On basicity of the degenerate trigonometric system with excess

Abstract

The basis properties (completeness, minimality and Schauder basicity) of systems of the form {ω(t)φ_n(t)}, where {φ_n(t)} is an exponential or trigonometric (cosine or sine) systems, have been investigated in several papers. Concrete examples of the weight function ω(t) are known for which the system itself is not complete and minimal but has excess – becomes complete and minimal in corresponding L_p space only after elimination of some of its elements. The aim of this paper is to show that if ω(t) is any measurable weight function such that the system {ω(t) sin nt}_n∊ℕ has excess, then neither this system itself, nor a system obtained from it by elimination of an element, is not a Schauder basis.