RIS ID
114783
Abstract
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 2−1[(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⁎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.
Publication Details
Nillsen, R. (2017). Vanishing Fourier coefficients and the expression of functions in L2(T) as sums of generalised differences. Journal of Mathematical Analysis and Applications, 455 1425-1443.