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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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