p-Sidon Sets and a Uniform Property

Authors

Gordon S. Woodward, University of Nebraska - LincolnFollow

Date of this Version

1976

Citation

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

Comments

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.

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.