and let ⁎ denote convolution. If b∈[0,2π]b∈[0,2π] let λbλb be the measure
A function of the form λb⁎fλb⁎f is called a generalised difference , and we let FF 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∈L2([0,2π]), g∈Fg∈F if and only if View the MathML sourcegˆ(α)=gˆ(β)=0. Consequently, FF is a Hilbert subspace of L2([0,2π])L2([0,2π]) and it is the range of D2−i(α+β)D−αβID2−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.