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On basicity of the degenerate trigonometric system with excess. (English) Zbl 1395.46024
Summary: The basis properties (completeness, minimality and Schauder basicity) of systems of the form $$\{\omega(t)\varphi_{n}(t)\}$$, where $$\{\varphi_{n}(t)\}$$ is an exponential or trigonometric (cosine or sine) systems, have been investigated in several papers. Concrete examples of the weight function $$\omega(t)$$ 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 $$\omega(t)$$ is any measurable weight function such that the system $$\{\omega(t) \sin nt\}_{n\in\mathbb{N}}$$ has excess, then neither this system itself, nor a system obtained from it by elimination of an element, is not a Schauder basis.
##### MSC:
 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces 42C30 Completeness of sets of functions in nontrigonometric harmonic analysis
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