On Banach frameness of degenerate weighted exponential system - Ismailov, Migdad I.; Simsir Acar, Kader (Fixed point theory and algorithms for sciences and engineering, 2025)
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 ) .