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

Journal Article


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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 Date


  • 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

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers1/330

Number Of Pages


  • 18

Start Page


  • 1

End Page


  • 19

Volume


  • 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.

Publication Date


  • 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

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers1/330

Number Of Pages


  • 18

Start Page


  • 1

End Page


  • 19

Volume


  • Online First

Place Of Publication


  • United States