On the fourier coefficients of functions concentrated on a subset of the circle

Abstract

We consider\(\mathbb{T} = \{ z \in \mathbb{C} :{\mathbf{ }}|z|{\mathbf{ }} = 1\} ,{\mathbf{ }}E{\mathbf{ }} \subset {\mathbf{ }}\mathbb{T}\),mE > 0,G(E) is a certain subspace of L1(E) consisting of functions concentrated on E and integrable, and {dk}, (k ∈ ℤ) in a summable sequence of positive numbers. It is proved that if G(E)=Lp(E), p≥2, then there exists f∈G(E) such that |f(n)|≥dn,\(\hat f(n)|{\mathbf{ }} \geqslant {\mathbf{ }}d_n ,{\mathbf{ }}n \in \mathbb{Z}\) (one of the questions involved in the majorization problem). Sufficient conditions are obtained for certain other function classes G(E). We study the question of partial majorization. Bibliography: 2 titles.