On basicity of the degenerate trigonometric system with excess

  • Aydin Sh. Shukurov Institute of Mathematics and Mechanics NAS of Azerbaijan, B. Vahabzade 9, Az1141, Baku
Keywords: Schauder bases, system of cosines with degenerate coefficients, system of sines with degenerate coefficients, Muckenhoupt condition


The basis properties (completeness, minimality and Schauder basicity) of systems of the form {ω(t)φn(t)}, where {φn(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 Lp 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.