The support of a function with thin spectrum

Kathryn Hare

ArticleOriginal scientific text

Title

The support of a function with thin spectrum

Authors ¹

Affiliations

Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Abstract

We prove that if

E \subseteq Ĝ

does not contain parallelepipeds of arbitrarily large dimension then for any open, non-empty

S \subseteq G

there exists a constant c > 0 such that

∥ f 1_{S} ∥_{2} \geq c ∥ f ∥_{2}

for all

f \in L^{2} (G)

whose Fourier transform is supported on E. In particular, such functions cannot vanish on any open, non-empty subset of G. Examples of sets which do not contain parallelepipeds of arbitrarily large dimension include all Λ(p) sets.

A. Bonami, Etude des coefficients de Fourier des fonctions de $L^{p} (G)$ , Ann. Inst. Fourier (Grenoble) 20 (2) (1970), 335-402.
J. Fournier, Uniformizable Λ(2) sets and uniform integrability, Colloq. Math. 51 (1987), 119-129.
K. Hare, Strict-2-associatedness for thin sets, ibid. 56 (1988), 367-381.
K. Hare, Arithmetic properties of thin sets, Pacific J. Math. 131 (1988), 143-155.
B. Host et F. Parreau, Sur les mesures dont la transformée de Fourier-Stieltjes ne tend pas vers 0 à l'infini, Colloq. Math. 41 (1979), 285-289.
G. W. Johnson and G. S. Woodward, On p-Sidon sets, Indiana Univ. Math. J. 24 (1974), 161-167.
I. Klemes, Seminar notes on the transforms of some singular measures, private communication, 1990.
N. Levinson, Gap and Density Theorems, Amer. Math. Soc. Colloq. Publ. 26, Amer. Math. Soc., Providence, R.I., 1940.
J. Lopez and K. Ross, Sidon Sets, Lecture Notes in Pure and Appl. Math. 13, Marcel Dekker, New York, 1975.
S. Mandelbrojt, Séries de Fourier et classes quasi-analytiques de fonctions, Gauthier-Villars, Paris, 1935.
I. Miheev, On lacunary series, Math. USSR-Sb. 27 (1975), 481-502; translated from Mat. Sb. 98 (140) (1975), 538-563.
I. Miheev, Trigonometric series with gaps, Anal. Math. 9 (1983), 43-55.
W. Rudin, Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203-227.
A. Zygmund, Trigonometric Series, Vol. I, Cambridge University Press, Cambridge, 1959.

Title

The support of a function with thin spectrum

Affiliations

Abstract

Bibliography