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

Journal Article

### 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.

• 2018

### Citation

• Nillsen, R. (2018). Vanishing Fourier coefficients and the expression of functions in L2(T) as sums of generalised differences. Journal of Mathematical Analysis and Applications, Online First 1-19.

### Scopus Eid

• 2-s2.0-85021300546

### Ro Full-text Url

• http://ro.uow.edu.au/cgi/viewcontent.cgi?article=1331&context=eispapers1

• http://ro.uow.edu.au/eispapers1/330

• 18

• 1

• 19

• Online First

### Place Of Publication

• United States

### 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.

• 2018

### Citation

• Nillsen, R. (2018). Vanishing Fourier coefficients and the expression of functions in L2(T) as sums of generalised differences. Journal of Mathematical Analysis and Applications, Online First 1-19.

### Scopus Eid

• 2-s2.0-85021300546

### Ro Full-text Url

• http://ro.uow.edu.au/cgi/viewcontent.cgi?article=1331&context=eispapers1

• http://ro.uow.edu.au/eispapers1/330

• 18

• 1

• 19

• Online First

### Place Of Publication

• United States