On Banach frameness of degenerate weighted exponential system

• Published in: Fixed point theory and algorithms for sciences and engineering Vol. 2025; no. 1; pp. 26 - 15
• Main Authors: Ismailov, Migdad I., Simsir Acar, Kader
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 23.09.2025; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; Atomic properties; Banach spaces; Basicity; Completeness; Decomposition; Differential Geometry; Frameness; Mathematical and Computational Biology; Mathematics; Mathematics and Statistics; Minimality; Muckenhoupt condition; Topology; Weighted exponential system; Weighting functions; 46B15; Completeness; 46E30; Minimality; Muckenhoupt condition; Basicity; 42C15; Weighted exponential system; Frameness
• ISSN: 2730-5422; 2730-5422
• DOI: 10.1186/s13663-025-00805-5

This work deals with the frameness of weighted exponential system E ( ω , Z ) = { ω ( t ) e i n t } n ∈ Z in the space L p ( − π , π ) , p > 1 , with the weight function ω ( t ) of general form. Basis properties of E ( ω , Z ) in L p ( − π , π ) , p > 1 , are studied, in other words, the criteria of completeness, minimality and basicity of the system E ( ω , Z ) in the space L p ( − π , π ) , p > 1 , are given. Sufficient conditions for the completeness and minimality of E ( ω , Z ∖ F ) in L p ( − π , π ) , p > 1 , are found, where F is an arbitrary finite nonempty subset of the set of integers Z . A different method to prove that the system E ( ω , Z ∖ F ) does not form a Schauder basis for L p ( − π , π ) , p > 1 , is given. Theorem on a property of expansion system and criterion of Banach frameness for E ( ω , Z ) in L p ( − π , π ) , p > 1 , are proved. In particular, it is proved that the system E ( ω , Z ) with defect cannot form atomic decomposition for L p ( − π , π ) , p > 1 . The obtained results are the generalizations of those on the atomic decomposition of power weighted exponential system in L p ( − π , π ) , p > 1 , and the frameness of weighted exponential system in L 2 ( − π , π ) .