## Mathematics, Department of

#### Date of this Version

1976

#### Citation

Indiana University Mathematics Journal, ©, Vol. 25, No. 10 (1976)

#### Abstract

Let *G *be a compact abelian group with dual group Γ. Denote by* L ^{p }*(

*E*) and

*M*(

*E*) the usual spaces of Haar-measurable functions and bounded regular Borel measures, respectively, which are supported on the subset

*E*of

*G*or Γ

*.*The Haar measure on

*G*is normalized and its dual is the Haar measure on Γ. Let ᵩ

^{˰}denote the Fourier or Fourier-Stieltjes transform of the function or measure ᵩ. A subset

*E*< Γ is said to be

*p-Sidon*for some 1 ≤

*p*< 2 (not interesting for

*p*≥ 2) if there is an

*α*> 0 such that ||ᵩ

^{˰}||

_{p }≤

*α*||ᵩ||

_{∞ }for all trigonometric polynomials ᵩ on

*G*with supp ᵩ

^{˰}<

*E*. This is equivalent to the dual statement:

*E*is

*p*-Sidon if and only if

*L*(

^{q }*E*) <

*M*(

*G*)̂ |

_{E }, where 1/

*p*+ 1/

*q*= 1 and "|

_{E}" denotes restriction to

*E*. Hereafter

*p*and

*q*will

*always*be as above.

## Comments

Let

Gbe a compact abelian group with dual group Γ. Denote byL^{p }^{}(E) andM(E) the usual spaces of Haar-measurable functions and bounded regular Borel measures, respectively, which are supported on the subsetEofGor Γ.The Haar measure onGis normalized and its dual is the Haar measure on Γ. Let ᵩ^{˰}denote the Fourier or Fourier-Stieltjes transform of the function or measure ᵩ. A subsetE< Γ is said to bep-Sidonfor some 1 ≤p< 2 (not interesting forp≥ 2) if there is anα> 0 such that ||ᵩ^{˰}||_{p }≤α||ᵩ||_{∞ }for all trigonometric polynomials ᵩ onGwith supp ᵩ^{˰}<E. This is equivalent to the dual statement:Eisp-Sidon if and only ifL^{q }^{}(E) <M(G)̂ |_{E }, where 1/p+ 1/q= 1 and "|_{E}" denotes restriction toE. Hereafterpandqwillalwaysbe as above.