SPARSE SYSTEMS OF FUNCTIONS CLOSED ON LARGE SETS IN ℝ^N

Abstract

Let E(ℤ) = {e^inx}_n∈ℤ: denote the trigonometrical exponential system. It is well known that E(ℤ) forms an orthogonal basis in the space L²(0, 2π). In 1964, H. Landau discovered that the trigonometrical system has the following property: certain small perturbations of E(ℤ) yield exponential systems which are complete in L² on any finite union of 2π-periodic translations of any interval (ε, 2π−ε), 0 < ε < π.